The Brexit Vote, Productivity Growth

and Macroeconomic Adjustments

in the United Kingdom∗

Ben Broadbenta,d, Federico Di Pacea, Thomas Drechselc,d,Richard Harrisona,d,†, and Silvana Tenreyroa,b,d,e

aBank of EnglandbLondon School of Economics

cUniversity of MarylanddCentre for Macroeconomics

eCEPR

May 15, 2020

Abstract

The UK economy experienced significant macroeconomic adjustmentsfollowing the 2016 referendum on its withdrawal from the European Union.To understand these adjustments, this paper presents empirical facts basedon novel UK macroeconomic data and estimates a small open economymodel with tradable and non-tradable sectors. We demonstrate that thereferendum outcome can be conceptualized as news about a future declinein productivity growth in the tradable sector. While aggregate growth slows,an immediate fall in the relative price of non-tradable goods induces atemporary “sweet spot” for tradable producers. UK interest rates declineand investment growth decelerates.

JEL Classifications: E32, F17, F41, F43, O16.Keywords: Brexit, Small open economy, Productivity, Real exchange rate.

∗The views expressed here are those of the authors, and not necessarily those of the Bank ofEngland. Without implication, we would like to especially thank Konstantinos Theodoridisfor very stimulating discussions. We are grateful to Yunus Aksoy, Juan Antolin-Diaz, JamesBullard, Ambrogio Cesa-Bianchi, Nikola Dacic, Pierre De Leo, Marco Garofalo, Pete Kyle,Nuno Limao, Enrico Longoni, Michael McMahon, Francesca Monti, Ivan Petrella, LukaszRachel, Morten Ravn, Peter Sinclair, Greg Thwaites and Jan Vlieghe for very helpful comments.Philip King, Oliver Ashtari Tafti and Mette Nielsen provided excellent research assistance.We also thank seminar participants at the Bank of England, Banque de France, BirkbeckUniversity, Washington University in St. Louis, the 50th MMF Conference at LSE, the 15thDynare Conference in Lausanne, the University of Maryland Smith School of Business and theSpanish Economics Association.†Corresponding Author. Address: Threadneedle Street, London EC2R 8AH, United King-

dom. E-mail: Richard.Harrison@bankofengland.co.uk

1. Introduction

In the momentous referendum on 23 June 2016, voters decided thatthe United Kingdom (UK) should leave the European Union (EU).While many of the details regarding the UK’s ultimate withdrawal(‘Brexit’) are still to be determined, the aftermath of the referendum hasbeen characterized by significant macroeconomic adjustments in the UKeconomy. UK economic activity has slowed relative to its long-run trend.Growth in the tradable sector has remained resilient in comparison tothe non-tradable sector. The British pound sterling has been subject to apronounced depreciation (and with it the relative price of non-tradableoutput has fallen). Exports have been growing robustly. At the sametime, UK interest rates have declined relative to their world counterpartand investment has slowed down materially, while employment hasremained strong. This paper documents these empirical patterns usingnewly constructed UK macroeconomic data and demonstrates that theyare consistent with what economic theory predicts for the effects of ananticipated productivity growth slowdown in the UK tradable sector.

Our analysis is motivated by the remarks of Broadbent (2017a),who conjectured that market participants may have interpreted theconsequences of the Brexit vote as a future slowdown in the tradablesector, prompting the depreciation of sterling following the referendum.We formalize and assess this idea through the lens of an estimated smallopen economy model with tradable and non-tradable sectors. Our modelallows us to characterize how firms and households respond to newsabout future productivity in the tradable sector by shifting resourcesacross expenditure components, sectors and time. We demonstrate thatthe macroeconomic dynamics triggered by such news are consistent withthe patterns in post-referendum UK data. We also explain how drivers ofgrowth in tradable sector productivity are linked to specific consequencesof Brexit, such as reduced goods, capital and labor mobility.

The paper proceeds in four steps. First, we document a numberof stylized facts about the UK economy in the period following the2016 referendum. This is based on a new quarterly macroeconomicdata set in which we construct key variables separately for the tradable

1

and non-tradable sectors.1 The stylized facts describe growth, exchangerate, interest rate, investment and employment dynamics following theBrexit vote. Second, we introduce a small open economy (SOE) modelwhich is composed of tradable and non-tradable sectors. Our modelcan encompass differential trend growth rates across these sectors underrestrictions on preferences and technology. Introducing these sector-specific trends allows us to conduct the relevant experiments. To aidexposition, we derive some key economic relationships of our modelanalytically in a simplified version. Third, we estimate the model atbusiness cycle frequency using the newly constructed data set. Theestimation enables us to pin down not only the structural parameters,but also the initial balanced growth path around which we simulateBrexit news scenarios. Fourth, we use the model to conduct simulationswhich shed light on the mechanism that generated key patterns in theUK economy following the Brexit vote. At the heart of our analysis is abenchmark experiment that assesses the economic impact of news thatthe growth rate of productivity in the tradable sector will be persistentlylow. The fall in productivity growth takes places 11 quarters after it isannounced, consistent with the broad contours of the legislative processfor EU withdrawal implemented following the referendum.

The mechanism works as follows. The Brexit news – conceptualizedas a persistent drop in the growth rate of future productivity in the UKtradable sector – generates a temporary boom in tradable production.This is driven by the response of the relative price of non-tradable output(an ‘internal’ real exchange rate) which jumps down when the news isrevealed. Consequently, there is an opportunity to sell tradable outputat a temporarily higher relative price before productivity in the sectoractually falls, a temporary “sweet spot” for producers of tradable output(Broadbent, 2017a,b). This generates the reallocation of capital and labortowards the tradable sector, a rise in tradable output growth and anincrease in net exports, all of which reverse after the productivity growthdecline in the tradable sector actually occurs. The Brexit news also

1The construction of this novel data set involves classifying industry data at the2-digit level into tradable and non-tradable sectors. Based on this classification, weconstruct time series of gross value added, hours worked, labor productivity as well asrelative prices for the tradable and non-tradable sector.

2

moves interest rates. In the model, we compute interest rates that areindexed to tradable and to non-tradable goods, respectively, permittingconsideration of relative interest rate developments. In response to theshock, the return on bonds denominated in non-tradable output (the‘domestic’ interest rate) drops sharply in the short run. Once productivitygrowth in the tradable sector actually falls, it prompts a reversal ofthe inter-sectoral resource flows towards the non-tradable sector. Thisgenerates a persistent and hump-shaped rise in the real return on non-tradable denominated bonds over the longer term. In addition, the newstriggers a material reduction in investment growth, while employmentremains resilient. These simulated patterns of adjustment are in line withthe empirical facts that we document for the post-referendum period.

We explore the robustness of our results along several dimensions,including the timing and profile of the news shock, and show that othertypes of shocks do not match the dynamics present in the data. Forexample, one notable insight is that the persistent response of interestrates can plausibly only be related to expectations about growth raterather level changes in relative productivities.

While the main contribution of this paper lies in providing newempirical facts and a formal framework to understand the short-runadjustments of the UK economy to the referendum outcome, we also dis-cuss how more specific consequences of Brexit are in fact key drivers ofproductivity growth in the tradable sector. In particular, we explain howincreased trade barriers, reduced capital flows and impediments to labormobility can all be linked to a looming slowdown in the productivity oftradable goods and services. This discussion of the underlying driversof productivity growth in the tradable sector lends further support toour choice of interpreting and modeling the referendum news.

Our work is related to several strands of research. First, there hasbeen a surge in papers exploring the impact of Brexit on the UK economyand beyond, from a variety of angles.2 Similar to us, Born et al. (2019)and Vlieghe (2019) focus on the period immediately after the referendum

2There are also various studies that focus on the reasons for the outcome of thereferendum rather than its economic impact. See for example Becker et al. (2017),Fetzer (2018) and further references provided in these papers.

3

from a macroeconomic angle. These studies apply a synthetic controlmethod to study the effects of Brexit on UK growth. While the aggregateeffects they find are very similar, our additional focus on sectoral activityand relative prices contributes to understanding the precise adjustmentmechanism. Other papers on the impact of Brexit focus on long-runtrade (Dhingra et al., 2017; Sampson, 2017), foreign direct investment(McGrattan and Waddle, 2020), financial market volatility and stockreturns (Davies and Studnicka, 2018), uncertainty (Steinberg, 2017; Bloomet al., 2018, 2019; Faccini and Palombo, 2019), as well as exchange ratepass-through (Forbes et al., 2018).3 Consistent with the notion of a“deglobalization shock” put forward by Gourinchas and Hale (2017), ourwork captures the referendum impact as one fundamental economicshock. We provide a novel interpretation of this shock as news aboutproductivity growth in the tradable sector. The suggested economicmechanism successfully matches the patterns observed in the newlyconstructed macroeconomic data that we present.

Second, our paper relates to work that has undertaken a seriouscalibration of models with tradable and non-tradable sectors, such asDe Gregorio et al. (1994), Betts and Kehoe (2006) and Lombardo andRavenna (2012). Similar to this line of research, we allocate 2-digit SICindustry level data into tradable and non-tradable categories. To the bestof our knowledge, we are the first to do so for the UK and the first touse such a classification to construct time series aggregates in order toestimate a structural model.

Third, we contribute to the broader SOE literature, which follows theclassic work of Mendoza (1991). We build on the contribution of Drechseland Tenreyro (2018) by allowing for productivity growth differentialsbetween sectors.4 While Drechsel and Tenreyro (2018) focus on emerging

3In particular the trade literature features many more studies that are helpful toanalyze Brexit and its effects. See for example Erceg et al. (2018) for an analysis of theshort-run macroeconomic effects of specific trade policies such as tariffs, as well asCaldara et al. (2020) and Graziano et al. (2018) for recent papers on the effects of tradepolicy uncertainty.

4Other contributions to the SOE literature include, but are not limited to, Kose(2002), Schmitt-Grohe and Uribe (2003), Aguiar and Gopinath (2007) and Garcia-Ciccoet al. (2010). Gourinchas and Rey (2014) survey research on external adjustments thatincludes both small open economy as well as large open economy models.

4

economies, we demonstrate that sectoral shocks to trend productivityare also a useful modeling device for advanced economies.5

Fourth, our paper relates to research on the role of economic news inbusiness cycles more generally, in particular Beaudry and Portier (2006),Jaimovich and Rebelo (2009) and Schmitt–Grohe and Uribe (2012). Ourpaper contributes to the literature that studies news in a open economysetting (Siena, 2014; Kamber et al., 2017) and in multi-sector models(Gortz and Tsoukalas, 2018; Vukotic, 2018). News shocks in our settingcapture information about the UK’s future relationship with the EU andthe structural composition of the UK economy.

The remainder of the paper is structured as follows. Section 2 docu-ments stylized facts about the UK economy following the referendum.Section 3 introduces our two-sector SOE model. Section 4 presents thedata and discusses our estimation to pin down the structural parametersand the initial balanced growth path. Section 5 simulates our Brexitscenario and provides a comprehensive description of the results. Section6 presents a discussion of the economic drivers underlying productivitygrowth in the tradable sector. Section 7 concludes.

2. UK Macroeconomic Adjustments to the Brexit Vote

This section documents a collection of stylized facts about the UKeconomy following the Brexit referendum. Some of these facts arebased on a novel quarterly macroeconomic data set, which we buildby classifying industry data at the 2-digit level into tradable and non-tradable sectors over time. We are the first to do such an exercise for theUK economy. Details on the data construction are provided in Section4.1, where we also document the industrial composition of the tradableand non-tradable sectors.

Figure 1 presents key UK macroeconomic time series for the years2010 to 2018. In each panel, the vertical line indicates the date of thereferendum, 23 June 2016, and the shaded area marks the period after the

5Incorporating differential growth rates in technologies across sectors in businesscycle models also relates to the literature that studies investment-specific technologyshocks alongside shocks to TFP, see Greenwood et al. (2000) and Justiniano et al. (2011).

