Ch5 Slides

A Closed Economy One-PeriodMacroeconomic Model

Chapter 5

Topics in Macroeconomics 2

Economics DivisionUniversity of Southampton

February 2010

Chapter 5 1/40 Topics in Macroeconomics

Competitive EquilibriumEconomic Efficiency

Using the Model

IntroductionDefinitionGraphical Analysis

Closing the Model

◮ We have seen the optimal behavior of the representativeconsumer and the representative firm

◮ We now want to impose consistent behavior◮ We will study a closed economy: markets are restricted to

a single country◮ We will impose market clearing in the country: resources

produced must be consumed by economic agents in themodel

◮ Other market clearing condition: labour inputs intoproduction must be supplied by consumers

◮ The result will be our first model of the Macroeconomy



Using the Model


From Chapter 4

◮ The representative consumer faces a tradeoff betweenconsuming and working (work/leisure)

◮ The consumer is paid labour income for hours worked, andbuys goods from the firm

◮ The firm hires labour to produce output (consumptiongoods) which it sells to the consumer

◮ Both markets (goods and labour) need to clear



Using the Model


The Government Budget Constraint

G = T

◮ G is the amount of government spending (these areconsumption goods that also need to be produced)

◮ T is the total amount of taxes collected by the government(these are also consumption goods)

◮ The government budget constraint imposes thatgovernment spending be financed by taxes

◮ We will assume that G is exogenous, that is, determinedoutside of the model

◮ Fiscal policy refers to the government’s choice overspending and taxes

◮ Other fiscal instruments: government debt and transfers



Using the Model


Endogenous and Exogenous Variables

◮ Exogenous variables are determined outside the model:they are taken as given

◮ Endogenous variables are determined by the model:we have to find those

◮ Exogenous variables in our model:G, z, and K

◮ Endogenous variables in our model:C, Ns, Nd , T , Y , π and w



Using the Model


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 ) suchthat, given exogenous variables (G, z, and K ), the followingconditions 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 ) maximizesprofits (π = Y − wNd is paid to consumers as dividend)

3. The government budget constraint is satisfied (G = T )4. Markets clear:

Labour market: Nd = Ns = NGoods market: C + G = Y



Using the Model


Redundance of One Market Clearing Condition

If the labour market clears, so will the goods market!

◮ The consumer’s budget constraint has to hold:C = wNs + π − T

◮ But dividend income is the firm’s profits:π = Y − wNd

◮ If we replace profits in the consumer’s budget constraint:C = wNs + Y − wNd

− T◮ Since Nd = Ns = N, these terms cancel out. And since the

government budget constraint holds, T = G◮ It follows that C = Y −G or C + G = Y – our goods market

clearing condition



Using the Model


Production Possibilities Frontier

◮ This is our good old productionfunction

◮ Notice that the maximum numberof hours that firms can hire is h,where the firm producesY ∗ = zF (K , h)

◮ When no labour is used, nothingis producedF (K , 0) = 0



Using the Model



◮ We can change the horizontalaxis from N to ℓ

◮ The production function can bewritten zF (K , h − ℓ)

◮ The minimum number of hours ofleisure (ℓ = 0) occurs whenN = h, where the firm stillproduces Y ∗

◮ As before, when no labour isused (ℓ = h), nothing is produced

◮ The slope is now equal to −MPN



Using the Model



◮ We can change the vertical axisfrom Y to C since C = Y − G(NOT Y ∗

− G)◮ The maximum amount of

consumption is Y ∗− G (point D)

◮ When nothing is produced,consumption is equal to −G

◮ The shaded area is theproduction possibilities set

◮ The arc DA is the productionpossibilities frontier



Using the Model



C = zF (K , h − ℓ) − G

◮ The production possibilities frontier (PPF) describes whatthe economy can produce as a whole, in terms ofproduction of consumption and leisure

◮ All points in the production possibilities set aretechnologically possible to produce

◮ The slope of the PPF (−MPN) is also called the marginalrate of transformation (MRT )

