lecture 3.1 an introduction to general equilibrium policy...
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Lecture 3.1
An Introduction to General Equilibrium Policy Modeling
David Roland-Holst, Sam Heft-Neal, and Anaspree Chaiwan UC Berkeley and Chiang Mai University
Training Workshop Economywide Assessment of High Impact Animal Disease
14-18 January 2013 InterContinental Hotel, Phnom Penh, Cambodia
Roland-Holst 2 16 January 2013
SAMs
ACT COM VA HH GOV INV ROW TOTALS
ACT Gross Output
Receipts
COM Int. Use Household Consumption
Government Expenditure
Gross Investment
Exports Demand
VA GDP at Factor Cost
Factor Income
HH GDP at Factor Cost
ROW Trans. to HH
Household Income
GOV Net Indirect Taxes
Household Taxes
Government Borrowing
Government Revenue
INV Household Saving
Government Saving
Current account balance
Savings
ROW Imports ROW
TOTALS Payments Supply Factor Allocation
Household Expenditure
Government Expenditure
Investment ROW
Roland-Holst 3 16 January 2013
SAM to CGE
• The SAM provides a snapshot of the economy at equilibrium (columns equal rows), but it is a static equilibrium with fixed prices, no substitution, and typically average behavior.
• On the contrary, in many cases what we are interested in examining is how economic actors respond to changes in relative prices.
• CGE allows for flexible prices, substitution, and marginal behavior, at the same time meeting the accounting constraints enforced by SAM structure.
Roland-Holst 4 16 January 2013
SAM to CGE
• To put this another way, CGE models overcome the shortcomings of a SAM by specifying a functional form for every cell in the SAM.
• Each cell in the SAM can be represented by a price and quantity, so the model must be able to determine both prices and quantities.
• Let’s start with a VERY simple CGE model, then work our way to something a bit more complicated.
Roland-Holst 5 16 January 2013
Very Basic CGE
• To see how we go from a SAM to a CGE model, let’s begin with a 2-sector, 2-factor really really simple SAM (RRSS):
Producers Factors Institutions ROWSUM
AG OTH L K HH
AG 150 150
OTH 500 500
L 100 200 150
K 50 300 150
HH 300 350 650
COLSUM 150 500 300 350 650
Roland-Holst 6 16 January 2013
Our Simple Economy
• Note that the government is not an economic actor, the economy is closed, factor costs are the only input to production, and households spend all their income.
• In this case, we have three economic actors – Producers (2; AG and OTH) – Factors (2; L and K) – Households (1)
• Let’s further assume that labor and capital are fully mobile across sectors (1 wage and rental rate).
Roland-Holst 7 16 January 2013
Side Note
• (Let’s maintain our convention of having i be rows and j be columns; this means that i will reflect the income side of the economy and j will reflect the expenditure side of the economy).
Roland-Holst 8 16 January 2013
Supply
• On the supply side, at a minimum we need to specify how producers behave (e.g., minimize costs), how they choose inputs (factor demands), and how their decisions determine aggregate supply. Using a Cobb-Douglas form, we can describe production within our economy as:
Total Supply
Labor Demand
Capital Demand
Roland-Holst 9 16 January 2013
Demand
• On the demand side, we need to specify the level of household income, and how households decide to spend that income. Household income is the sum of factor incomes:
(Remember that we are decomposing SAM transactions into prices and, in this case, volumes.)
Roland-Holst 10 16 January 2013
Demand
• Household consumption is modeled with a constant elasticity of substitution (CES) utility function:
Maximizing U s.t. a budget constraint gives us the two reduced form consumption functions:
Roland-Holst 11 16 January 2013
Equilibrium
• Lastly, we need to define some sort of equilibrium conditions for the economy, which in our case we can represent by supply = demand in product and factor markets.
Commodity Market
Labor Market
Capital Market
Roland-Holst 12 16 January 2013
Endogenous Variables
• In 13 equations we have built a simple general equilibrium model.