5

2010 2012 2014 2016 2018 2020 2022 20241

1.5

2

2.5

3

3.5A: Annual growth forecasts by the IMF (%)

Apr-2016

Oct-2017

Apr-2019

2010Q1 2015Q112.05

12.1

12.15

12.2

12.25

12.3B: Gross value added (log level)

Tradable

Non-tradable

2010Q1 2015Q14.3

4.4

4.5

C: Relative prices (log level)

4.58

4.6

4.62

REER Rel. price across sectors

Figure 1: Adjustments of the UK economy following the Brexit voteNotes. Panel A presents the 5-year ahead IMF forecasts for annual UK GDP growth asof April 2016, October 2017 and April 2019 (source: World Economic Outlook). PanelB plots the log level of Gross Value Added in the tradable and non-tradable sector(source: ONS and own calculations, see details in Section 4.1). Panel C displays the loglevel of the real effective exchange rate (left scale) and the relative price of non-tradablevis-a-vis tradable goods (right scale) (source: BIS, ONS and own calculations, see detailsin Section 4.1). In all panels, the vertical line indicates the date of the referendum andthe shaded area marks the period from the Brexit referendum until 2018Q4.

referendum. In this post-referendum period, no actual implementationof the withdrawal from the EU took place. As the figure reveals, however,meaningful adjustments in the economy are visible. Panel A shows thechange in aggregate UK growth relative to pre-referendum expectations.Specifically, it reveals how the IMF successively revised its UK GDPgrowth forecasts following the referendum. A marked decline in theseforecasts is visible over time, indicating a significant negative impact of

6

2010Q1 2015Q1

-2.2

-2

-1.8

-1.6

-1.4

D: Exports and trade balance (%)

26

27

28

29

30

31

2010Q1 2015Q110.4

10.45

10.5

10.55

10.6

10.65

10.7

10.75

10.8E: Aggregate investment (log level)

2010Q1 2015Q1-6

-5

-4

-3

-2

-1

0

1

2

3

4F: Aggregate hours (% dev. from mean)

Jan2010 Jan20150

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5G: Interest rates (%)

US 10-year yield

UK 10-year yield

Figure 1 (continued): Adjustments of the UK economy following the Brexit voteNotes. Panel D shows the 3-year moving average of the ratio between the trade balanceand GDP (left scale) and the ratio between exports and GDP (right scale) (source: ONSand own calculations). Panel E plots the log level of aggregate investment from 2010Q1

(source: ONS and own calculations). Panel F displays the (demeaned) ratio betweentotal hours and labour force participation (source: ONS and own calculations). Panel Gshows the 10-year zero coupon yields for the US and the UK (source: Bank of England).In all panels, the vertical line indicates the date of the referendum and the shaded areamarks the period from the Brexit referendum until 2018Q4.

the Brexit vote on overall growth prospects relative to the pre-referendumperiod. Panel B shows a decomposition of gross value added (GVA) intotradable and non-tradable sectors. The two sectors show a similar trendprior to the referendum, after which there is a sharp break in the growthrate for the non-tradable sector. With Brexit commonly understood tonegatively affect trade, it is notable that growth in the non-tradable

7

sector slows immediately. To explain this, relative price developmentsneed to be accounted for. Panel C presents the relative price of non-tradable to tradable output together with the real effective exchangerate (REER). As we will highlight in the exposition of our two-sectormodel, these concepts are closely related. It is evident that the UKreal exchange rate, while displaying movements throughout the periodshown, drops sharply and relatively permanently around the time of thereferendum. While overall UK growth appears to be negatively affectedby the referendum outcome (Panel A), the relative price effect (PanelB) generates relatively beneficial conditions – a “sweet spot” – for theproducers of tradable goods and services (Panel C). This interpretationis supported by Panel D, which presents exports and the trade balance,both measured as a percentage of GDP. While the patterns in this panelare less stark, they show that UK trade developed relatively robustlyfollowing the Brexit vote. The fact that relative price developmentsgenerate favorable conditions for the tradable sector in the short run willbe a cornerstone of our interpretation of the UK economy’s adjustmentsto the Brexit vote.

Panels E and F show the evolution of aggregate factors of production.While total investment weakened following the referendum, total laborinput (measured relative to labor force participation) has continued toincrease. This difference between investment and employment effectsis consistent with the firm-level survey evidence in Bloom et al. (2019).Panel H shows ten-year zero coupon yields for the UK and the US. Theseyields closely track each other prior to the Brexit vote but a meaningfulspread opens up thereafter, with UK yields remaining persistentlybelow their US counterpart. Omitting inflation risk and term premiaconsiderations, this pattern is already indicative of a mechanism bywhich market participants may have perceived a fall in productivityin the UK relative to the US. The response of interest rates will beparticularly important to verify our interpretation of Brexit relative toalternative explanations.

In summary, while many of the details regarding the UK’s ultimatewithdrawal from the EU are still highly uncertain, the aftermath ofthe referendum has been characterized by significant macroeconomic

8

adjustments in the UK economy. UK economic activity has slowedrelative to trend. Growth in the tradable sector has remained resilientrelative to the non-tradable sector. The British pound has been subject toa pronounced depreciation (and with it the relative price of non-tradablegoods). Exports have been growing robustly. At the same time, UKinterest rates have declined relative to their international counterpart.Our model simulations will jointly explain these facts.

3. The Model

The setting is a real small open economy with a tradable (T) and anon-tradable (N) sector. As in Drechsel and Tenreyro (2018), labour-augmenting productivity in each sector grows at its own rate, denotedby gTt and gNt. The presence of two stochastic trends implies thatthe levels of different variables may grow at different rates along abalanced growth path. To aid exposition, we use lower-case letters todenote stationary variables and upper-case letters to denote variablesthat contain a stochastic trend. The economy is small in the sense thatthe real interest rate is exogenous and the rest of the world absorbsany trade surplus or deficit fully elastically. To capture both domesticand international interest rate variation, we introduce different bonds,denominated in terms of tradable and non-tradable goods, where thelatter is traded by domestic households only. Following Schmitt-Groheand Uribe (2003), we close the model with a debt elastic premium onexternal borrowing. After laying out the agents’ problems and marketclearing conditions, we discuss the key economic forces of the model bypresenting analytical results in a simplified version.

3.1. The firms’ problem

Firms in both sectors combine labor and physical capital using a Cobb-Douglas technology to produce final output. Physical capital is sector-specific and previously accumulated capital cannot be reallocated across

9

sectors. Labor is sector-specific.6 Formally, sector M = T, N producesa final good YMt by combining capital KMt and labor nMt according to

YMt = aMtKαMMt(XMtnMt)

1−αM . (1)

Here, aMt denotes a stationary TFP disturbance and XTt the non-stationarycomponent of labor-augmenting productivity. TFP in sector M follows

ln aMt = $aM ln aMt−1 + εa

Mt, with εaMt ∼N (0, ςa

M) . (2)

where $aM is the persistence (stationary) sectoral TFP and ςa

M the varianceof the shock. The growth rate of sectoral labor-augmenting productivityis defined as

gMt =XMt

XMt−1, (3)

and follows an autoregressive process of the form:

ln (gMt/gM) = $gM ln (gMt−1/gM) + ε

gM, with N

(0, ς

gMt)

, (4)

where $gM captures the persistence and ς

gMT denotes the dispersion

of shocks to its process. Transitory shocks to gMt capture changes tothe growth rate of labor-augmenting productivity in sector M, whichpermanently affect the level of productivity. gM denotes the steady statevalue of the growth rate in sector M. In the remainder of the paper, werefer to the labor-augmenting productivity growth process in each sectorsometimes simply as “productivity growth”.

Firms rent capital and labor in competitive factor markets at rentalrate rk

Mt and real wage wMt, respectively. Profits are given by

YTt −WTtnTt − rkTtKTt (5)

in the tradable sector and

PtYNt −WNtnNt − PtrkNtKNt (6)

6Given that we focus on short-run adjustments, assuming that labor and capital arefreely mobile would likely generate implausibly rapid inter-sectoral reallocation.

10

in the non-tradable sector. Under the assumption of perfect competition,firms make zero profits in equilibrium. The variable Pt denotes therelative price of the non-tradable vis-a-vis tradable goods. This price canbe interpreted as an ‘internal’ measure of the real exchange rate. Froma conceptual point of view, this interpretation goes back to the work ofSamuelson (1964) and Balassa (1964), who have studied internationalproductivity differences and their implications for real exchange rates.7

3.2. The household’s problem

From the perspective of the representative household, while tradable andnon-tradable consumption are assumed to be gross complements, theconsumption of home tradable goods and their foreign counterpart canbe perfectly substituted (the law of one price for tradable goods holds).Following Drechsel and Tenreyro (2018), we specify the period utilityfunction as in Greenwood et al. (1988). In this instance, we scale thedisutility of labor supply by tradable labor-augmenting productivity toensure that both consumption and labor elements of the utility functiongrow at the same rate along the balanced growth path. Formally,

Ut (Ct, XTt−1, XNt−1, nTt, nNt) =

[Ct − XTt−1

ω

(θTnω

Tt + θNnωNt)]1−γ

1− γ, (7)

where θM denotes the disutility of labor in sector M and ω the elasticityof labor supply. The variable Ct is a CES consumption aggregator,expressed in tradable units, that combines tradable and non-tradableconsumption (denoted by CTt and CNt)

Ct =

[ζ1−σCσ

Tt + (1− ζ)1−σ(

XTt−1

XNt−1CNt

)σ] 1σ

, (8)

7The Harrod-Balassa-Samuelson effect is the empirically observed tendency forcountries with stronger productivity in tradable goods relative to non-tradable goodsto have higher prices levels overall. The basic mechanics of this effect feature in ourmodel, where a weakness in productivity growth in the tradable sector introduces afall in the domestic price level.

11

where γ > 1 is the inverse inter-temporal elasticity of substitution (IES)and η = 1/ (1− σ) the elasticity of substitution between tradable andnon-tradable consumption.8 The representative household seeks tomaximize

E0

∞

∑t=0

νtβt[Ct − XTt−1ω−1 (θTnω

Tt + θNnωNt)]1−γ

1− γ, (9)

subject to the budget constraint (expressed in tradable units)

CTt + PtCNt + B∗t + PtBt + ITt + Pt INt + PtYNtsy

st

+φT

2

(KTt+1

KTt− gT

)2

KTt + PtφN

2

(KNt+1

KNt− gN

)2

KNt

= rkTtKTt + Ptrk

NtKNt + WTtnTt + WNtnNt +B∗t+1

1 + r∗t+ Pt

Bt+1

1 + rt.

(10)

β ∈ [0, 1) denotes the subjective discount factor and the variable νt

denotes a risk-premium shock given by:

ln νt = $ν ln νt−1 + ενt with ενt ∼N (0, ςν) . (11)

Sectoral physical capital depreciates at the rate δM and φM captureshow costly is to adjust capital in sector M. Physical investment (IMt)results in the following law of motion:

KMt+1 = (1− δM)KMt + IMt. (12)

One important aspect of the budget constraint is the presence of twodifferent assets, B∗t and Bt with corresponding interest rates r∗t andrt. These are risk-free bonds that pay one unit of tradable goods andnon-tradable goods in the following period, respectively. They can bethought of as bonds that are indexed to different types of inflation ratesin practice. While a bond that pays tradable units – a standard ingredientof SOE models – allows the economy to achieve a trade balance that is

8Note that XTt−1 and XNt−1 enter the utility function to ensure balanced growth.The parameters θT and θN will allow us to match the relative quantities of labor usedin the two sectors.

12

different from zero, the bond that pays non-tradable units remains inzero net supply. Introducing it allows us to determine its interest rate rt,which will move differently from r∗t , shedding light on how “domestic”relative to “world” interest rates can diverge in response to the Brexitnews. This is motivated by the different movement of UK and US ratesobserved in the data, as shown in Section 2. The interest rate on thebonds denominated in tradable goods is given by

r∗t = r∗ + ψ(

eB∗t+1/XTt−b∗ − 1)+ (eµt−1 − 1), (13)

where r∗ is the steady state world interest rate, and the term multipliedby ψ captures a country risk premium, which is increasing in the amountof external debt. The latter assumption follows Schmitt-Grohe and Uribe(2003) and ensures a stationary solution of the model after detrending.9

Finally, the term (eµt−1− 1) captures an interest rate shock, which follows

ln µt = $µ ln µt−1 + εµt with εµt ∼Nt(0, ςµ

). (14)

The variable st in the budget constraint is a government expenditureshock, which can be thought of as a broader aggregate demand shifter,and which follows

ln st = $s ln st−1 + εst with εst ∼N (0, ςs) , (15)

The ratio s/y in (10) is the steady state share of government expendi-ture to non-tradable output.

Given preferences, the relative price of the aggregate consumptionbundle (in terms of tradable units) is

Pct =

[ζ + (1− ζ)

(XNt−1

XTt−1Pt

) σσ−1] σ−1

σ

. (16)

Note that Pct is a stationary variable.

9As we discuss in Section 5.4 and show formally in Appendix F, the conclusionswe draw in this paper are robust to alternative assumptions to ensure the model’sstationary solution.

13

3.3. Market clearing and equilibrium

The market clearing conditions are

YTt = CTt + ITt +φT

2

(KTt+1

KTt− gT

)2

KTt + TBt (17)

in the tradable sector and

YNt = CNt + INt +sy

YNtst +φN

2

(KNt+1

KNt− gN

)2

KNt (18)

in the non-tradable sector. We define the trade balance as

TBt = B∗t −B∗t+1

1 + r∗t. (19)

The model exhibits two stochastic trends and is de-trended accordinglyto characterize a stationary equilibrium. Following Aguiar and Gopinath(2007), Garcia-Cicco et al. (2010) and Drechsel and Tenreyro (2018), we di-vide the sectoral variables by the corresponding technology level XM,t−1.We then calculate the deterministic steady state of this normalized model.All details are given in Appendix A.

3.4. The model’s main economic forces

Using the model, we will show that the empirical facts about the UKeconomy after the referendum can be understood as an adjustmentto news about a lower future productivity growth rate in the tradablesector gT. We do so using careful simulation experiments in an estimatedversion of the model. Before we turn to the estimation and simulationsteps, it is useful to highlight some key economic relationships inthe model. This helps to guide intuition, but also gives grounds forour choice of a real model, which abstracts from frictions such asnominal rigidities and relies purely on relative price and intertemporalsubstitution forces.

What is central to our mechanism are the movements of the relativeprice and the relative returns across the non-tradable and tradable sector

14

in response to news about productivity developments. It is insightfulto think through how these unfold across sectors and over time. Acrosssectors, the relative price of output is directly inversely related to therelative productivities of the sectors. This is a standard feature ofmulti-sector production economies: lower efficiency in making a goodimplies that it is relatively more expensive. Through time, the household’sforward-looking behavior implies that today’s consumption-saving andinvestment allocation decisions depend on the full path of future returnsin both sectors, which in turn are driven by relative productivity. Inequilibrium, a future reduction in productivity growth in the tradablesector therefore lowers the path of the relative price and relative returnin the non-tradable sector from today until the change materializes.

To illustrate this reasoning formally, a simplified version of the modelallows us to show analytically that the current relative price of non-tradables, as well as the return on the bond denominated in non-tradablegoods move in the same direction as the future productivity growthrate in the tradable sector. As a consequence, both the exchange rateas well as the domestic interest rate fall in response to an expectedslowdown in tradable productivity (remember that the internationalinterest rate is exogenous). We present this simplified model and theanalytical derivations in Appendix B. Specifically, we show that thereis a two-period endowment economy structure in which the followingholds:

∂r∂g′T

> 0 (20)

and

∂p∂g′T

> 0, (21)

where r and p are the current home interest rate and current relativeprice of non-tradable goods (the real exchange rate) and g′T is the futuregrowth rate in the tradable sector. Keeping this basic economic forcein mind will be helpful when interpreting the results of our full-blownsimulation exercise.

