econ 3020 intermediate macroeconomics - xiaohui...

ECON 3020 Intermediate MacroeconomicsChapter 5 A Closed-Economy One-Period

Macroeconomic Model

Instructor: Xiaohui Huang

Department of EconomicsUniversity of Virginia

c© Copyright 2014 Xiaohui Huang. All Rights Reserved. Macroeconomics, Williamson 5/E, Chapter 5 1 / 51

Main Topics

Government sectorCompetitive equilibrium

How do consumers and firms interact in markets?How is equilibrium obtained in their actions?How can unconstrained markets produce socially efficientoutcomes?

Analyze macroeconomic issues:Effects of an increase in government spending.Effects of an increase in total factor productivity.

An example of inefficient equilibrium: distortionarygovernment tax.


Model Structure

Closed economy: markets are restricted to a singlecountry; no trade.The representative consumer

Buys goods from the firm.Works for the firm and earns labor income.Trade-off between consuming and working.

The representative firmHires labor to produce consumption goods.Sells consumption goods to the consumer.Trade-off between revenue and cost.

GovernmentIn equilibrium, both markets (goods and labor) need to becleared under the given prices.


Government Sector

Government spending is financed by taxes.Government budget constraint (in real terms):

G︸︷︷︸government spending

= T︸︷︷︸tax revenue

Fiscal policy in our one-period model: government choicesover spending and taxes.Assume: G is fixed and determined outside the model.G is exogenous.


Exogenous Variables vs. Endogenous Variables

Exogenous variables: determined outside the model;taken as given.Endogenous variables: determined within the model.Use the model: how changes in exogenous variableschange endogenous variables.

In our one-period model:Exogenous: G, z,K .Endogenous: C,Ns,Nd ,T ,Y , π,w .


Competitive Equilibrium (CE)

Competitive: consumers and firms are price-takers.Equilibrium: interaction among all sectors reaches astable and consistent state.

Consumers and firms have no incentive to change theirbehavior under given prices. (Optimal decisions)Price does not change. (All markets clear)


Definition: Competitive Equilibrium

A competitive equilibrium is a set of endogenous variables(C,Ns,Nd ,T and Y ) and an endogenous real wage rate (w),such that, given exogenous variables (G, z and K ), thefollowing conditions are satisfied:

1 Given w , T and π, the bundle (C,Ns) maximizes theconsumer’s utility subject to the budget constraint.

2 Given w , z and K , the labor demand Nd maximizes thefirm’s profits.

3 Government has a balanced budget: G = T .4 Markets clear:

Labor market: Nd = Ns = N.Goods market: C + G = Y . (I = 0, NX = 0 here)


Walras’ Law

In our model, if labor market clears, so will the goodsmarket, and vice versa! Show this.Generally, if there are N markets, and N − 1 markets clear,then the Nth market clears automatically. This is called theWalras’ Law.


Study the Competitive Equilibrium

Throughout this book, we work with a model in graphicalforms (i.e. supply curve and demand curve derived fromoptimal behavior of consumers and firms).Examine the consumer’s decision and the firm’s decision inthe same diagram.Now, for the one-period model, put consumer’s problemand firm’s probelm in the same (l,C) plane.


Production Possibilities Frontier (PPF)

Figure 5.2(a) Production Function

Our production functionY = zF (K ,Nd ).Maximum level of output isY ∗ = zF (K ,h).F (K ,0) = 0.The slope is MPN .



Figure 5.2(b) Output as a Function of Leisure

Change the horizontal axisfrom Nd to l, using l = h − Nd .Production function:Y = zF (K ,h − l).When l = 0, Nd = h,maximum output Y ∗ achieved.When l = h, Nd = 0, nothingis produced.The slope is −MPN .



Figure 5.2(c) The Production Possibilities Frontier

Change the vertical axis fromY to C, using C = Y −G.When l = 0, C∗ = Y ∗ −G.When l = h, C = −G.The shaded area is theproduction possibilities set.The arc DA is the productionpossibilities frontier.Points on arc BA are notfeasible.



In equilibrium, C = Y −G, we can write PPF as

C = zF (K ,h − l)−G

The PPF describes what the economy can produce interms of production of consumption and leisure.Marginal rate of transformation (MRT)

MRTl,C is called marginal rate of transformation of leisureinto consumption.MRTl,C is the rate at which leisure can be converted intoconsumption goods.MRTl,C = MPN = − slope of the PPF.



Firm’s PPF: curve FH.If w is an equilibrium wagerate and AD with slope −w istangent to the PPF, then ADBis the budget constraint ofconsumers. Why?Optimizing conditions:

Consumer: MRSl,C = w .Firm: MPN = w .