• Our 13 endogenous variables include: – Pi – prices for AG and OTH goods – r – rate of return on capital – w – wage rate – LDj – labor demand for AG and OTH producers – KDj – capital demand for AG and OTH producers – XSj – aggregate supply – Ci – household consumption of AG and OTH goods – Y – household income
Roland-Holst 13 16 January 2013
Exogenous Variables
• We have left 2 variables exogenous: – LS – Aggregate labor supply – KS – Aggregate capital supply
Roland-Holst 14 16 January 2013
Initializing Prices
• Prices are going to be endogenous in our simple CGE model, but we are going to represent prices in a price index rather than as absolute values. Prices can be initialized to any level, but 1 is generally the most obvious choice. – PAG = 1 POTH = 1
• We select PAG as the numeraire, which fixes our economy-wide relative price as – P = POTH / PAG
Roland-Holst 15 16 January 2013
Initializing Prices
• We represent factor prices in the same way (as an index). In contrast to goods, however, we might want to initialize wages and rental rates at different levels to represent a factor price ratio that differs from unity – w = 0.8 – r = 1 (i.e., capital is more expensive in relative terms than labor)
Roland-Holst 16 16 January 2013
Initializing Endogenous Variables
• We can assign values to endogenous variables based our SAM: – LDAG0 = 100 LDOTH0 = 200 – KDAG0 = 50 KDOTH0 = 300 – XSAG0 = 150 XSOTH0 = 500 – CAG0 = 150 COTH0 = 150 – Y0 = 650
Roland-Holst 17 16 January 2013
SAM Check
Producers Factors Institutions ROWSUM
AG OTH L K HH
AG 150 150
OTH 500 500
L 100 200 150
K 50 300 150
HH 300 350 650
COLSUM 150 500 300 350 650
Roland-Holst 18 16 January 2013
Initializing Endogenous Variables
• LD, KD, XS, and C are volumes, so we need to convert them to volumes by dividing by the appropriate initialized price – LDAG0/w0 = 100/0.8 =125 LDOTH0/w0 = 200/0.8 = 250 – KDAG0/r0 = 50/1 = 50 KDOTH0/r0 = 300/1 =
300 – XSAG0/pAG0 = 150/1 = 150 XSOTH0/pOTH0 = 500/1
= 500 – CAG0/pAG0 = 150/1 = 150 COTH0/pOTH0 = 500/1
= 500
• We can also initialize LS and KS volumes – LS0 = LDAG0+ LDOTH0
– KS0 = KDAG0+ KDOTH0
Roland-Holst 19 16 January 2013
Model Calibration
• We can use SAM data to determine the baseline values of some of our parameters; in this case: – Cobb-Douglas scaling factors (Aj) – Cobb-Douglas share parameters (αj) – CES utility function share parameters (δ)
Roland-Holst 20 16 January 2013
Model Calibration
• From our aggregate output equation
• We can calculate the Cobb-Douglas scaling factors as
Roland-Holst 21 16 January 2013
Model Calibration
• Similarly, from labor demand
we can calculate the Cobb-Douglas share parameters as
Roland-Holst 22 16 January 2013
Model Calibration
• The CES share parameters are derived from
with a less than tidy result of
Roland-Holst 23 16 January 2013
Model Calibration
• Alternatively, the CES utility function’s substitution elasticity (σ) cannot be determined with SAM data.
• We can either specify σ heuristically (e.g., a 0 if we determine that the goods are perfect complements, or a high value if they are perfect substitutes) or through econometrics.
• In this case, let’s arbitrarily assign σ with a value of 0.3.
Roland-Holst 24 16 January 2013
Model Simulation
• Let’s walk through what happens when we perturb one of the exogenous variables in the model. Say we have an exogenous increase in labor supply (LS). From
we know this exogenous increase in LS will be accompanied by an increase in aggregate LD.
Roland-Holst 25 16 January 2013
Model Simulation
• But it isn’t clear how this change will affect our other variables:
Roland-Holst 26 16 January 2013
To a New Equilibrium
• We need a way to move from our initial equilibrium, in which all of our model equations held (i.e., our markets cleared), to a new equilibrium, in which all of our equations hold again. This shift from an old equilibrium to a new equilibrium is what is usually meant by “adjustment.”
• To find our new equilibrium solution, our endogenous variables will have to adjust so that both our equations hold, and our exogenous shock is accounted for.