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4. Novel Data Set and Estimation Strategy

Our strategy of bringing the model to the data and carrying out Brexitsimulations consists of two steps. First, we estimate the model upto the quarter of the EU referendum (2016Q2), using our novel UKmacroeconomic data set. By exploiting variation in this data at businesscycle frequency, this estimation pins down the structural parameters andbalanced growth path of the model, which determine the starting pointfrom which the Brexit simulations are conducted. Since we interpretBrexit as a unique and unprecedented event, this estimation is based onunanticipated disturbances. Second, we simulate the impact of Brexitfrom 2016Q3 onwards. This is done by feeding an anticipated shock intoa model that is calibrated to the parameters and balanced growth pathobtained in the first step. This section describes the first step. We presentthe construction of the data, how we select observables for estimation, aswell as the estimation algorithm and settings. The second step followsin Section 5, which forms the core of the analysis in this paper.

4.1. Data and sectoral classification

We construct a new UK macroeconomic data set from 1987Q3 to 2016Q2,the period during which the UK was a full member of the EU. A keycontribution of this paper is that we construct time series data fortradable and non-tradable GVA, hours, labor productivity and relativeprices. These sectoral time series complement standard macroeconomicvariables, such as aggregate GDP, consumption and investment. To thebest of our knowledge, we are the first to apply such a classification onindustry-level data from the UK and the first to generate sectoral timeseries aggregates that are used to estimate a structural model. Specifically,we use detailed industry-level GVA data from the UK Office of Nationalstatistics (ONS) and treat a given 2-digit SIC industry as tradable if morethan 10% of its final demand is traded, a standard cutoff suggested in theliterature, for example in Lombardo and Ravenna (2012). We chain-linkthe data using the standard national accounts methodology employedby the ONS and also compute series for sectoral total hours by adding

16

up hours data using the same industry classification. The time-seriesfor sectoral labor productivities are then constructed by taking the ratiobetween sectoral GVA and total hours. Having aggregated detailed GVAdata, we calculate the relative price of non-tradable goods by dividingthe resulting implicit price deflators.

Table 1: Industries shares in non-tradable and tradable sector (%)

Non-tradable TradableAgriculture 0.07 1.35

Mining and Quarrying 0.00 2.29

Manufacturing 0.89 20.99

Electricity, Gas, Steam Air Conditioning 3.19 0.00

Water Supply, Sewage, Waste Mgmt 1.07 0.91

Construction 10.93 0.00

Services 83.85 74.46

Notes. Nominal output shares of each industry (under the SIC 2007 classification) inthe classification into tradable and non-tradable sectors. This is shown for the year2016. The supply and use tables are used to calculate the tradability index; nominalGVA data at factor prices are taken from low level aggregates published by the ONS.

Our classification into tradable and non-tradable sectors leads to aroughly 50-50 of total UK hours worked on average over the sample(our model will be calibrated accordingly). The same is approximatelythe case for the shares of nominal GVA. Table 1 shows that in 2016,21% of tradable GVA was produced by manufacturing sectors, 74% byservices. The corresponding numbers for the non-tradable sector are1% and 84%. The most important tradable manufacturing industries aremotor vehicles, wearing apparel and alcoholic beverages and tobaccoproducts, with food and beverage services, insurance services andfinancial services, representing key tradable services. For robustness, wealso exclude government-related sectors from the non-tradable sector.The resulting dynamics in non-tradable GVA are very similar, with acorrelation of 0.93 between non-tradable GVA and non-tradable GVAexcluding government. More details on the classification methodology,including a full list of all 2-digit industries and their classifications, areprovided in Appendix C.

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4.2. Mapping the model to observable variables

As observable variables for the estimation, we use the following set oftransformed time-series: the quarterly growth rates of sectoral laborproductivity (available from 1994Q1), the quarterly growth rate of therelative price of non-tradable goods (only available from 1997Q1), thequarterly growth rate of the real effective exchange rate, total hours(demeaned) and the ratios of nominal consumption, investment andtrade balance to GDP (available from 1987Q3). We make use of theKalman filter to handle missing observations in the time-series of thesectoral labor productivities and the relative price of non-tradable goods.In the estimation step, we allow for measurement error for each of theconstructed observable variables.10

Constructing observables to estimate our model poses two key chal-lenges. The first one entails the use of implicit price deflators to derivereal quantities. Model consistent consumption and investment can becomputed by deflating nominal consumption and investment by thetradable GVA implicit price deflator. However, since the resulting GVAdeflators exhibits significant amount of noise (and are only availablefrom 1997Q1), using them to calculate model consistent aggregateswould imply discarding useful information (and having to rely on theintroduction of additional measurement errors). To circumvent this issue,we use the ratios of nominal aggregates, rather than the growth rate ofreal quantities, as observable variables following Christiano et al. (2015).To estimate the structural parameters more precisely, our proceduresrequires that the values of the steady state ratios implied by the modelmatch the averages in the data.

The second challenge is that there are two exchange rates conceptsin the model: the relative price of non-tradables vis-a-vis tradables(an ‘internal exchange rate’) and the relative price of aggregate homeconsumption with respect to its foreign equivalent (an ‘external exchangerate’). In the data, the internal exchange rate is calculated using the

10We do so for two reasons. First, aggregation of detailed industry level datainevitably gives rise to measurement error. Second, although the growth rates ofGVA and GDP are highly correlated, the measures of GVA and GDP are not exactlyequivalent.

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implicit price deflators. Mapping the external real exchange rate to thedata requires assumptions about the rest of the world. First, preferencesin the rest of the world are assumed to be the same as those in the homeeconomy. Second, at business cycle frequencies, we further assumethat stochastic trends of the tradable sectors at home and abroad arecointegrated. We define the real effective exchange rate as:

Qt =EtP c

tP c,∗

t,

where Et denotes the nominal exchange rate, P ct the nominal price

level of the home consumption bundle and P c,∗t its foreign equivalent.

Under the Law of One Price (LOOP), it follows that P∗Tt/Et = PTt.Using this relation, and dividing the numerator and denominator in thedefinition of the real effective exchange rate by PTt to obtain a ratio ofreal consumption bundles, we get

Qt =Pc

tPc,∗

t=

Pct

ξt.

where ξt captures exogenous movements in foreign prices (Pc,∗t ) and is

governed by the following stochastic process

ln ξt = $ξ ln ξt−1 + εξt with εξt ∼N(0, ςξ

). (22)

This shock captures variation in the exchange rate that arises in the restof the world. We emphasize that the exchange rate is an endogenousobject and the shock will play a minor role. It allows us to use moreinformation in the estimation to pin down structural parameters moreprecisely and can be interpreted as a persistent measurement error.

4.3. Estimation procedure, calibration and priors

The model is estimated using Bayesian techniques. We select structuralshocks that are widely accepted in the literature. The model has arelatively small number of parameters due to its parsimonious structure.We set σ to −0.5, which corresponds to an elasticity of substitution equal

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to η = 11−σ = 0.67, within the range of estimates in the literature. The

chosen value gives rise to gross complementarity across consumptionaggregates; a feature that helps generating unconditional output co-movement across sectors. The depreciation rates are assumed to be equalacross sectors and match the sample average of the ratio of nominalinvestment to GDP (18.12%). We choose θN and θT to equally distributehours worked across sectors, which is true in the data based on ourclassification of tradable and non-tradable sectors discussed above. Wecalculate the ratios s

y = 0.184 and tby = −0.015 using ONS data. We

also compute gT and gN directly from the data. The discount factor (β)is set to match a quarterly foreign real interest rate of 1%. Finally, theelasticity of the foreign interest rate with respect to debt (ψ) is set toa small number (5× 10−6) following the literature (Schmitt-Grohe andUribe, 2003). The model calibration is summarized in Table 2.

Table 2: Calibrated parameter values

Description Source Period Value

γ inverse of the IES Drechsel and Tenreyro(2018)

2

σ elast. of subst. mid-range estimate −0.5

δM depreciation in M ONS 1987− 2016 i/y = 0.181

φN capital adjustmentcost (N)

4

θT disutil. of labor (T) ONS/own calc. 1994− 2016 nT/n = 0.5

θN disutil. of labor (N) ONS/own calc. 1994− 2016 nN/n = 0.5

sy govt exp./GDP own calculations 1994− 2016 0.184

tby trade balance/GDP own calculations 1994− 2016 −0.015

gT trend growth rateof productivity (T)

ONS/own calc. 1990− 2016 1.83% ann.

gN trend growth rateof productivity (N)

ONS/own calc. 1990− 2016 1.02% ann.

β discount factor r∗ = 0.01

ψ debt-elasticity ofpremium

5× 10−6

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The locations of the prior means largely correspond to those in Smetsand Wouters (2007) (see Table 3). Using the ONS supply-and-use tablesfor the period 1997-2014/5, we compute the annual shares of tradablesinto aggregate consumption and then pin down the prior mean of ζ

that targets the sample average of 0.59). The prior mean of ζ is set tomatch this sample average. We also calculate the sample means of thesectoral capital shares to be αT = 0.315 and αN = 0.245 in the tradableand non-tradable sectors. We center the prior means around the sampleaverages and then compute their posterior distributions.11 Note thatwe set the value of the investment adjustment cost parameter in N toφN = 4, in line with the chosen prior mean for sector T.

Table 3: Priors and posteriors for estimated parameters

Description Distribution Mode Mean 90% HPDIStructural parameters

cT/C share T consumption Gaussian 0.59 0.59 0.57 0.61

ω elasticity of labor supply Gaussian 1.99 1.99 1.85 2.13

αT capital share in T Gaussian 0.31 0.31 0.30 0.32

αN capital share in N Gaussian 0.25 0.25 0.24 0.26

φT capital adjustment cost in T Gaussian 9.65 9.65 8.45 10.85

Shocksς

gN st.dev. of prod. growth shock in N Inv. Gamma 0.014 0.014 0.012 0.016

ςgT st.dev. of prod. growth shock in T Inv. Gamma 0.014 0.014 0.012 0.016

ςs st.dev. of expenditure shock Inv. Gamma 0.036 0.036 0.031 0.04

ςµ st.dev. of foreign interest rate shock Inv. Gamma 0.01 0.01 0.009 0.011

ςν st.dev. of risk-premium shock Inv. Gamma 0.035 0.036 0.03 0.042

ςaT st.dev. of TFP level shock in T Inv. Gamma 0.013 0.013 0.011 0.015

ςaN st.dev. of TFP level shock in N Inv. Gamma 0.013 0.013 0.011 0.012

ςξ st.dev. of labor supply shock in N Inv. Gamma 0.026 0.026 0.022 0.029

$gN persistence of prod. growth shock in N Beta 0.23 0.25 0.07 0.43

$gT persistence of prod. growth shock in T Beta 0.12 0.15 0.04 0.25

$s persistence of expenditure shock Beta 0.88 0.86 0.79 0.94

$µ persistence of foreign interest rate shock Beta 0.03 0.04 0.01 0.08

$ν persistence of risk-premium shock Beta 0.94 0.93 0.88 0.98

$aN persistence of TFP level shock in N Beta 0.80 0.75 0.58 0.93

$aT persistence of TFP level shock in T Beta 0.97 0.97 0.95 0.99

$ξ persistence of exchange rate shock Beta 0.95 0.94 0.91 0.99

Measurement errorsιN labor productivity in N Inv. Gamma 0.013 0.013 0.011 0.015

ιT labor productivity in T Inv. Gamma 0.014 0.014 0.012 0.016

ιP relative price Inv. Gamma 0.014 0.015 0.012 0.017

11Setting the right priors for the capital shares is very important not only becausethey affect the value of the depreciation rate that matches the investment to GDPratio but also because they influence the estimated value of adjustment costs. Inaddition, both capital shares and the adjustment cost parameters are key parametersfor understanding the dynamics of the returns on bonds denominated in tradable andnon-tradable units.

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The posterior mean of the elasticity of labor supply ω is estimatedto be 1.99, in line with standard values in the literature. The meanestimate of the investment adjustment cost in the tradable sector (9.65)is somewhat higher than in related studies, but plausible given therelatively low depreciation rates. The discount factor, the temporarysectoral TFP, expenditure and foreign price are estimated to be highlypersistent stochastic processes (ρν = 0.93, ρa

N = 0.75, ρaT = 0.97, ρs =

0.85 and ρξ = 0.95 respectively). A common finding in most modelsfeaturing stochastic trends in labour-augmenting productivity is thatthe estimated persistence of the growth shocks tends to be relativelylow (ρg

N = 0.25 and ρgT = 0.15). The foreign interest rate shock displays

very little persistence (ρµ = 0.04). The posterior standard deviationof measurement errors (denoted by ι) for sectoral labor productivitiesand the relative price of non-tradable goods are similar and statisticallydifferent from zero. The estimation results are detailed in Table 3.

5. The Brexit Simulation

In this section we present our main Brexit simulation, which shows thatthe economy’s response to news about a reduction in tradable sectorproductivity growth matches the empirical facts for the post-referendumUK economy. Naturally, the interpretation of Brexit as productivitynews abstracts from a wide range of specific implications of Brexit. Wededicate Section 6 to a detailed discussion on how these relate to areduction in tradable sector productivity growth.