Same tangent point J. Why?



Maximum profit of the firm:π∗ = zF (K ,h− l∗)−w∗(h− l∗)

π∗: distance DH. Why?

Maximum after-tax dividendincome paid to the consumer:π∗ − T

π∗ − T : distance DB. Why?



In equilibrium (at point J), we have

MRSl,C = MRTl,C = MPN = w

Interpretation:The consumer and the firm face the same market realwage in equilibrium, so the rate at which the consumer isjust willing to trade leisure for consumption (MRSl,C) is thesame as the rate at which leisure can be converted intoconsumption goods (MRTl,C).Graphically, the PPF, the budget line, and the indifferencecurve are tangent to each other at the same point!


Competitive Equilibrium and Efficiency

Why we study this connection?How free markets can produce socially optimal outcomes.If CE is equivalent to a socially optimal/efficient outcome,then we can study the social optimum instead of CE, whichis much easier!

Criterion of efficiency — Pareto optimality.


Pareto Optimality

Pareto OptimalA CE is Pareto optimal if there is no way to rearrangeproduction or to reallocate goods so that someone is madebetter off without making someone else worse off.Question: Whether the CE is Pareto optimal?

To find efficient allocations, we use a benevolent socialplanner.Then compare the CE in our model with the social planneroutcome.

Attention: social planner 6= government.


Pareto Optimum: Social Planner’s Choice

The social planner chooses aconsumption bundle that is

On or within the PPF (feasiblefor the firm to produce)On the highest possibleindifference curve (best for theconsumer)

Pareto optimum: point B, whereMRSl,C = MRTl,C = MPN .To the left of B:MRSl,C > MRTl,C . Could makethe consumer better off by ↑ l.To the right of B:MRSl,C > MRTl,C . Could makethe consumer better off by ↓ l.


Competitive Equilibrium vs. Pareto Optimum

Social planner’s choice is the same as the outcome of freemarkets: under certain conditions, CE⇔ PO.Two fundamental theorem of welfare economics.


Fundamental Theorems of Welfare Economics

First fundamental theorem of welfare economics:Under certain conditions, a competitive equilibrium isPareto optimal. (CE⇒ PO)Second fundamental theorem of welfare economics:Under certain conditions, a Pareto optimum is acompetitive equilibrium. (PO⇒ CE)Attention: efficiency 6= equity!


Fundamental Theorems of Welfare Economics

First welfare theorem holds:The CE allocation (C∗, l∗) isPareto optimal. Why?Second welfare theorem holds:The Pareto efficient allocation(C∗, l∗) can be decentralized as aCE. Why?


Sources of Social Inefficiencies

A CE may fail to be Pareto optimal due to:Externalities

The social planner would take externalities into account, butindividuals and firms don’t.Externalities can be negative or positive.Essentially roots in missing markets. (Market failure)

Distortionary taxesFor example, a propotional labor income tax t .Consumers and firms do not face the same price(consumers facing w(1− t); firms facing w).The competitive equilibrium is not Pareto optimal, why?In practice, all taxes cause distortions.

Market powerFirms with market power do not take price as given, as theyknow they can influence them.Typically leads to under-production.


Comments on Efficiency

Why should we analyze an economy that is efficient in thesense that a CE is Pareto optimal?Why don’t we model sources of inefficiencies here?With inefficiencies existing in the real world, is it alwaysbetter to have government regulations?


Use the Model: Working with Planner’s Problem

In general, the planner’s problemis easier to work with.

No worry about prices.Consumption and productionchoices are determined bypreferences and technology.

Pareto optimum: tangent point ofindifference curve and PPF.We can always find prices todecentralize the allocation as aCE.PO⇔ CE.From now on, we work with thesocial planner’s problem insteadof CE.


Two Experiments

How a change in an exogenous variable (G, z or K ) affectsthe key endogenous variables (C, Y , N and w).

The effects of a change in government purchases, G.The effects of a change in TFP, z.

Model mechanism:Changes in exogenous variables⇒ Shift or twist PPF in some ways⇒ Changes in the PO/CE⇒ Changes in endogenous variables


Experiment 1: G ↑

G ↑ ⇒ PPF parallelly shifts down.At each l, the slope of PPF is thesame as before.This is exactly like a negativeincome change to theconsumers, since G ↑⇔ T ↑.Pareto optimum: A→ B.Both C and l are normal goods,we should expect C ↓ and l ↓.Now we study the effects of G ↑on C, Y , l/N, and w .