Roland-Holst 27 16 January 2013
Model Solutions and Consistency
• CGE models require numerical solutions, which means that you will need to use some sort of solver package to generate a solution.
• To ensure that the model is consistent and you have not made errors in coding, in general your first step after building a CGE model is to make sure that you can reproduce the base solution (i.e., with no exogenous shock).
Roland-Holst 28 16 January 2013
A Quick Thought on Model Building
• Before we get into more complex models, a bit of advice. It is always useful to start any research project with a quick theoretical model that maps relationships among the variables that you wish to examine.
• By doing this kind of exercise, you can get a good sense of where you can make simplifications, where you should be more detailed, and how much you can leave out of your model.
Roland-Holst 29 16 January 2013
Toward more Complex Models
• Our next model will be significantly more complex, but still simple as far as CGE models go.
• MINI_CGE will address several of the oversimplifications of our previous model: – Producers typically have non-factor intermediate
inputs and non-uniform substitution elasticities – Households are more complex than CES utility
describes – Most economies have an active government and
capital markets – Most economies have ROW interactions
Roland-Holst 30 16 January 2013
1. Producer Behavior
• Producers choose inputs to minimize costs; with two inputs we can represent this mathematically as: Min(wL+rK) s.t. V=F(K,L) where w and r are the wage and rental rates, L and K are labor and capital, and V is the level of output. Producers choose K and L; w, r, and V are typically determined by market equilibrium conditions.
Roland-Holst 31 16 January 2013
Producer Behavior
• The Lagrangean for the producer’s optimization problem is L = wL + rK + P[V – F(K,L)]
• Setting the partial derivatives with respect to K, L and P equal to zero, we have the following three first order conditions:
∂∂
∂∂
LK
r P FK
= ⇒ =0
LFPw
L ∂∂
∂∂
=⇒= 0L ( )LKFVP
,0 =⇒=∂∂L
Roland-Holst 32 16 January 2013
CES Production
• Let’s take this one step further by assigning a functional form to F: a CES (constant elasticity of substitution) function, the most ubiquitous functional form used in GE models.
• The primal form of the CES function is
where the coefficients al and ak are called the labour and capital share parameters, respectively, and ρ is the CES exponent (which will be related to the CES substitution elasticity).
( ) [ ]V F K L a L a Kl k= = +,/ρ ρ ρ1
Roland-Holst 33 16 January 2013
CES First Order Conditions
• Differentiating the primal form CES yields
• Substituting back into our original problem, this implies
[ ]∂
∂ ρρρ ρ ρ ρ
ρFL
a L a K a L a LVl k l l= + =⎛⎝⎜
⎞⎠⎟
− −−1 1 1 11
/
∂∂
ρFK
a KVk= ⎛⎝⎜
⎞⎠⎟
−1
1−
⎟⎠
⎞⎜⎝
⎛==ρ
∂∂
VLaP
LFPw l
1−
⎟⎠
⎞⎜⎝
⎛==ρ
∂∂
VKaP
KFPr k
Roland-Holst 34 16 January 2013
CES Factor Demand
• Three simplifications:
then give us the following derived factor demands
σ
ρρ
σσ
=−
⇔ =−1
11
( )α ρ σl l la a= =−1 1/
( )α ρ σk k ka a= =−1 1/
L Pw
V
K Pr
V
l
k
= ⎛⎝⎜
⎞⎠⎟
= ⎛⎝⎜
⎞⎠⎟
α
α
σ
σ
(i.e., the relationship between the CES exponent and CES substitution elasticity.)
Roland-Holst 35 16 January 2013
CES Unit Cost and Pricing
• Using the total cost function
And substituting reduced for expressions for L and K Gives us the unit price-cost equivalence from duality
PV wL rK= +
[ ]PV w P
wV r P
rV VP w rl k l k=
⎛⎝⎜
⎞⎠⎟ +
⎛⎝⎜
⎞⎠⎟ = +− −α α α α
σ σσ σ σ1 1
[ ] ( )P w rl k= +− − −
α ασ σ σ1 1 1 1/
Roland-Holst 36 16 January 2013
Generalized CES
• The CES optimization problem can be generalized to i = 2,…,n inputs as
where Xi are the inputs to production and Pi are their prices. A is a uniform shift parameter that can be applied to all inputs, and λ is an input-specific shift parameter. So, for instance, neutral productivity growth could be applied by shifting the A parameter. Hicks neutral productivity growth could be applied by shifting the λ parameter.
min P Xi ii∑
( )V A a Xi i i
i
=⎡
⎣⎢
⎤
⎦⎥∑ λ
ρρ1/
s.t.