5.1. Settings of the Brexit simulation

We model the news shock to gTt as persistent, but temporary. As aresult, there is a permanent effect on the level of future tradable sectorproductivity, but the growth rate eventually fully recovers. In the short-run, a fully permanent reduction in the growth rate delivers qualitativelysimilar results. In contrast, as we will show in alternative simulations,a temporary news shock about only the level of productivity does notsuffice to explain the empirical patterns in UK macroeconomic data. To

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implement the persistent reduction, we modify the process for gTt in theestimated version of the model. While that estimated process capturesbusiness cycle movements during the period of EU membership, it is lesssuitable for analyzing a structural change of the type we are investigating.Specifically, in the simulation gTt is determined by

ln (gTt) = $gT ln (gTt−1) +

(1− $

gT)

ln (gTt) ,

ln (gT,t) =$gT ln (gTt−1) +

(1− $

gT)

ln (gT) + εgTt,

where we set $gT = 0.95 and $

gT = 0.8. This implies that the initial

fall in tradable sector productivity growth is gradual and that the totalreduction in the level of productivity in the tradable sector is completeafter about 30 years.

Table 4: Estimates of long-run effects of WTO rules on UK trade and GDP

Estimated reduction inStudy trade (%) GDP (%)Ebell and Warren (2016) 21–29 2.7–3.7IMF (2018) 5.2–7.8Kierzenkowski et al. (2016) 10–20 2.7–7.5UK Government (2018) 13–18 6.3–10.7

The experiment is configured so that the future reduction in produc-tivity growth is fully anticipated. The economy starts on a balancedgrowth path in quarter 0. In quarter 1, it is revealed that there willbe a persistent reduction in productivity growth in the tradable sectorfrom quarter 11 onward. This anticipation horizon broadly mimics theplanned timeline for EU exit following the referendum.12 We calibratethe scale of the shock with reference to existing studies of the potentialeffects of Brexit on trade. We use those that focus on the move to tradingarrangements governed by World Trade Organization (WTO) rules. Thisis not necessarily because we believe this is the most likely outcome,

12The referendum was held on 23 June 2016. The UK government triggered Article50 of the Lisbon treaty on 30 March 2017, with the United Kingdom’s membership ofthe European Union to end within two years of that date. The end date of the UK’s EUmembership was subsequently postponed as the negotiation process developed.

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but because the underlying assumptions about the eventual tradingarrangements are more consistent across studies. Table 4 summarizesrecent estimates. We calibrate our experiment so that trade falls by 10%in the long run, in line with the smaller estimates of the effects of movingto WTO rules. Our results could therefore be regarded either as a lowerbound estimate of a transition to WTO rules or as a simulation of atransition to a relatively closer trading relationship with the EU.

Our simulation abstracts from uncertainty about the news shock andits timing. Of course, the event of the Brexit vote is likely to have affectedboth first and second moments. Our simulations demonstrate that firstmoment effects go a long way in rationalizing the adjustments in theeconomy that we document in the data. We refer to the work of Bloomet al. (2019) for a firm-level analysis that focuses explicitly on uncertainty.Using a structural model Steinberg (2017) finds that uncertainty plays arelatively small role in the context of Brexit. Caldara et al. (2020) providea model for US economy that features both news and uncertainty abouttrade. We will present various robustness checks of our simulation withrespect to the profile and the timing of the news shock.

5.2. Main results of the Brexit simulation

Figures 2 and 3 present our main simulation results. In each panel, theshaded area (quarters 1 to 10) marks the phase after the news aboutthe shock have been revealed, but before the shock materializes (inquarter 11). This mimics the shaded area after the referendum but beforeBrexit implementation in Figure 1. The black dashed line represents thecounterfactual balanced growth path in which the Brexit shock does notoccur. The simulation shows that the anticipated fall in productivitygrowth in the production of tradables leads to an immediate fall inthe relative price of non-tradable output. This encourages a near-termreallocation of resources towards the tradable sector and an export boom(the “sweet spot” effect). Overall activity in the economy, as measuredby GDP, slows down. The return denominated in terms of non-tradables,our notion of the domestic interest rate, declines. In the longer-term,resources are reallocated towards the non-tradable sector.

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Figure 2: Headline responses to benchmark Brexit scenario

Unpacking the results in more detail reveals the key economic forcesat play. Panel A of Figure 2 shows the trajectory of tradable sector labor-augmenting productivity growth that we feed into the model. Duringthe anticipation phase, tradable sector growth is unchanged from thebaseline balanced growth path. In quarter 11, productivity growth fallsfor several quarters before starting to recover gradually. The cumulativeeffect of the shock is a permanent reduction in the level of tradable sectorproductivity of around 10%. The news about this future productivitygrowth trajectory leads to an immediate fall in the relative price of non-tradable output, which will become relatively more efficient to produce.This is one of the core economic forces of the model, as highlightedanalytically in Section 3.4. Panel F shows that the price of non-tradable

25

output falls immediately, even before productivity growth has changed.During the anticipation phase, tradable goods are relatively profitableto produce because productivity growth has not yet begun to fall. Asshown in panels C and D, this effect encourages production of tradablegoods and the trade balance rises in the near term. Once tradable sectorproductivity falls, however, the incentives to produce tradable goodsdecline and output and the trade balance fall in the longer term. Theprofile of non-tradable output is the mirror image of tradable output, asshown in panel D. During the anticipation phase, non-tradable outputis relatively unprofitable and output declines. Once tradable sectorproductivity falls, non-tradable output becomes relatively profitable andoutput increases in the longer term. Eventually non-tradable outputconverges back to the pre-shock trajectory, since the balanced growthpath for the non-tradable sector is unaffected by the change in tradablesector productivity. The net effect of the opposing forces on the tradableand non-tradable sectors gives rise to a muted response of GDP (panelB).13 The initial response of GDP is small, but the effect builds over time.Its long-run level is around 3% lower. This is towards the smaller end ofthe range of estimates in Table 4, consistent with the fact that the scaleof the shock we study generates a relatively small reduction in tradecompared to the studies cited.

Figure 3 focuses on factors of production and rates of return. Theinter-sectoral reallocation is consistent with the main mechanism un-derpinning our results: during the anticipation phase, the tradablesector becomes relatively profitable, but this effect is reversed oncetradable sector productivity actually falls. Panels A and B show thatlabor moves from the non-tradable sector to the tradable sector duringthe anticipation period, to support increased production of tradablegoods. Overall, total employment rises during the anticipation phase.This pattern starts to reverse once productivity growth in the tradable

13The chain-linked GDP growth rate is computed as:

gGDPt = ωT,t

yTtyTt−1

gTt−1 + (1−ωT,t)yNt

yNt−1gNt−1,

where ωT,t is computed as a one-year rolling average of the expenditure share ontradable goods, yTt

yTt+ptyNt. This approximates a national accounts treatment.

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Figure 3: Sectoral responses to benchmark Brexit scenario

sector actually falls. Panel C shows that investment in the tradable sectorfalls abruptly before slowly converging to a new, lower, level. Investmentprospects in the tradable sector are dominated by the longer-term outlookfor productivity. In contrast, panel D shows that, while non-tradableinvestment initially falls, it subsequently rises above the baseline path.In aggregate, there is a significant near-term fall in investment whileemployment increases. Panels E and F show the real bond returnsin both sectors. The small decline in the bond rate denominated intradable goods is driven entirely by the debt elastic premium.14 Thereturn on bonds denominated in non-tradable output, our notion of the

14The near term rise in exports reduces foreign debt and hence the interest rate. Bycalibrating ψ to be small, the effect on the r∗t is restricted to a few basis points.

27

domestic UK interest rate, falls during the anticipation phase, beforerising above the steady-state level. These dynamics reflect the behaviorof the marginal product of capital, which falls in the near term becausereturns to production in the non-tradable sector are temporarily lower.This confirms our analytical derivations in Section 3.4.

5.3. Comparison with empirical patterns

Although the Brexit process is still underway, our simulation is broadlyconsistent with the macroeconomic dynamics of the UK economy sincepresented in Section 2. The simulation predicts a long-run reduction ofGDP of around 3%, relative to the baseline path. As shown in SectionFigure 1, the IMF forecasts of UK GDP were reduced by roughly 0.5%per year following the referendum, amounting to a reduction in the levelof UK GDP (relative to the pre-referendum forecast) of around 2.5%over the five-year forecast horizon.15 The simulated path of quantitiesand prices in the tradable and non-tradable sectors also match the datafairly well, with a sharp decline in both the relative price of non-tradableoutput and the real effective exchange rate around the referendum date,as well as a marked slowdown in GVA growth in the non-tradablerelative to the tradable sector. The simulation implies that the news ofBrexit triggers a fall in interest rates, consistent with the decline in UKlong-term government bond yields following the referendum shown inFigure 1, though the fall in the data is larger and more protracted.16 Thesimulation predicts a temporary boom in exports, mirroring the UK’srelatively strong export performance following the referendum. Whilethe movement in the trade balance in the data is relatively modest, thepickup in the export to GDP ratio is around 2 percentage points, similar

15Born et al. (2019) provide an estimate of 2% based on constructing a no-Brexitcounterfactual.

16The comparison with the data is complicated by the range of factors affectinggovernment bond yields that are omitted from the model. The model abstracts entirelyfrom nominal prices and risk.

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to the response of the trade balance to GDP ratio in our simulation.17 Thevolatility in the trade data makes it difficult to draw strong conclusions,but according to the latest vintage of data, calendar-year export growthin 2017 was 5.6%, which is substantially above the Bank of England’s(pre-referendum) May 2016 Inflation Report forecast of 1.25% (see Bank ofEngland, 2016, Table 5.E, page 34). Moreover, as our simulation predicts,there was a substantial fall in UK investment following the referendumresult in the data. Estimates presented by Carney (2019) suggest thatthe effects of the referendum result may have reduced UK businessinvestment by around 25%. At the same time, employment has beenstrong in the data, consistent with the pick-up in total hours in oursimulation. Our model simulations suggest an economy-wide shift fromcapital towards labor, a phenomenon directly in line with firm-levelsurvey evidence provided by Bloom et al. (2019).18

Our overall conclusion is that the model performs well in matchingthe broad contours of UK macroeconomic performance since the refer-endum. The interpretation of the referendum outcome as news about areduction future productivity growth in the tradable sector is consistentwith the UK’s adjustment visible in the data.

5.4. Robustness

Appendix D shows that our main results are robust with respect to threeimportant assumptions. First, the results are qualitatively similar fora range of plausible variation in the timing of the decline in tradable

17The model abstracts from gross trade flows (differentiated imports and exports)making it less straightforward to map from model concepts to the data. Mechanically,the fact that the trade balance increases by less than exports suggests that imports rosefollowing the referendum. In the absence of strong expenditure switching effects, thevalue of imports may increase because of the higher price of imports associated withthe depreciation of sterling.

18These observations suggest that the source of the shock matters for the patternof sectoral reallocations. Though a formal comparison is beyond the scope of thispaper, it is instructive to compare the simulation with the behavior of the UK economyfollowing the depreciation of sterling associated with the UK’s exit from the ExchangeRate Mechanism in 1992. The period following that depreciation saw a significantinvestment boom, more apparent in tradables than non-tradables. Our simulation doesnot have these properties, because the depreciation is the result of the anticipation of anegative shock that depresses the returns on investment.

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sector productivity growth. Appendix D.1 reports results for casesin which the shock is anticipated to occur 5 and 15 quarters in thefuture, alongside the baseline assumption of 11 quarters. The timingof reversals in inter-sectoral allocation changes, but the dominant forceunderpinning the scenario is the long-run decline in the level of tradablesector productivity. Since the long-run decline is independent of thetiming of the productivity growth reduction, the results from the variantsconsidered are very similar. Second, the responses are robust to theassumption that productivity falls more sharply than the benchmarkcase. Again, this reflects that the dominant force is the effect on thelong-run level of tradable sector productivity. Holding the scale ofthis effect constant, a faster decline in tradable sector productivityhas relatively little effect on the dynamic responses, even in the nearterm. Appendix D.2 provides the details. Third, the responses are notsensitive to the assumption used to close the model, that is, to ensurea determinate return to the steady-state net foreign asset position (seethe discussion below equation (13)). Our model achieves this throughthe presence of a debt-elastic premium on foreign borrowing. AppendixD.3 demonstrates that the simulations are almost identical in a variantof the model in which the r∗t is fixed and a determinate net foreign assetposition is achieved by the assumption that there is constant growth inthe population of households.

5.5. Comparison with other structural shocks

Simulations using other structural shocks are inconsistent with theempirical patterns in UK data. We demonstrate this by repeating oursimulation for two different news shocks. The first is news about areduction in the level (rather than the growth rate) of productivity in thetradable sector. The second generates news about an acceleration in thegrowth rate of productivity growth in the non-tradable sector (ratherthan a deceleration in the tradable sector). Note that in principle, we canrepeat the simulation for any of the various shocks present in our model.We chose to focus on these two, as they enter as a very similar wedge asour benchmark shock in the equations of the model. This means they

30

are likely “competitors” in terms of generating comparable dynamics.19

The first alternative shock we study reduces aTt in the future. Again,we focus on a shock that arrives with certainty 11 quarters ahead. Wecalibrate the shock to generate the same reduction in trade as in ourbenchmark simulation in Section 5.2. This experiment is obviously arelevant comparison, as it works through the same economic margin butreduces the level rather than the growth of productivity in the productionof traded output. The full simulation results are provided in AppendixE. The simulation shows that the level shock indeed generates dynamicsin activity across sectors that are qualitatively similar. We do see a boomin tradable and a reduction in non-tradable production, accompanied bya fall in the relative price. However, what stands out as the importantdifference is the response of relative returns. A level shock generatesa short-lived interest rate differential at the exact time when the shockmaterializes. This is clearly inconsistent with the persistent decouplingof UK from world interest rates in the data starting immediately after thereferendum outcome. The Euler equations of the model need variationin the technology growth rate to imply persistent changes in returns. Thistells us that the adjustment of the UK economy is more plausibly relatedto expectations about growth rates than about levels.