Experiment 1: G ↑Effects of G ↑ on C, N and Y .

Effects on N and Y :Leisure is a normal good.l ↓ ⇒ N ↑ ⇒ Y ↑

Effects on C:C = Y −G so ∆C = ∆Y −∆G.∆Y > 0⇒ ∆C > −∆GC ↓ by less amount than G ↑.In graph, AE<AD.

Consumption is crowded outby government spending, butnot completely.


Experiment 1: G ↑Effects of G ↑ on w

How can firms hire more labor?It must be w ↓.N ↑ ⇒ MPN ↓ ⇒ w ↓.Individuals still want less leisureeven if it is cheaper, why?Exercise:Decompose the total effect intoan income effect and asubstitution effect for theconsumer.


Government Spending and Business Cycles

Is fluctuation in G a likely cause of business cycles?Can our model replicate those key business cycle facts inresponse to a change in G?

In our model, if G ↑:Output ↑

Employment ↑ (pro-cyclical)

Consumption ↓ (counter-cyclical)

Real wage rate ↓(counter-cyclical)

Key business cycleco-movements (Ch3):

Employment is pro-cyclical

Consumption is pro-cyclical

Real wage rate is pro-cyclical

Conclusion:Business cycles are not likely to be the results ofgovernment spending fluctuations.


Experiment 2: z ↑

Figure 5.8 Increase in Total Factor Productivity

z ↑ ⇒ production function shiftsup.z ↑ ⇒ MPN ↑ at each N.Maximum amount of output ishigher: z ↑ ⇒ zF (K ,h) ↑.Still starts from the origin point.


Experiment 2: z ↑

Figure 5.9 Competitive Equilibrium Effects of an Increase inTFP

z ↑ ⇒ the PPF shifts outwardfrom BA to DA.More consumption goods can beproduced at any level of leisure.z ↑ ⇒ MPN ↑, PPF is steeper.Equilibrium (efficient) allocation:F → H.


Experiment 2: z ↑

Effects of z ↑ on C, N, Y and w

Effects on C:C ↑ (normal good).

Effects on Y :Y = C + G, G fixed, so Y ↑.

Effects on w :w ↑ (see later).

Effects on N:z ↑⇒ MPN ↑⇒ w ↑⇒ N ↑.w ↑⇒ income ↑ ⇒ N ↓.Total effect on N is uncertain.In the figure, N is unchanged.


Experiment 2: z ↑

Figure 5.10 IE and SE of an Increase in TFP

Draw a PPF3: parallel to PPF2and tangent to I1 at D.Substitution effects:

A→ DC ↑, l ↓, N ↑

Income effects:D → BC ↑, l ↑, N ↓

Total effects:C ↑l or N uncertainWelfare ↑


Experiment 2: z ↑

We want to show: z ↑⇒ w ↑ regardless the change of l

From A to D (SE):MRSD

l,C > MRSAl,C

wD > wA

From D to B (IE):PPF2 and PPF3 are parallel –same MPN for each l.lB > lD (normal good)NB < ND ⇒ MPB

N > MPDN ⇒

wB > wD

From A to B, w ↑ no matter SEdominates or IE dominates.Intuition: z ↑ ⇒ MPN ↑ ⇒ demandfor labor ↑ ⇒ w ↑.


TFP and Long-Run Trends

There have been many important technological innovationssince World War II. Should they be responsible for themain economic changes over time?Can our model with an increase in TFP match the keyobservations in the long-run data?

In our model, if z ↑:Y ↑

C ↑

w ↑

N ambiguous

Since WWII, we have observed:

A rise in output

A rise in consumption

A rise in the real wage rate

Roughly constant hours worked

Conclusion:IF IE and SE roughly cancel out with each other, the modelis consistent with technological innovations having beenkey to changes in these economic variables in the long run.


TFP and Business Cycles

Could fluctuations in TFP be a cause of business cycles?Can our model replicate those key business cycle facts inresponse to an increase in z?