Roland-Holst 37 16 January 2013
Generalized CES
• The generalized CES has first order conditions
where the shift and share parameters have been merged so that
111
1
−−−
−
=⎥⎦
⎤⎢⎣
⎡= ∑ ρρρ
ρρ
ρiiii
iiii XcVPXcXcPP
ρρ
1
⎥⎦
⎤⎢⎣
⎡= ∑
iii XcV
( )ρλiii Aac =
Roland-Holst 38 16 January 2013
Generalized CES
• We can rewrite as
substituting back into the second FOC gives
and with a bit of manipulation, we get unit costs
VPPcXi
ii
ρ−
⎥⎦
⎤⎢⎣
⎡=
11
11 −− ρρii XcVP
ρ
ρρρ /1
1
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎦
⎤⎢⎣
⎡= ∑
−
i i
ii VPPccV
( ) ( ) ( )σσ
σ
σσ
σσ
σσ
λλ
−−−−−
−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛=⎥
⎦
⎤⎢⎣
⎡= ∑∑∑
1/111/111/11 1
i i
ii
i i
ii
iii
PaAA
PaPcP
Roland-Holst 39 16 January 2013
Generalized CES
• So our unit cost of production is determined by
• Again using the relationship between the CES exponent and CES substitution elasticities, we get reduced form input demands
( )X A PP
Vi i ii
=⎛
⎝⎜
⎞
⎠⎟
−α λ
σσ
1
( )
PA
Pi
i
ii
=⎛
⎝⎜
⎞
⎠⎟
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
− −
∑11 1 1
αλ
σ σ/
Roland-Holst 40 16 January 2013
Generalized CES
• In most applications, A is typically set to 1, and the exponent on the share parameter is merged into the primal share parameter to yield
and
VPPXi
iii
σ
σλα ⎟⎟⎠
⎞⎜⎜⎝
⎛= −1
( )σσ
λα
−−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛= ∑
1/11
i i
iiPP
Roland-Holst 41 16 January 2013
Generalized CES
• The first of these two equations
defines the CES dual price, which is an average of the input prices. The CES dual price function is the aggregator, with the share and productivity parameters providing the appropriate weights.
( )σσ
λα
−−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛= ∑
1/11
i i
iiPP
Roland-Holst 42 16 January 2013
Generalized CES
• The second equation
represents optimal demand for each input. Individual demand equals a constant share of the level of output, V, adjusted by a term in the relative price of the input (compared to the aggregate cost of inputs). Hence, if an input’s price increases (relative to overall costs), then demand for that factor will decrease. The percentage decrease will depend on the elasticity of substitution.
VPPXi
iii
σ
σλα ⎟⎟⎠
⎞⎜⎜⎝
⎛= −1
Roland-Holst 43 16 January 2013
CES Substitution Elasticities
• By dividing input demands, we can calculate the ratio of demand for any two inputs (e.g., i and j) as:
• Taking the partial derivative of the above with respect to the ratio Pj/Pi and multiplying the resulting expression by (Pi/Pj)/(Xi/Xj) yields the elasticity of substitution, or…
σ
σ
σ
λαλα
⎟⎟⎠
⎞⎜⎜⎝
⎛= −
−
i
j
jj
ii
j
i
PP
XX
1
1
Roland-Holst 44 16 January 2013
CES Substitution Elasticities
• Which is the percent change in the ratio of two inputs with respect to a percentage change in their relative prices.
σ−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
j
i
j
i
j
i
j
i
XXPP
PP
XX
Roland-Holst 45 16 January 2013
Special Case: Leontief
• In the case where σ = 0, there is no input substitution and we have the Leontief fixed coefficient, linear production technology that we used in I/O analysis
• Output price is simply a weighted average of input prices
VA
Xi
ii λ
α=
V A
Xi
i
i=
⎛
⎝⎜
⎞
⎠⎟min λ
α
ii i
i PA
P ∑=λα1
Roland-Holst 46 16 January 2013
Stratified Input Substitution
• Up until now we have been assuming a single elasticity of substitution among all input types, which is obviously not realistic.