The motivation for studying the second alternative shock, an ac-celeration the growth rate of non-tradable productivity, is to verifywhether our mechanism is symmetric in the sense that it only requires avariation in the ratio gTt/gNt but does depend on whether numeratoror denominator are changed. The corresponding results in Appendix Eshow that for a calibration of the experiment that generates the sameon-impact reduction in the relative price of non-tradable goods, a very(perhaps implausibly) large increase in productivity growth in the non-tradable sector is required. More importantly, the longer run projection ofthis simulation is vastly different from our benchmark. The accelerationin future gNt implies a large long-run expansion in GDP as well as a largeincrease in the real rate on bonds denominated in non-tradable goods.While the unobserved longer run outcomes cannot be used to reject the

19Simulations based on other shocks, such as expenditure, labor supply, etc., areavailable upon request.

31

explanation with certainty, we are not aware of any theoretical argumentwhich would link Brexit with a productivity growth improvement in thenon-tradable sector. On the other hand, a number of mechanisms linkBrexit to a reduction in tradable productivity growth, as we discuss inthe next section.

6. Drivers of Tradable Sector Productivity Growth

We conceptualize the 2016 referendum outcome as news about a futureslowdown in productivity growth in the UK’s tradable sector. Oursimulations show that forward-looking decisions of households andfirms in a general equilibrium environment lead to aggregate and sectoraleconomic dynamics that closely resemble the UK economy immediatelyafter the Brexit vote. The appeal of our exercise is that it providesa simple framework to understand the short-run adjustments to thereferendum news: when nothing fundamental has actually materialized,but when the economy is already responding strongly.

Naturally, the drawback of our analysis is that it cannot speak to themore specific underlying changes in the economy that Brexit is expectedto entail. We now consider more detailed structural adjustments thatfirms and households may expect, and explain how they may directlyaffect productivity growth in the tradable sector. We focus on threeexpected consequences of Brexit: barriers to trade in goods and services,reduced capital flows and declining labor mobility. All three havefeatured prominently in the public debate around the referendum.20

Importantly, a large-scale survey of UK firms by Bloom et al. (2019)reveals that these potential consequences feature prominently as self-reported sources of Brexit uncertainty of economic decision-makers. Foreach aspect, we explain how they link to a fall in gTt, by highlightingrelevant research.

20For an up-to-date description of the specifics of the EU Withdrawal Agreement, seehttps://www.bankofengland.co.uk/eu-withdrawal/eu-withdrawal-agreement-act.

32

6.1. Barriers to trade in goods and services

Leaving the single market is commonly understood to bring aboutimpediments to trade, most notably tariffs. There is a large strand ofresearch that has provided integrated theories of international tradeand economic growth, for example the seminal work of Grossmanand Helpman (1989, 1991). While we simulate an exogenous shockto the productivity growth rate, these authors provide a theory ofhow trade barriers endogenously determine the rate of growth of aneconomy.21 This line of research highlights distortions in the allocation ofresources towards technical change as central to the mechanism by whichtrade in goods and services affects growth endogenously. For example,if importing human-capital intensive goods becomes more difficult,substituting these imports with domestic production will absorb factorinputs that would otherwise be used for innovative activities. Hence,if Brexit is expected to harm trade in a way that growth-enhancingresources in the UK’s tradable sector have to be diverted to importsubstitution, decision makers will indeed expect growth in the tradablesector to fall.

6.2. Reduced capital flows

Trade barriers and regulations may reduce cross-border capital mobilityand hinder in particular inflows of foreign direct investment (FDI) bymultinational corporations that operate in the UK and the EU. Thelonger run effects of this are explicitly studied in a recent paper byMcGrattan and Waddle (2020). Using a multi-country dynamic equi-librium model, these authors present a Brexit simulation in which theeconomy converges to a new balanced growth path with lower aggregategrowth. This is largely driven by a reduction in investment in technology

21More recent contribution include Sampson (2016) and Impullitti and Licandro(2018), who show that trade liberalization increases growth via firm selection. Note thattrade barriers are frequently studied in an influential class of trade models sometimesreferred to as New New Trade Theory (NNTT). Prominent examples include Eaton andKortum (2002) and Melitz (2003). The effect of tariffs in these models is typicallystudied by means of comparison between different steady states, so do not allow us todirectly think about the impact on the economy’s (expected) growth dynamics.

33

capital, which in their baseline experiment is shown to fall by almostone third. While the model of McGrattan and Waddle (2020) doesnot feature separate sectors for tradable and non-tradable output, it isconceivable that this technology capital channel operates largely throughinvestments made by tradable producers. Along these lines, Benignoet al. (2020) provide a recent discussion as well as a host of references onthe role of technological spillovers through capital flows. The prospect ofdiminished FDI as part of leaving the EU therefore provides a rationalefor diminished productivity growth expectations in the tradable sector.

6.3. Lower labor mobility

Portes and Forte (2017) provide an assessment of the potential restrictionson movement of workers after Brexit, and conclude that they willlikely have a significant negative impact on UK growth and produc-tivity. Bloom et al. (2019) find that firms with a larger share of EUmigrants in their workforce are significantly more concerned aboutthe uncertain prospects of the withdrawal process. To the extent thatBrexit may reduce the UK’s ability to attract labor that is employedin productivity-enhancing activities in the tradable sector, the newsabout Brexit generates expectations of a lower trajectory for productivitygrowth in tradable production. This is again akin to the factor allocationaspect at the heart of frameworks linking trade and growth such asGrossman and Helpman (1991).

In brief, expected impediments to the free flow of goods and services,capital and labor provide a range of mechanisms that could reduce futureproductivity growth in the tradable sector. As phrased by Gourinchasand Hale (2017), the common component of these fundamental channelsis a “deglobalization shock” that reduces specialization and efficiency.While each of the channels deserves greater attention in their own right,the contribution of our work lies in summarizing the mechanism in aconcise formal framework that sheds light on the short-run macroeco-nomic response of the UK economy to Brexit. The direct link betweenthe setup of our experiments and the underlying drivers of growth inthe tradable sector makes this a credible and powerful exercise.

34

7. Conclusion

Motivated by empirical facts, we develop a model to analyze the im-pact of the Brexit referendum on the UK economy. Negative news toproductivity growth in the tradable sector – our way to conceptualizeBrexit – have a long-lasting negative effect on the price of non-tradablegoods. This creates a “sweet spot” for exporters during the periodleading up to EU withdrawal, with a subsequent reversal after Brexit isimplemented. The news also give rise to a sharp fall in domestic interestrates and a material slowdown in investment, accompanied by relativelystrong employment, an unprecedented combination in UK historicaldata. While Brexit encompasses a variety of economic mechanisms, thispaper provides a concise framework through which macroeconomicadjustments to this momentous historical event can be interpreted.

35

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40

Appendix For Online Publication

Appendix A contains details about the model. It presents the optimality conditions(A.1), describes the de-trending and stationary equilibrium of the model (A.2), anddetermines its steady state (A.3). Appendix B presents analytical results in a simplifiedversion of the model. Appendix C contains details on the data construction. AppendixD examines the sensitivity of the Brexit simulations to alternative assumptions about thetiming of the shock and the speed of the fall in LAP. Appendix E presents simulationsbased on alternative news shocks. Appendix F lays out the alternative model withpopulation growth.

A. Model Details

A.1. First order conditions

The optimality conditions of firms are:

rkTt = αTaTtK

αT−1Tt (XTtnTt)

1−αT , (23)

WTt = (1− αT) aTtKαTTt (XTtnTt)

−αT XTt, (24)

rkNt = αNaNtK

αN−1Nt (XNtnNt)

1−αN , (25)

andWNt = Pt (1− αN) aNtK

αNNt (XNtnNt)

−αN XNt. (26)

These conditions state that sectoral factors of productions are paid their marginalproducts.

The household’s optimality conditions with respect to CTt, CNt, nTt, nNt, KTt+1,KNt+1, B∗t+1, Bt+1 are:

[Ct − XTt−1ω−1 (θTnω

Tt + θNnωNt)]−γ

(CTt

ζCt

)σ−1

= X−γTt−1λt, (27)

[Ct − XTt−1ω−1 (θTnω

Tt + θNnωNt)]−γ

[CNt

(1− ζ)Ct

XTt−1

XNt−1

]σ−1 XTt−1

XNt−1= X−γ

Tt−1λtPt, (28)

[Ct − XTt−1ω−1 (θTnω

Tt + θNnωNt)]−γ

θTXTt−1nω1−1Tt = X−γ

Tt−1λtWTt, (29)

1

[Ct − XTt−1ω−1 (θTnω

Tt + θNnωNt)]−γ

θNXTt−1nω−1Nt = X−γ

Tt−1λtWNt, (30)

X−γTt−1λtνt

[1 + φT

(KTt+1

KTt− gT

)]= X−γ

Tt βEtλt+1νt+1

[rk

Tt+1 + (1− δ)+

φT(KTt+2

KTt+1− gT)

KTt+2

KTt+1− φT

2(

KTt+2

KTt+1− gT)

2]

,(31)

X−γTt−1λtνtPt

[1 + φN

(KNt+1

KNt− gN

)]= X−γ

Tt βEtλt+1νt+1Pt+1

[rk

Nt+1 + (1− δ) +

φN

(KNt+2

KNt+1− gN

)KNt+2

KNt+1− φN

2

(KNt+2

KNt+1− gN

)2 ],

(32)

X−γTt−1λtνt = X−γ

Tt β (1 + r∗t )Etλt+1νt+1, (33)

andX−γ

Tt−1λtνtPt = X−γTt β (1 + rt)Etλt+1νt+1Pt+1. (34)

Equations (27)-(28) pin down the optimal tradable and non-tradable consumptionchoices, equations (29)-(30) state the labor supply choices as increasing functions ofsectoral wages, equations (31)-(32) denote the Euler equations associated to sectoralphysical capital and equations (33)-(34) the Euler equations for bonds. Note that, bytaking the ratio between (28) and (27), we can determine the relative price of non-tradable output (the internal exchange rate).

A.2. Stationary equilibrium

We now proceed to characterize the stationary equilibrium by introducing ”lower-case”variables, denoting the detrended counterparts of non-stationary variables. Definect =

CtXTt−1

, cTt =CTt

XTt−1, cNt =

CNtXNt−1

, KTt =KTt

XTt−1, KNt =

KNtXNt−1

, pt = PtXNt−1XTt−1

.The household first order conditions in normalized forms become

ct =[ζ1−σcσ

Tt + (1− ζ)1−σ (cNt)σ] 1

σ , (35)

(ct −

(θT

ωnω

Tt +θN

ωnω

Nt

))−γ ( cTt

ζct

)σ−1

= λt, (36)

(ct −

(θT

ωnω

Tt +θN

ωnω

Nt

))−γ ( cNt

(1− ζ) ct

)σ−1

= ptλt, (37)

2

(ct −

(θT

ωnω

Tt +θN

ωnω

Nt

))−γ

θTnω−1Tt = λtwTt, (38)

(ct −

(θT

ωnω

Tt +θN

ωnω

Nt

))−γ

θNnω−1Nt = λtwNt, (39)

λtνt

[1 + φT

(kTt+1

kTtgTt − gT

)]= βg−γ

Tt Et

λt+1νt+1

[rk

Tt+1 + (1− δ)

+ φT

(kTt+2

kTt+1gTt+1 − gT

)kTt+2

kTt+1− φT

2

(kTt+2

kTt+1gTt+1 − gT

)2 ],

(40)

λtνt pt

[1 + φN

(kNt+1

kNtgNt − gN

)]= β

g1−γTtgNt

Et

λt+1νt+1pt+1

[rk

Nt+1 + (1− δ)

+ φN

(kNt+2

kNt+1gNt+1 − gN

)kNt+2

kNt+1− φN

2

(kNt+2

kNt+1gNt+1 − gN

)2 ],

(41)

λtνt = β (1 + r∗t ) g−γTt Etλt+1νt+1, , (42)

and

λtνt pt = β (1 + rt)g1−γ

TtgNt

Et pt+1λt+1νt+1. (43)

The firms’ first order conditions become

rkTt = αTaTtk

αT−1Tt (nTtgTt)

1−αT , (44)

wTt = (1− αT)aTtkαTTt (nTt)

−αT g1−αTTt , (45)

rkNt = αNaNtk

αN−1Nt (nNtgNt)

1−αN (46)

andwNt = pt(1− αN)aNtk

αNNtn−αNNt g1−αN

Nt . (47)

The normalized constraints are

yTt = cTt + iTt +φT

2

(kTt+1

kTtgTt − gT

)2

+ tbt, (48)

yNt

(1− s

yst

)= cNt + iNt +

φN

2

(kNt+1

kNtgTt − gN

)2

, (49)

3

iTt = kTt+1gTt − (1− δ) kTt, (50)

iNt = kNt+1gNt − (1− δ) kNt, (51)

yTt = aTtkαTTt g1−αT

Tt n1−αTTt , (52)

yNt = aNtkαNNt g1−αN

Nt n1−αNNt , (53)

and

tbt = b∗t −b∗t+1

1 + r∗tgTt. (54)

A.3. Steady state

A.3.1 Analytical derivation of the steady state (for Brexit simulation purposes)

We remove time subscripts to compute the steady state values. From equations (40)-(43),it follows that

β =1

(1 + r∗) g−γT

, (55)

r =gN

βg1−γT

− 1, (56)

rkT =

1

βg−γT

− (1− δ) , (57)

rkN =

gN

βg1−γT

− (1− δ) . (58)

From the rental rates of capital in (44) and (46) we recover the sectoral capital to laborratios

kT

nT=

(rk

TαT

) 1αT−1

gT, (59)

kN

nN=

(rk

NαN

) 1αN−1

gN. (60)

Steady state wages can be calculated from equations (45) and (47) as

wT = (1− αT) g1−αTT

(kT

nT

)αT

(61)

4

andwN

p= (1− αN) g1−αN

N

(kN

nN

)αN

. (62)

By calibrating syN

, and using equations (49) and (53), we can express non-tradableconsumption as:

cN =

(kN

nN

)αN

g1−αNN

(1− s

yN

)− [gN − (1− δ)]

kN

nN

nN = ANnN. (63)

By calibrating the ratio tby and using equation (48) and (52), we can express consumption

in T in terms of nN,

cT =

(kT

nT

)αT

g1−αTT

(1− tb

y

)− [gT − (1− δ)]

kT

nT

nT = ATnT. (64)

Dividing equations (37) by (36), we can get an expression for p. We use equations (63)and (64) to recover the ratio of sectoral hours,

p =

[ζcN

(1− ζ) cT

]σ−1

=

[ζ ANnN

(1− ζ) ATnT

]σ−1

⇒ nN

nT= p

1σ−1

(1− ζ) AT

ζ AN. (65)

We divide equation (39) by equation (38) and substitute for nN/nT as above to get

wN

pwT=

1p

θN

θT

(nN

nT

)ω−1

=1p

θN

θT

(p

1σ−1

(1− ζ) AT

ζ AN

)ω−1

⇒ p =

[pwT

wN

θN

θT

((1− ζ) AT

ζAN

)ω−1] 1−σ

ω−σ

.