◮ MRTℓ,C is the rate at which leisure can be converted intoconsumption goods

◮ So we have: MRTℓ,C = MPN = −slope of the PPF



Using the Model


Equilibrium Wage Rate

◮ Recall that firms optimize whenMRTℓ,C = MPN = w

◮ So if w is an equilibrium wagerate, then AD with slope −w istangent to the PPF

◮ But if w is an equilibrium wagerate, then ADB is the budgetconstraint

◮ Recall that consumers optimizewhen the budget line is tangentto the indifference curve, i.e.when MRSℓ,C = w



Using the Model


Equilibrium Profits: Firm’s Perspective

π∗ = zF (K , h − ℓ∗) − w∗(h − ℓ∗)

◮ Y ∗ = zF (K , h − ℓ∗) = C∗ + G◮ At point J: C = A − w∗ℓ∗

At point D: C = A − w∗h◮ The vertical distance between

these two points isA − w∗ℓ∗ − (A − w∗h), which isequal to the wage bill w∗(h − ℓ∗)

◮ Subtracting w∗(h − ℓ∗) from Y ∗

give us profits: distance DH



Using the Model


Equilibrium Profits: Consumer’s Perspective

◮ Recall that at point D, theconsumer does not work at all soconsumption must equal π∗

− T◮ Since π∗

− T = π∗− G, the

distance DH is total profits again



Using the Model

Welfare TheoremsSources of Inefficiencies

Pareto Optimality

DefinitionAn allocation is Pareto Optimal if there is no way to rearrangeproduction or to re-allocate resources so that someone is madebetter off without making someone else worse off

◮ To find efficient allocations we use a benevolent SocialPlanner

◮ The Planner wants to make the representative consumeras well off as possible

◮ The Planner does not face markets – it chooses quantities:◮ How many hours the consumer works ⇒ Y = zF (K , N)◮ G is given to the government◮ Y − G is given to the consumer



Using the Model


Pareto Optimum Allocation

◮ Point B is the Pareto optimum:MRSℓ,C = MRTℓ,C = MPN

◮ To the left of point B:MRTℓ,C < MRSℓ,C

Could make the consumer betteroff by giving him more leisure(less work) and less consumption

◮ To the right of point B:MRTℓ,C > MRSℓ,CCould make the consumer betteroff by giving him less leisure(more work) and moreconsumption



Using the Model


Welfare Theorems

First Welfare TheoremThe first fundamental theorem of welfare economics states that,under certain conditions, a competitive equilibrium allocation isPareto optimal

Second Welfare TheoremThe second fundamental theorem of welfare economics statesthat, under certain conditions, a Pareto optimal allocation canbe decentralized as a competitive equilibrium

How does this look on a graph?



Using the Model


Welfare Theorems

◮ The competitive equilibriumallocation (C∗, ℓ∗) is Paretooptimal since moving away frompoint B makes the consumerworse off

◮ The Pareto efficient allocationcan be decentralized as acompetitive equilibrium underprice w∗, the slope of the line thatis tangent to the indifferencecurve and the PPF



Using the Model


Sources of Inefficiencies

◮ ExternalitiesWhereas the Planner would take externalities into account,individuals and firms DO NOT

◮ Distortionary taxesFor example, a proportional labour income tax – consumersand firms do not face the same price (the competitiveequilibrium will have MRSℓ,C < MRTℓ,C and is therefore notPareto optimal)

◮ Firms may have market powerFirms with market power do not take price as given, as theyknow they can influence them, which typically leads tounder-production



Using the Model

Making use of the Welfare TheoremsEffects of a Change in Government SpendingEffects of a Change in TFP

Working with the Planner’s Problem

◮ Since the welfare theorems hold,we can either work with the CEconcept or the Planner’s problem

◮ In general, the Planner’s problem isconsiderably easier to work withthan the CE since we need notworry about prices

◮ Efficiency dictates having a point oftangency between the indifferencecurve and the PPF (point B)

◮ We can always find prices todecentralize the allocation as a CE



Using the Model


Increase in G

◮ With G2 > G1, the PPF shiftsdown from PPF1 to PPF2 (samevertical distance everywhere)