• To get around this problem, we can group inputs into sub-groups that each have different substitution elasticity properties.
Roland-Holst 47 16 January 2013
Nested CES
• A common solution is to use nested CES production, which uses stratified input substitution
σ = 0
σm σk
σv
σp XP
ND VA
L KF
K F
XAp
XM XDd
Roland-Holst 48 16 January 2013
CES Nest
• The top nest determines demand for intermediate demand and value added
with unit cost of production
σ is the ND-VA substitution, which is set to 0 in THAIMINI.
ii
indii XPPNDPXND
piσ
α ⎟⎟⎠
⎞⎜⎜⎝
⎛= i
i
ivaii XPPVAPXVA
piσ
α ⎟⎟⎠
⎞⎜⎜⎝
⎛=
[ ] ( )pipi
pi
ivaii
ndii PVAPNDPX
σσσ αα−
−− +=1/111
Roland-Holst 49 16 January 2013
CES Nest
• The value added sub-nest includes three factors
( ) iil
ili
di VA
WPVAL
viv
i
σσ
λα ⎟⎠
⎞⎜⎝
⎛=−1
ii
ikfii VAPKFPXKF
viσ
α ⎟⎟⎠
⎞⎜⎜⎝
⎛=
( )vivi
vi
ikfil
i
lii PKFWPVA
σ
σ
σ
αλ
α
−
−
−
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡+⎟⎟
⎠
⎞⎜⎜⎝
⎛=
1/1
11
Roland-Holst 50 16 January 2013
2. Household Behavior
• THAIMINI uses a linear expenditure system (LES), also known as a Stone-Geary demand system, which is widely used in CGE models.
• LES is also a fairly general functional form that is based on a number of tenuous assumptions (e.g., all goods are gross substitutes, no good is inferior).
• Nevertheless, LES has 3 advantages: – Relatively small number of parameters to calibrate
(2n) – Flexible income elasticities – Ease of use
Roland-Holst 51 16 January 2013
LES Utility
• The LES utility function is given by
or equivalently
with a condition that
( )∑ −=i
iii xxU θµ ln)(
( )∏=
−=n
iii
ixU1
µθ Yxpn
iii =∑
=1s.t.
Yxpn
iii =∑
=1s.t.
11
=∑=
n
iiµ
Roland-Holst 52 16 January 2013
LES Utility
• The Lagrangean for the LES utility function is
• The LES Lagrangean has FOCs
rearranging dL/dx gives the LES demand function
µθ
λi
i iixp
−− = 0 Y p xi i
i
− =∑ 0
xpi ii
i= +θ
µλ
( )L = − + −
⎛
⎝⎜
⎞
⎠⎟∑∑µ θ λi i i i i
ii
x Y p xln
Roland-Holst 53 16 January 2013
LES Utility
• To express the LES as a reduced form in prices, note that
(remember our condition on µ)
and the Lagrange multiplier can be expressed as
Y p
pp p
ppi i
i
iii i
ii
i
iii i
i
= +⎛
⎝⎜
⎞
⎠⎟ = +
⎛
⎝⎜
⎞
⎠⎟ = +∑ ∑ ∑ ∑θ
µλ
θµλ λ
θ1
∑−=
jjjpY θ
λ1
Roland-Holst 54 16 January 2013
LES Elasticities
Income elasticities can be obtained directly from
and, by extension
In words, LES income elasticities are the ratios of marginal (µi) to average (si) expenditure shares.