(66)We take the ratio between equation (35) and cT to obtain the ratio

ccT

=

(ζ1−σ + (1− ζ)1−σ

(cN

cT

)σ) 1σ

. (67)

Finally, we divide equations (38) and (36) and then substitute for ccT

to obtain nT

[wT

θT

(cT

ζc

)σ−1] 1

ω−1

= nT. (68)

Once the value of nT is pinned down, the remaining algebra is simple.

A.3.2 Numerical computation of steady state (for estimation purposes)

The steady state values of β, r, rkT, rk

N , kTnT

and kNnN

are given by equations (55)-(60). Giventhe values of sectoral hours (nN and nT), we can compute the steady state values of

5

sectoral physical capital

kN =kN

nNnN, (69)

andkT =

kT

nTnT. (70)

Sectoral outputs are therefore

yN = kαNN (nM gN)

1−αN (71)

andyT = kαN

T (nT gT)1−αT . (72)

Given the ratios s/y, tb/y and cT/C, we solve numerically for p, cN, cT and y .

cN +sy

yp= yN

1− [gN − (1− δN)]

αN

rkN

(73)

cT +tby

y = yT

1− [gT − (1− δT)]

αT

rkT

(74)

pcN = cT1− cT

CcTC

(75)

y = yT + pyN. (76)

We then compute the constants which allow us to exactly match the analytic steadystate in the previous section as follows:

syN

=

sy

pyN, (77)

tbyT

=tby

yyT

, (78)

ζ =p

1σ−1

p1

σ−1 + cNcT

, (79)

θN =

wNp (1− ζ)1−σ ( cN

c )σ−1

nω−1N

(80)

and

θT =wTζ1−σ( cT

c )σ−1

nω−1T

, (81)

where wN and wT are given by (61) and (62).

6

B. Simplified Model with Analytical Results

In this Appendix we lay out a simplification of our macroeconomic model which allowsus to derive some key economic relationships analytically. In particular, we presentcomparative statics that show that the domestic interest rate as well as the relative priceor non-tradable output fall with a future reduction in tradable productivity growth.Formally, we show that

∂r∂g′T

> 0 (82)

and∂p

∂g′T> 0. (83)

B.1. An endowment version of the model

We first simplify the technological structure of the economy. Suppose there is no capitaland no labor, and at time t income in both sectors is given by an exogenous endowmentflow XTt and XNt, respectively. Suppose the economy starts of with XT0 = XN0 = 1 andthe growth rates of the sectors are defined as gTt = XTt/XTt−1 and gNt = XNt/XNt−1.In what follows we also abstract from any stochastic disturbances other than thosedriving variation in gTt. The stationary equilibrium in this economy is defined by thefollowing set of equations:

ct =[ζ1−σcσ

Tt + (1− ζ)1−σ (cNt)σ] 1

σ , (84)

c−γt

(cTt

ζct

)σ−1

= λt, (85)

c−γt

(cNt

(1− ζ) ct

)σ−1

= ptλt (86)

λtνt = β (1 + r∗t ) g−γTt Etλt+1νt+1, (87)

λtνt pt = β (1 + rt)g1−γ

TtgNt

Et pt+1λt+1νt+1, (88)

cTt = gTt − tbt, (89)

cNt = gNt, (90)

where r∗t is still exogenously given as in the main text.

7

B.2. A two-period characterization

Suppose at time t the economy is growing along its balanced growth path (the stationaryversion is in steady state) and there is only one future period t′. At time t, agents findout that at time t′ a lower growth rate in the tradable sector materializes with certainty.Without loss of generality, we normalize gTt = gNt = gNt′ = 1. The realization of gTt′

is denoted g′T < 1. For ease of notation, we also normalize the predetermined level ofdebt b∗t = 0.

Now consider the case σ = 0, so that ct = cζTc1−ζ

N (Cobb-Douglas consumption

aggregation). Using the fact that tbt = b∗t − gTtb∗t+11+r∗t

, the equilibrium in this two periodstructure is characterized by the following set of equations:

cT = 1 +b∗′

1 + r∗, (91)

c′T + b∗′ = g′T, (92)

λ = β(1 + r∗)

g′Tλ′, (93)

λp = β(1 + r)λ′p′, (94)

λp = (1− ζ)cζ(1−γ)T , (95)

p = ζcζ(1−γ)−1T , (96)

λ′p′ = (1− ζ)(c′T)ζ(1−γ), (97)

p′ = ζ(c′T)ζ(1−γ)−1. (98)

Note that we have dropped t subscripts and adopted the notation x = xt, x′ = xt′ .This is a system in 8 equations and 8 unknowns (cT, c′T, p, p′, λ, λ′, b∗′, r). This defineseach variable as a function of g′T.

In this setting it is insightful to derive an analytical expression for the domesticinterest rate (the return on bonds paying non-tradable units) as

r = βζ(1−γ)−1(1 + r∗)ζ(1−γ)(g′T)γ(1−ζ(1−γ))−1 − 1 (99)

8

If γ > 1 and given that 0 < ζ < 1, is evident that

∂r∂g′T

> 0. (100)

Similarly, it is possible to implicitly differentiate p with respect to g′T. This becomesa relatively tedious derivation. For ease of notation, we make the further simplificationγ = 1 (log-utility), which allows for an explicit derivation. Specifically, the relative priceof non-tradable goods can be derived as

p =

ζg′Tβ(1+r∗) + ζ(1 + r∗)

g′T + 1 + r∗. (101)

This gives us∂p

∂g′T=

1/β + ζ(1 + r∗)(g′T + 1 + r∗)2 > 0. (102)

9

C. Data Construction

We use ONS supply and use tables for 1997− 2015 to calculate, for each 2-digit SICindustry, a tradability index at basic prices (the ratio of exports plus imports to finaldemand). Exports denote exports of domestic output only (i.e. excluding re-exportsof imported goods). Following Lombardo and Ravenna (2012), we define a sector as‘tradable’ if more than 10% of its total demand is traded using the 2-digit SIC industrylevel classification. This threshold is arbitrary, but it coincides with those suggestedby De Gregorio et al. (1994) and Betts and Kehoe (2006).1 Table C.1 at the end of thesection shows the full list of industries corresponding to the tradable and non-tradablecategories. Figure C.1 shows the industry classification that yields from using the 10%cut-off. In particular, around 0.54 of aggregate GVA is classifies as non-tradable andthe remaining as tradable. As is evident in Table 1 in the main text, service industriestend to have lower ratios (although there are many exceptions), and manufacturingindustries higher ratios.

0.1 0.45 0.55 0.65 0.75 0.78 0.85 0.86 0.9 0.950

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Agriculture

Mining and Quarrying

Manufacturing

Electricity, Gas and Steam Air Conditioning

Water Supply, Sewage and Waste Management

Construction

Services

Cut-off

Figure C.1: Industry classification using 2016 Supply and Use Tables

After classifying each of the 114 industries into the tradable and non-tradable cate-gories, we add the consumption expenditure of households and non-profit institutionsserving households. We then divide sectoral expenditure by aggregate consumptionfor the years 1997 to 2016 to calculate the share of tradable consumption into totalconsumption. We then compute the sample mean and retrieve a value of 0.59. As

1An alternative definition is proposed by De Gregorio et al. (1994) that classify a sector as ‘tradable’ if10% of its total supply is exported.

10

shown in the Figure C.2, this share is rather constant over time. This value is in linewith the estimate for the UK in Lombardo and Ravenna (2012), who calculated a valueof 0.64 (based on 2000− 2005 data) and with a previous internal Bank of England’sestimate of 0.5− 0.6. Using this arbitrary threshold, we also find that around half ofthe economy by GVA can be classified as tradable. It is worth noting that the shareof tradable output in aggregate output is lower than the tradable share in aggregateconsumption because non-tradable services, such as construction, public administrationand defense and compulsory social security services, have a much higher weight inoutput than in household consumption.

The factor shares are computed using the supply and use tables from 1997 to 2016. Inline with Goodridge et al. (2018), we use partial appropriation of self-employed incometo labor income. This assumes a fraction of self-employed income accruing to laborincome. The labor share in sector i in year t is then defined as the sum of compensationof employees and the fraction of self-employed income accrues to labor divided by totalGVA (at basic prices). In computing the labor share in the N sector, we exclude imputedrents as they tend to bias the estimates. The capital share is residually determined asone minus the labor share. The sample means of the capital shares are 0.315 and 0.245in the T and N sectors sector respectively. Note that assigning self-employed incometo labor income tends to increase the values of the labor shares. Figure C.2 shows theevolution of the consumption share of T goods into aggregate consumption and thelabor shares in the T and N sectors respectively.

1998 2000 2002 2004 2006 2008 2010 2012 2014

0.6

0.65

0.7

0.75

0.8

Figure C.2: Labor Shares in T and N and Consumption T Shares

Once we have classified each 2-digit industries into the tradable or the non-tradablecategory, we use the associated ONS detailed industry-level GVA data to construct a

11

1995Q1 2000Q1 2005Q1 2010Q1 2015Q1

-10

-8

-6

-4

-2

0

2

4

6

8

Figure C.3: Gross Value Added in T and N

time-series for tradable output consistent with aggregate GVA, by aggregating GVAover the set of industries in each category, using ONS’s standard national accountschain-linking methodology. The resulting time-series for GVA growth are shown inFigure C.3. The growth rates of GVA output do not display significant differences acrossthe two categories since total hours in the non-tradable sector display an upward trend.

We also construct tradable and non-tradable total hours, using the published industryhours data underlying ONS labor productivity estimates, together with our classificationof industries.2 Hours worked by sector are measured following Tenreyro (2018). Wethen compute the average labor productivity growth rate in each sector from 1994-2017.Over this period labor productivity growth in the tradable sector averaged about 1.8%on an annual basis.

2The data on hours are available at a slightly higher level of aggregation than 2-digit level; therefore,we need to make a judgement about the tradability of each grouping of 2-digit industries in the hoursdata, based on the tradability of the underlying 2-digit industries.

12

Table C.1: Detailed industry classification

Tradable Industries Non-Tradable IndustriesEmployment services Sewerage services, sewage sludgeRepair and maintenance of ships and boats Remediation serv. and other waste management serv.Computer programming, consultancy and related services Retail trade serv., except of motor vehiclesServices of head offices, management consulting services Veterinary servicesGambling and betting services Services furnished by membership organisationsFood and beverage serving services Residential Care & Social Work ActivitiesAccounting, bookkeeping and auditing services, tax consulting services Natural water, water treatment and supply servicesWarehousing and support services for transportation Gas, distr. of fuels, steam and air cond. supplyRail transport services Printing and recording servicesLegal services Households as employers of domestic personnelLibraries, archives, museums and other cultural services ConstructionTelecommunications services Public admin. and defence, compulsory soc. securityMining support services Wholesale & retail trade & repair of motor vehiclesPrepared animal feeds Electricity, transmission and distributionBakery and farinaceous products Services to buildings and landscapeScientific research and development services Other personal servicesSoft drinks Owner-Occupiers’ Housing ServicesAdvertising and market research serv. Travel agency, tour operator and other rel. serv.Architectural and engineering serv.; technical testing and analysis serv. Real estate serv.& imputed rentInsurance, except compulsory social security & Pension funding Human health serv.Repair and maintenance of aircraft and spacecraft Security and investigation servicesGrain mill products, starches and starch products Real estate activities on a fee or contract basisFinancial services, except insurance and pension funding Rest of repair, InstallationMotion Picture, Video & TV & Music & Programming And Broadcasting Education servicesProducts of forestry, logging and related services Rental and leasing servicesDairy products Manufacture of cement, lime, plaster (and articles of)Information services Land transport and transport via pipelines, excl. railWeapons and ammunition Sports serv. and amusement and recreation serv.Publishing services Postal and courier serv.FurnitureAlcoholic beverages & Tobacco productsRepair services of computers and personal and household goodsPreserved meat and meat productsFinancial Services (and Auxiliary) And Insurance ActivitiesFabricated metal products, excl. machinery& ammunitionWaste collection, treatment and disposal, materials recovery serv.Other food productsProducts of agriculture, hunting and related servicesProcessed and preserved fish, crustaceans, molluscs, fruit and vegetablesSoap and detergents, cleaning and polishing, perfumes and toiletPaper and paper productsCoke and refined petroleum productsWood and (products of), except furnitureTextilesGlass, refractory, clay, other porcelain & ceramic, stone and abrasivePaints, varnishes and similar coatings, printing ink and masticsOther transport equipmentAccommodation servicesRubber and plastic productsVegetable and animal oils and fatsFish and other fishing productsOther manufactured goodsWearing apparelShips and boatsIndustrial gases, inorganics and fertilisers (all inorganic chemicals)Creative, arts and entertainment servicesCoal and ligniteAir transport servicesBasic iron and steelLeather and related productsOffice administrative, office support and other business support servicesServices auxiliary to financial services and insurance servicesElectrical equipmentMotor vehicles, trailers and semi-trailersCrude Petroleum And Natural Gas & Metal OresDyestuffs, agro-chemicalsBasic pharmaceutical products and pharmaceutical preparationsOther professional, scientific and technical servicesWater transport servicesOther basic metals and castingPetrochemicalsComputer, electronic and optical productsOther chemical productsOther mining and quarrying productsMachinery and equipment n.e.c.Air and spacecraft and related machineryWholesale trade services, except of motor vehicles and motorcycles

Notes. Industry categorization computed using the supply used tables from 1997-2016. Tradability index is calculated as the ratio between the trade and

final demand and then averaged over 1997-2016. Cut-off is set to 10%.