In our model, if z ↑:Y ↑

C ↑ (pro-cyclical)

w ↑ (pro-cyclical)

N ambiguous(N ↑ if SE dominates)

Key business cycleco-movements (Ch3):

Consumption is pro-cyclical

Real wage rate is pro-cyclical

Employment is pro-cyclical

Conclusion:Fluctuations in TFP may be the primary cause of businesscycles if the substitution effect dominates the income effectin the short run. (Real Business Cycle Theory)


Example of Inefficiency: Distorting Wage Income Tax

Consider a proportional tax on wage income.Now, CE 6= PO. So we can no longer use the socialplanner’s problem to study CE.We study the CE directly:

Firm’s behaviorConsumer’s behaviorMarket clearing

Incentive effects of wage income tax on labor supply.Relationship between government’s tax revenue and taxrates. ("Laffer curve")


A Simplified Model with Proportional Income Tax

Assume: labor is the only input for production.Production function for the representative firm:

Y = zNd

No lump-sum tax (T = 0).Income tax rate t .Budget constraint for the representative consumer:

C = w(1− t)(h − l) + π

w(1− t): the effective wage rate facing the consumer.


Firm Optimization

Firm’s profit maximization problem:

maxNd

π = Y − wNd = (z − w)Nd

Firm’s behavior:If w < z, firm wants to hire infinite labor.If w > z, firm would hire no labor.As long as w = z, any Nd is optimal.

Requirement for existence of equilibrium:Equilibrium exists if and only if w = z.If w 6= z, no equilibrium.

In equilibrium, w = z, π = 0, Nd is set at the value thatclears the labor market.


Labor Demand Curve

Firm’s demand for labor is infinitely elastic at wage w = z.Figure 5.16 The Labor Demand Curve


Consumer Optimization

Consumer’s budget constraint inequilibrium:

C = z(1− t)(h − l)

Note thatw = z in equilibrium.π = 0 and T = 0.

Budget line: DF .Optimal bundle: H.(MRSl,C = z(1− t))


PPF

PPF: C = Y −G = z(h − l)−G.Now the PPF is linear (line AB).


Competitive Equilibrium

Figure 5.17 Competitive Equilibrium with a Proportional IncomeTax

Competitive equilibrium: H.Check conditions for CE:

Firm optimizes. Why?Consumer optimizes. Why?Government’s balanced budget:G = zt(h − l), which implies thatAB intersects DF at H.Labor market clears. Why?Goods market clears. Why?


Inefficiency of Competitive Equilibrium

Pareto optimum: E .Competitive equilibrium: H.E is better than H.Reason for inefficiency:

Income tax distorts privatedecisions.Disincentive to work: labor supplyis lower in CE than in PO.Both C and Y are lower in CE.

Welfare loss: utility differencebetween I1 and I2.


Income Tax Revenue

Tax revenue collected by the government in equilibrium:

REV (t) = tw [h − l(t)] = tz[h − l(t)]

In equilibrium, w = z.l(t): consumer’s choice of leisure depends on t .z[h − l(t)] is called the tax base.


Tax Rate and Tax Revenue REV (t)

When t ↑, there are two opposing effects on l(t):Substitution effect:t ↑ ⇒ after-tax real wage z(1− t) ↓ ⇒ leisure is relativelycheaper⇒ l(t) ↑ (disincentive to work)Income effect:t ↑ ⇒ after-tax income ↓ ⇒ l(t) ↓ (normal good)Total effect on l(t) is ambiguous.

Total effect on z[h − l(t)] is also ambiguous.If SE > IE , higher tax rate might lead to lower tax revenue!


Laffer Curve

Laffer curve: graphically shows therelationship between t and REV (t).(curve AB)Shape of Laffer curve depends onlabor supply behavior.End points A and B will always bethe intersection points.

Point A: t = 0, so REV = 0.Point B: t = 1, no one would work,so REV = 0.

Maximum tax revenue REV ∗ isreached at t∗.


Government’s Choice of Tax Rate

Government’s budget: G = REV (t).If G > REV ∗, impossible forgovernment to collect enough taxrevenue.If G < REV ∗, there are at least twofeasible tax rates.Suppose there are two possibleequilibrium tax rates to financeG < REV ∗, t1 and t2.Which tax rate would thegovernment choose?


Government’s Choice of Tax Rate

Suppose t2 > t1, andREV (t1) = REV (t2) = G.With low tax rate t1, CE is F .With high tax rate t2, CE is H.F must be on a higherindifference curve than H. Why?The consumer is better off in theequilibrium with lower tax rate t1.Government should choose taxrate t1 rather than t2.


Laffer Curve and Supply-Side Economists

Supply-side economists’ view about income tax:Large incentive effects of income taxes on labor supply.Tax rate reductions will largely increase labor supply andthus the tax base.

Example: Ronald Reagon’s campaign in 1980 election.

Question against this argument:Whether the U.S. economy in 1980 was on theupward-sloping side (the "good" side) or thedownward-sloping side (the "bad" side) of the Laffer curve?In general, economists believe that U.S. economy is on thegood side of the Laffer curve.


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