Price elasticities follow from
∂∂
µxY pi i
i=
η
∂∂
µ µi
i
i
i
i i
i
i
xYYx p
Yx s
= = =
( )∂
∂µ µ
θµ
θxp p
Yp p
Yp
i
i
i
i
i
ii
i
i ii= − + − = − +
⎛
⎝⎜
⎞
⎠⎟2
**
( ) ( ) ( )
ε∂∂
µθ
µθ
µ θθ
θ µi
i
i
i
i
i
i
i
ii
i
i
ii i
i i
i ii i
i i
i
xppx
px p p
px
x xx
x= = − + −
⎛
⎝⎜
⎞
⎠⎟ = − − − =
−−
1 1 11
Roland-Holst 55 16 January 2013
3. Other Final Demand – Government
• We assume that the volume of government expenditure is fixed, i.e.,
• Government is assumed to have a CES expenditure function
• Where government expenditure price is given by
0XGXG =
( ) XGPA
PGXAgg
iitgi
gii
σ
τα ⎟⎟
⎠
⎞⎜⎜⎝
⎛
+=
1
( )[ ]( )g
g
ii
itgi
gi PAPG
σσ
τα−
−
⎥⎦
⎤⎢⎣
⎡+= ∑
1/111
Roland-Holst 56 16 January 2013
3. Other Final Demand – Investment
• Investment is savings determined. The investment-savings closure rule is given by
where XI is the aggregate volume of investment, PI is an investment price deflator, Sh and Sg represent domestic savings, ER.Sf is foreign savings adjusted by the exchange rate (which we’ll discuss later), and DeprY is a depreciation allowance term.
DeprYSERSSXIPI fgh +++= ..
Roland-Holst 57 16 January 2013
Investment
• As with government expenditure, a CES expenditure function is assumed to allocate aggregate investment into sectoral demand XAi
( ) XIPA
PIXAii
iitii
iii
σ
τα ⎟⎟
⎠
⎞⎜⎜⎝
⎛
+=
1
( )[ ]( )i
i
ii
itii
ii PAPI
σσ
τα−
−
⎥⎦
⎤⎢⎣
⎡+= ∑
1/111
Roland-Holst 58 16 January 2013
4. Trade
• Trade is the final key component of demand. • If there are no differences between imports and
domestic products, imports are a residual between domestic production and domestic demand.
• In reality, there are few commodities (e.g., oil) that are truly homogeneous, and most models assume some degree of differentiation between imports and domestically produced goods.
Roland-Holst 59 16 January 2013
Trade Stratification
• Demand is thought to combine domestic and imported goods in each product category with a nested CES aggregation
• Output is modeled symmetrically with a dual nested CET structure
Imports/Exports Domestic Goods
Aggregate Demand/Supply
CES/CET
Roland-Holst 60 16 January 2013
Trade Analytically – Imports
Denoting domestic demand by XD and imports by XM, total demand is modeled with the CES preference function
Min(PD•XD+PM•XM) subject to
where PD and PM denote prices for domestic and imported goods and XA is aggregate demand. Passing over derivations from the production analytics, we have the following reduced forms
where and denotes the price index of XA.
[ ]XA a XD a XMd m= +ρ ρ ρ1/
XAPDPAXD d
σ
α ⎟⎠
⎞⎜⎝
⎛= XM PA
PMXAm=
⎛⎝⎜
⎞⎠⎟ασ
[ ] ( )PA XD XMd m= +− − −
α ασ σ σ1 1 1 1/
α σd da=
α σm ma=
ρ
σσ
=−1
Roland-Holst 61 16 January 2013
Trade Analytically – Exports
Denoting domestic supply by XD and export supply by XE, total supply is modeled with the CET production frontier
Max(PD•XD+PE•XE) subject to
where PD and PM denote prices for domestic and imported goods and XA is aggregate demand. Passing over derivations from the production analytics, we have the following reduced forms
where and denotes the price index of XP.
XD PD
PPXPd= ⎛
⎝⎜
⎞⎠⎟γν
[ ] ωωω /1XEgXDgXP ed +=
XE PE
PPXPe= ⎛
⎝⎜
⎞⎠⎟γν
[ ] ( )( )PP XD XE PD XD PE XE XPd e= + = ++ + +
γ γν ν ν1 1 1 1/. . /
γ νd dg=
−
γ νe eg=
−
ω
νν
=+1
Roland-Holst 62 16 January 2013
Trade Schematically
Domestic Goods/Services
Exports
PPF
slope=-PD/PE
CET
Domestic Goods/Services
Impor t s
Indifference Curve
slope=-PD/PM
CES