13

D. Sensitivity Experiments

This appendix presents results from variations of our benchmark Brexit scenario ofSection 5. Appendix D.1 shows that the results are similar for alternative assumptionsabout the timing of the decline in tradable sector productivity growth. AppendixD.2 presents simulations that assume that productivity falls more sharply than thebenchmark case. Appendix D.3 demonstrates that the responses are not sensitive to theassumption used to close the model.

D.1. The anticipation horizon

Figures D.1 and D.2 show the results of the baseline experiment (solid blue lines) anda variant in which the decline in tradable sector productivity growth starts after fivequarters (dot-dashed red lines). The long-run effects of the shock are the same, butalternative timing assumptions are likely to affect short-term dynamics.

Unsurprisingly, the dynamics of the alternative assumption are slightly differentwhen the decline in productivity growth occurs earlier. The initial export boom is lesslong-lived and requires a larger reallocation of labor to deliver higher tradable sectoroutput. The switch in factor flows (from the tradable sector to the non-tradable sector)occurs earlier, commensurate with the earlier reduction in tradable sector productivitygrowth.

14

Figure D.1: Headline responses to Brexit scenario with alternative timing

15

Figure D.2: Sectoral responses to Brexit scenario with alternative timing

16

Figures D.3 and D.4 show the results of the baseline experiment (solid blue lines)and a variant in which the decline in tradable sector productivity growth is delayed forfifteen quarters (dot-dashed red lines).

Figure D.3: Headline responses to Brexit scenario with alternative timing

The relative effect of a longer anticipation horizon is, unsurprisingly, the opposite ofthe previous case of a shorter anticipation horizon. The adjustment dynamics are moreprotracted and the near-term reallocation of labor during the anticipation horizon ismore muted relative to the baseline case. With a longer anticipation horizon, tradablesector investment falls by less, as the reduction in tradable sector productivity occurs inthe more distant future.

17

Figure D.4: Sectoral responses to Brexit scenario with alternative timing

18

D.2. Faster fall in tradable sector productivity

Figures D.5 and D.6 show the results of the baseline scenario (solid blue lines) alongsidea case in which the decline in tradable sector productivity growth occurs more rapidly(red dot-dashed lines). The alternative scenario is constructed by assuming that theparameter controlling the persistent component of tradable sector productivity growthis set to $

gT = 0.9 (compared with the baseline assumption of 0.95).

The alternative scenario implies that tradable sector productivity reaches its new,lower, level in roughly half the time of the baseline scenario. The scale of the productivitygrowth shock is roughly doubled to ensure that the long-run effect on tradable sectorproductivity is identical to the baseline scenario.

Figure D.5: Headline responses to Brexit scenario with alternative persistence

19

Figure D.6: Sectoral responses to Brexit scenario with alternative persistence

Unsurprisingly, the dynamic responses to the more rapid productivity growthshock variant are somewhat faster in some cases. However, the broad contours of themacroeconomic responses are very similar in both cases. This demonstrates that thedominant effect is the anticipation of permanently lower tradable sector productivity inthe long run. This effect drives the key relative price in the model: the impact effect onthe relative price of non-tradable output is very similar (Figure D.5, panel B).

20

D.3. Population growth variant

Figures D.7 and D.8 show results for a variant of the model that incorporates populationgrowth. A derivation of this variant is presented in Appendix F. However, the innovationcompared with the baseline model is straightforward. In the variant, we assume thathouseholds are infinitely lived, but that new households are born each period. Thepopulation growth rate is constant. Individual households have identical preferencesto those that we have assumed in previous versions of the model. Therefore their firstorder conditions identical to the baseline model.

Figure D.7: Headline responses to Brexit scenario in alternative model version

21

However, population growth means that aggregate consumption is not characterizedby the same Euler equation as the one that holds for each individual household. Thisis because new households are born with no financial wealth. Accounting for theheterogeneity in financial wealth delivers an aggregate consumption equation thatdepends on the distribution of wealth. The simple population structure implies that thedistribution of wealth can be summarized by aggregate stocks of wealth (ultimately, thestock of foreign debt).

Figure D.8: Sectoral responses to Brexit scenario in alternative model version

The dependence of the aggregate consumption Euler equation on wealth means thatthe steady state net foreign asset position is pinned down, even if the economy mayfreely borrow and lend at a fixed world interest rate. The steady state NFA position ispinned down by the (im)patience of domestic agents relative to the (growth adjusted)world real interest rate.

22

While population growth is just a device to close the model, rather than a plausiblemodel of demographics, we set the constant population growth rate to be consistentwith 0.5% annual population growth (broadly consistent with 1997-2016 UK data).

The results show that the dynamics are virtually identical in the two variants ofthe model, despite the fact that the tradable bond rate remains fixed in the populationgrowth variant. The slight differences in relative bond rates generate small differences inthe returns to capital across the two variants, with minor implications for the dynamicresponses of hours and investment. However, the broad contours of the simulation arevery similar in the two variants.

23

E. Simulations Based on Other Structural Shocks

This appendix presents simulations results for two other structural shocks. In bothcases, the simulations are carried out following the methodology described in the text.Figures E.1 and E.2 show the results for a news shock about a reduction in the level(rather than the growth rate) of productivity in the tradable sector.

Figure E.1: Headline responses to a shock to the level of tradable sector productivity

24

Figure E.2: Sectoral responses to a shock to the level of tradable sector productivity

25

Figure E.3 and E.4 present the results for a shock containing news about accelerationin the growth rate of productivity growth in the non-tradable sector (rather than adeceleration in the tradable sector). This is calibrated to generate the response of therelative price of non-tradable goods on impact.

Figure E.3: Headline responses to a shock to non-tradable sector productivity growth

26

Figure E.4: Sectoral responses to a shock to non-tradable sector productivity growth

27

F. Model with Population Growth

F.1. Overview

This variant of the model involves a small adjustment to the household sector. Themodel is essentially an open economy variant of the Weil (1989), Weil (1991) model withGHH preferences.3

In this variant, we assume that households are infinitely lived, but that newhouseholds are born each period. The population growth rate is constant. Individualhouseholds have identical preferences to those in the baseline model, so their firstorder conditions identical to the ones derived in the main text. However, populationgrowth means that aggregate consumption is no-longer characterized by the same Eulerequation as the one that holds for each individual household. This is because newhouseholds are born with no financial wealth and accounting for the heterogeneityin financial wealth delivers an aggregate consumption equation that depends on thedistribution of wealth. The simple population structure implies that the distribution ofwealth can be summarized by aggregate stocks of wealth (ultimately, in our model, thestock of foreign debt).

The dependence of the aggregate consumption Euler equation on wealth meansthat the steady state net foreign asset position is pinned down, even if the economymay freely borrow and lend at a fixed (tradable-good denominated) interest rate. Thesteady-state net foreign asset position is pinned down by the (im)patience of domesticagents relative to the (growth adjusted) world real interest rate.

F.2. Households

The number of households alive in period t is Zt. Population evolves according to:

Zt+1 = (1 + ϑ) Zt (103)

As before, to convert the model into stationary units, aggregate quantities must bedetrended by sectoral growth rates. However, the introduction of population growthmeans that the quantities of interest in this variant are measured in per capita terms. Thismeans that the detrending factors for quantities (though not prices must also accountfor population growth). With this in mind, sectoral growth rates for this variant aredefined as:

gMt = (1 + ϑ)XMt

XMt−1, (104)

for M = N, T.3A closed economy version of the Weil model with GHH preferences is analyzed by Ireland (2005),

which also forms a guide for our approach.

28

A household born in period s maximizes the following utility function:

∞

∑t=s

βt−s

[Cs

t − XTt−1ω−1(

θT(ns

Tt)ω

+ θN(ns

Nt)ω)]1−γ

1− γ

so that the preferences of an individual household are identical to those in the baselinemodel. The s superscript indexes the date of birth of the household.

The household budget constraint, denominated in traded goods, is given by:

Pct Cs

t + B∗,st + PtBst = WTtns

Tt + WNtnsNt + Πt +

B∗,st+11 + r∗t

+ PtBs

t+11 + rt

(105)

where it is assumed that households are born with no financial wealth or debt, so thatBs

s = B∗,s = 0. As in previous derivations, positive values of B and B∗ represent debt.Relative to the baseline model, two adjustments are made to facilitate the subsequent

derivation. First, the budget constraint is written in terms of the total consumptionbundle, incorporating the price of consumption in terms of tradable output, Pc

t . Second,households are assumed to receive lump sum profits (allocated from both tradableand non-tradable firms) denoted by Π. These profits are distributed equally to allhouseholds (including newborns). This means that households do not own the capitalstock in this model. Instead, firms are assumed to own the capital stock, discussedbelow.

Writing the budget constraint in terms of the aggregate consumption bundle simpli-fies the derivations of the consumption function considerably. The total expenditure onconsumption satisfies:

Pct Ct = PtCNt + CTt (106)

and the consumption bundle (as in the baseline model) is given by

Ct =

[ζ1−σCσ

Tt + (1− ζ)1−σ(

XTt−1

XNt−1CNt

)σ] 1σ

(107)

The allocation of consumption between tradable and non-tradable consumption is astatic problem and the optimality conditions imply:

[CNt

CTt

ζ

1− ζ

XTt−1

XNt−1

]σ−1 XTt−1

XNt−1= Pt (108)

which is the ratio of the first two first order conditions for the household in the previousderivation.

These equations provide solutions for CNt, CTt, Pct given a solution for Ct. The rest

of the subsection derives a representation of the aggregate consumption function (and

29

hence a solution for Ct).The first order conditions for the household can be written as:[

Cst − XTt−1ω−1 (θT (ns

Tt)ω + θN (ns

Nt)ω)]−γ

=

β (1 + r∗t ) Pct

Pct+1

[Cs

t+1 − XTtω−1(

θT(ns

Tt+1)ω

+ θN(ns

Nt+1)ω)]−γ

1 + rt = (1 + r∗t )Pt

Pt+1

XTt−1θT (nsTt)

ω−1 =WTt

Pct

XTt−1θN (nsNt)

ω−1 =WNt

Pct

The household’s inter-temporal budget constraint is:

∞

∑j=0Dt+jPc

t+jCt+j =∞

∑j=0Dt+j

(WTt+jns

Tt+j + WNt+jnsNt+j + Πt+j

)−

∞

∑j=0Dt+j

(Pt+j

Pt+j+1(1 + rt+j

) − 11 + r∗t+j

)Pt+j+1Bs

t+j+1 − B∗,st − PtBst

where the discount factor satisfies

Dt+j ≡

Dt+j−1

1+r∗t+j−1for j ≥ 1

1 for j = 0

and the usual transversality condition

limj→∞Dt+j+1

B∗,st+j+1

1 + r∗t+j= 0

has been applied.The inter-temporal budget constraint says that the present value of consumption

expenditures equals non-financial wealth (the top line on the right hand side), net ofexpected debt revaluation effects and existing debts (second line).4

The first order condition for asset allocations implies that expected debt revaluations

4The debt revaluation effects measure the difference between the expected returns on non-tradableand tradable bonds, given that the tradable bond rate is chosen to value the intertemporal resourceconstraint.

30

are zero in all future periods, so that the inter-temporal budget constraint is:

∞

∑j=0Dt+jPc

t+jCt+j =∞

∑j=0Dt+j

(WTt+jns

Tt+j + WNt+jnsNt+j + Πt+j

)− B∗,st − PtBs

t

A household’s non-financial wealth is given by:

Ωst ≡

∞

∑j=0Dt+j

(WTt+jns

Tt+j + WNt+jnsNt+j + Πt+j

)= WTtns

Tt + WNtnsNt + Πt +

11 + r∗t

Ωst+1

where the first line is a definition and the second line exploits the properties of thediscount factor and employs a transversality condition.5

The household’s first order conditions for labor supply demonstrate an importantresult: labor supply is determined entirely by aggregate conditions (productivity, wagesand prices). This means that all households will supply the same labor, independentlyof their consumption. As a result the non-financial wealth of all households is identicaland given by:

Ωt = WTtnTt + WNtnNt + Πt +1

1 + r∗tΩt+1 (109)

where

nTt =

(WTt

θTXT,t−1Pct

) 1ω−1

(110)

nNt =

(WNt

θNXT,t−1Pct

) 1ω−1

(111)

To simplify the Euler equation, define the disutility of labor supply as:

N st ≡ XTt−1ω−1 (θT (ns

Tt)ω + θN (ns

Nt)ω)

=XTt−1ω−1

(θT

(WTt

θTXT,t−1Pct

) ωω−1

+ θN

(WNt

θNXT,t−1Pct

) ωω−1)

(112)

where the second line substitutes for the equilibrium levels of labor supply. Once again,the disutility of labor supply is identical for all households: so N s

t = Nt, ∀s.This means that the Euler equation can be written as:

Cst+1 −Nt+1 = β

1γ (1 + rt)

1γ

(Pc

tPc

t+1

) 1γ

[Cst −Nt]

5Specifically that human wealth does not grow faster than the interest rate:limj→∞

(1 + rt+j

)−1 Ωst+j+1 = 0.

31

which implies that

Pct+1Cs

t+1 − Pct+1Nt+1 = β

1γ (1 + rt)

1γ

(Pc

tPc

t+1

) 1γ−1

[Pct Cs

t − PctNt]

Iterating the Euler equation forward implies that

Pct+jC

st+j = Pc

t+jNt+j + βjγD

−1γ

t+j

(Pc

tPc

t+j

) 1γ−1

[Pct Cs

t − PctNt]

Using this expression in the household’s inter-temporal budget constraint gives:

Ωt − PtBst − B∗,st =

∞

∑j=0Dt+j

Pct+jNt+j + β

jγD

−1γ

t+j

(Pc

tPc

t+j

) 1γ−1

(Pct Cs

t − PctNt)

= (Pc

t Cst − Pc

tNt)∞

∑j=0

βjγD

1− 1γ

t+j

(Pc

tPc

t+j

) 1γ−1

+∞

∑j=0Dt+jPc

t+jNt+j

This implies that the household’s consumption function can be written as:

Pct Cs

t = PctNt + Ψ−1

t Ωt −Ψ−1t (PtBs

t + B∗,st )

where Ψt is the inverse of the marginal propensity to consume and Ωt is adjustednon-financial wealth, given respectively by:

Ψt = 1 + β1γ

(Pc

t+1Pc

t (1 + r∗t )

) γ−1γ

Ψt+1 (113)

Ωt = WTtnTt + WNtnNt + Πt − PctNt +

11 + r∗t

Ωt+1 (114)

Aggregation across households is straightforward. The consumption function is anaffine function of financial wealth. The non-financial wealth components are commonacross all households. So the aggregate consumption function is also an affine functionof (aggregate) financial wealth.

Aggregate consumption is equal to:

Pct Cagg

t ≡ Zt−1Pct Co

t + (Zt − Zt−1) Pct Cn

t

where Cot and Cn

t are per capita consumption levels of ‘old’ households (i.e., those alivein period t− 1) and newborn households respectively.

32

Since newborn households enter the model with no financial wealth, we have:

Cnt = Pc

tNt + Ψ−1t Ωt

The consumption functions of all old agents are affine in financial wealth/debt, so

Cot = Pc

tNt + Ψ−1t Ωt −Ψ−1

t (PtBt + B∗t )

where Bt and B∗t are per capita debt stocks (since the date t ‘old’ households representthe entire population in period t− 1).

This implies that:

Pct Cagg

t = Zt

(Pc

tNt + Ψ−1t Ωt

)− Zt−1Ψ−1

t (PtBt + B∗t )

and dividing both sides by Zt gives the per capita consumption function:

Pct Ct = Pc

tNt + Ψ−1t Ωt −Ψ−1

tPtBt + B∗t

1 + ϑ(115)

where C, B and B∗ are per capita consumption and debt stocks.Note that the case of log utility, γ = 1, implies that the expression for Ψt simplifies

substantially to:Ψt = (1− β)−1 , ∀t

which implies that the consumption function under log utility is

Pct Ct = Pc

tNt + (1− β) Ωt −1− β

1 + ϑ(PtBt + B∗t )

The aggregate household budget constraint is given by:

Pct Ct +

PtBt + B∗t1 + ϑ

= WTtnTt + WNtnNt + Πt + PtBt+1

1 + rt+

B∗t+11 + r∗t

(116)

reflecting the same logic as above.6

We can eliminate non-financial wealth from the consumption function to derive anaggregate Euler equation. Rearranging the consumption function gives:

Ωt = Ψt (Pct Ct − Pc

tNt) +PtBt + B∗t

1 + ϑ

6The household budget constraint holds for all households, but newborns have no initial financialwealth/debt: Bt

t = B∗,tt = 0. So the per capita value of previously accumulated debt is equal to the percapita value of debt held last period, divided by the change in population.

33

which we can substitute into the difference equation for Ω to give:

Ψt (Pct Ct − Pc

tNt) +PtBt + B∗t

1 + ϑ= WTtnTt + WNtnNt + Πt − Pc

tNt

+1

1 + r∗t

(Ψt+1

(Pc

t+1Ct+1 − Pct+1Nt+1

)+

Pt+1Bt+1 + B∗t+11 + ϑ

)

Using the aggregate budget constraint to substitute for PtBt+B∗t1+ϑ gives:

Ψt (Pct Ct − Pc

tNt) + PtBt+1

1 + rt+

B∗t+11 + r∗t

= Pct Ct − Pc

tNt

+1

1 + r∗t

(Ψt+1

(Pc

t+1Ct+1 − Pct+1Nt+1

)+

Pt+1Bt+1 + B∗t+11 + ϑ

)which can be rearranged to give:7

(Ψt − 1) (Pct Ct − Pc

tNt) =Ψt+1

1 + rt

(Pc

t+1Ct+1 − Pct+1Nt+1

)− ϑ

1 + ϑ

Pt+1Bt+1 + B∗t+11 + r∗t

Finally, noting from (113) that Ψt − 1 = β1γ

(Pc

t+1Pc

t (1+rt)

) γ−1γ

Ψt+1, gives

Pct Ct − Pc

tNt =

(Pc

t+1Pc

t

) 1−γγ

(β (1 + rt))− 1

γ(

Pct+1Ct+1 − Pc

t+1Nt+1)

− ϑ

1 + ϑ(Ψt − 1)−1 Pt+1Bt+1 + B∗t+1

1 + r∗t

This demonstrates that the aggregate Euler equation depends on total asset holdings.In the case of no population growth, ϑ = 0, the aggregate and individual householdEuler equations coincide.

F.3. Firms

Firms maximize dividends over an infinite horizon and distribute them lump sum tohouseholds.

The non-tradable firm maximizes the present discounted value of dividend payments(expressed in units of tradable output):

max∞

∑i=0

λt,t+i

Pt+iaNt+iKαNNt+i(XNt+inNt+i)

1−αN − Pt+iφN2

(KNt+i+1

KNt+i− gN

)2KNt+i

−Pt+i INt+i − wNt+inNt+i

subject to: KNt+i+1 = (1− δN)KNt+i + INt+i

7The no-arbitrage condition for asset returns implies that Pt1+rt

= Pt+11+r∗t

, which allows us to collectterms in both types of bonds.

34

where λt,t+1 is a (compound) discount factor (discussed below) and the term in bracketsis the per-period dividend.

Substituting for INt+1 implies that the firm maximizes:

max∞

∑i=0

λt,t+i

Pt+iaNt+iKαNNt+i(XNt+inNt+i)

1−αN − Pt+iφN2

(KNt+i+1

KNt+i− gN

)2KNt+i

−Pt+iKNt+i+1 + (1− δN) Pt+iKNt+i − wNt+inNt+i

The first order conditions are:

0 = WNt − Pt (1− αN) aNtKαNNt (XNtnNt)

−αN XNt (117)

0 = − λt,tPt

(1 + φN

(KNt+1

KNt− gN

))

+ λt,t+1Pt+1

αNaNt+1KαN−1Nt+1(XNt+1nNt+1)

1−αN + (1− δN)

−φN2

(KNt+2KNt+1

− gN

)2+ φN

(KNt+2KNt+1

− gN

)KNt+2KNt+1

(118)

The first order condition for labor is identical to the baseline model. The secondequation looks slightly different because of the different ownership structure (ie firmsare now assumed to own the capital stock).

The first order condition for capital can be written as:

αNaNt+1KαN−1Nt+1(XNt+1nNt+1)

1−αN + 1− δN =λt,tPt

λt,t+1Pt+1

(1 + φN

(KNt+1

KNt− gN

))+

φN

2

(KNt+2

KNt+1− gN

)2

− φN

(KNt+2

KNt+1− gN

)KNt+2

KNt+1

The left hand side is the return on capital, net of depreciation, measured in units ofnon-tradable output. The right hand side captures the inter-temporal cost of substitutingnon-tradable output across time and adjustment costs. The right hand side dependsonly on the ratio of discount factors between time periods. This is the same for allhouseholds and is given by the inverse of the rate of return on intermediary bonds.This means that the first order condition for non-tradable capital is:

αNaNt+1KαN−1Nt+1(XNt+1nNt+1)

1−αN + 1− δN =Pt (1 + r∗t )

Pt+1

(1 + φN

(KNt+1

KNt− gN

))+

φN

2

(KNt+2

KNt+1− gN

)2

− φN

(KNt+2

KNt+1− gN

)KNt+2

KNt+1

The tradable firm solves an isomorphic problem. So the labor demand equation is

35

the same as the baseline model. The first order condition for capital is:

αTaTt+1KαT−1Tt+1 (XTt+1nTt+1)

1−αT + 1− δT = (1 + r∗t )(

1 + φT

(KTt+1

KTt− gT

))+

φT

2

(KTt+2

KTt+1− gT

)2

− φT

(KTt+2

KTt+1− gT

)KTt+2

KTt+1

F.4. Market clearing

The market clearing conditions are the same as in previous derivations. Substitutinginto the per capita budget constraint gives:

B∗t1 + ϑ

= TBt +B∗t+1

1 + r∗t

where we impose that non-tradable bonds are in zero net supply.

F.5. Stationary units

The redefinition of sector-specific growth rates to include deterministic populationgrowth rates means that the transformations into stationary units in the baseline modelcontinue to hold in almost all cases. The main exception is the consumption Eulerequation and household budget constraints, which we have written in per capita terms,but without adjusting for non-stationary tradable productivity. Adjusting the Eulerequation for productivity gives:

XTt−1 (Pct ct − Pc

t nt) =

(Pc

t+1Pc

t

) 1−γγ

(β (1 + r∗t ))− 1

γ XTt(

Pct+1ct+1 − Pc

t+1nt+1)

− ϑXTt

1 + ϑ(Ψt − 1)−1 Pt+1bt+1 + b∗t+1

1 + r∗t

where lower case letters denote stationary units (as in the baseline model), which inthis variant means adjusted for both productivity and population.

In particular,

nt = X−1Tt Nt = ω−1

(θT

(wTt

θT pct

) ωω−1

+ θN

(wNt

θN pct

) ωω−1)

36

These considerations imply that the Euler equation in stationary units is given by:8

pct (ct − nt) =

(pc

t+1pc

t

) 1−γγ

(β (1 + r∗t ))− 1

γgTt

1 + ϑpc

t+1 (ct+1 − nt+1)

− ϑgTt

1 + ϑ(Ψt − 1)−1 Pt+1bt+1 + b∗t+1

1 + r∗t

Similar arguments apply to the flow budget constraint:

XTt−1b∗t1 + ϑ

= XTt−1tbt +XTtb∗t+11 + r∗t

so thatb∗t

1 + ϑ= tbt +

gTtb∗t+1

(1 + ϑ) (1 + r∗t )

F.6. Steady state

In steady state (also imposing market clearing) the Euler equation implies:

pc (c− n)[

1− (β (1 + r∗))−1γ

gT

1 + ϑ

]= − ϑgT

1 + ϑ

b∗

(Ψ− 1) (1 + r∗)

Under the assumption that c > n (so that marginal utility is positive in steady state),this expression reveals that the sign of the economy’s foreign debt position depends onthe relative patience of households. Specifically, it depends on the size of the discountfactor β in relation to the (productivity) growth adjusted real interest rate. In the

particular case in which β = β0 ≡(

1+ϑgT

)−γ1

1+r∗ , the economy will hold no foreigndebt or assets (and the trade balance will be zero) in steady state. If the economy isrelatively less patient (so β < β0) then the economy will be a net debtor, with b∗ > 0, insteady state. Conversely, if households are more patient, then the economy will holdforeign bonds in steady state (b∗ < 0 means that debt is negative and the economy holdspositive assets).

These observations mean that we can calibrate β to deliver a desired steady-stateforeign debt position (conditional on the values of the other model parameters). Firstnote that the steady-state (inverse) marginal propensity to consume is given by:

Ψ =1

1− β1γ (1 + r∗)

1γ−1

8Recall that Pct is cointegrated with tradable productivity, so pc

t = Pct .

37

Plugging this into the steady-state Euler equation and rearranging gives:

− ϑgT

1 + ϑ

b∗

pc (c− n) (1 + r∗)=

(1

1− β1γ (1 + r∗)

1γ−1− 1

)[1− (β (1 + r∗))−

1γ

gT

1 + ϑ

]

=β

1γ (1 + r∗)

1γ−1

1− β1γ (1 + r∗)

1γ−1

[1− (β (1 + r∗))−

1γ

gT

1 + ϑ

]=

B1 + r∗ −B

[1−B−1 gT

1 + ϑ

]where the final line makes use of the following definition:

B ≡ β1γ (1 + r∗)

1γ

Rearranging the final equation allows us to solve for B:

B =gT

(1− ϑb∗

pc(c−n)

)(1 + ϑ)

(1− ϑgTb∗

(1+ϑ)(1+r∗)pc(c−n)

)as a function of other parameters, steady-state allocations and the desired steady-stateforeign debt position, b∗.

Finally, we can solve for β = Bγ (1 + r∗)−1.

38

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