◮ Note that at each ℓ the slope ofthe PPF is the same as before

◮ Since G = T , an increase in Gmeans an increase in taxes forconsumers

◮ This part is exactly like a pure(negative) income change: sincegoods are normal, we shouldexpect both C and ℓ to decrease



Using the Model


Increase in G: Crowding Out

◮ Notice that ℓ2 < ℓ1, or N2 > N1

◮ Production must increase(∆Y = Y2 − Y1 > 0)

◮ Since C1 = Y1 − G1

and C2 = Y2 − G2,C2 − C1 = Y2 − G2 − (Y1 − G1),or ∆C = ∆Y − ∆G

◮ Since ∆Y > 0 we must have∆C > −∆G (AE vs AD)

◮ Consumption is crowded out bygovernment spending, but notcompletely



Using the Model


Increase in G: Equilibrium Effect

◮ How can we make firms hiremore labour?

◮ The real wage rate must fall◮ As N increases, the MPN

decreases, so w2 < w1

◮ But individuals still want less ℓ

(even if it is cheaper) becausethe pure income effect fromhigher taxes dominates

Exercise: Decompose the totaleffect into an income and asubstitution effect



Using the Model


Increase in G: Predictions of the Model

In the model, an increase ingovernment spending:

◮ Increases employment◮ Increases output◮ Decreases consumption

(counter-cyclical)◮ Decreases the real wage

rate (counter-cyclical)

Key business cycles co-movements (from Chap. 3):

◮ Employment is pro-cyclical◮ Consumption is

pro-cyclical◮ Real wage rate is

‘pro-cyclical’

We conclude that business cycles are not likely to be the resultof government spending fluctuations



Using the Model


Increase in z and the Production Function

◮ With z2 > z1, the productionfunction shifts up

◮ An increase in z also increasesMPN at each quantity of labourinput

◮ Notice that the maximum amountof output (zF (K , h)) is higher withz2 than with z1

◮ The amount of output producedwith no labour is zero regardlessof the value of z



Using the Model


Increase in z and the PPF

◮ With z2 > z1, the PPF shiftsoutward from BA to DA

◮ More consumption can beproduced at any level of leisure

◮ Since MPN is higher, the PPF issteeper under z2 than under z1

◮ The equilibrium (or efficient)allocation moves from F to H



Using the Model


Increase in z: Total Effect

◮ Consumption increases from C1

to C2

◮ Leisure could increase,decrease, or stay the same –here it remains ℓ1

◮ Production increases by thesame amount as C (G did notchange)

◮ The wage rate increases to w2 (Ndid not change and MPN ishigher at all levels of N)→ That will to be true in general



Using the Model


Increase in z: Substitution Effect

◮ We can construct an artificialPPF3 such that it is tangent to theoriginal indifference curve I1

◮ Just like we did in chap. 4, this istaking away consumption (orincome) away from the consumerin order to concentrate on thesubstitution effect

◮ Notice that PPF3 implies thesame wage rate as PPF2 (sameMPN)



Using the Model


Increase in z: Substitution Effect

◮ With PPF3, the efficient allocationis given by D

◮ The substitution effect is themove from A to D

◮ Consumption increases from A toD

◮ Leisure decreases from A to D◮ This is just like an increase in the

wage rate in Chap. 4



Using the Model


Increase in z: Income Effect

◮ The income effect is the movefrom D to B

◮ Consumption increases from D toB

◮ Leisure increases from D to B◮ This is just like an increase in

dividend income in Chap. 4

◮ Total effect:Consumption: must go upLeisure: uncertain



Using the Model


Increase in z: Long-Run Predictions of the Model

In the model, increases in TFP:

◮ Increase output◮ Increase consumption◮ Increase the real wage rate◮ Have an ambiguous effect

on hours worked

Since WWII, we have observed:

◮ A rise in output◮ A rise in consumption◮ A rise in the real wage rate◮ Roughly constant hours

worked

We conclude that if the income and substitution effects roughlycancel each other out over the long run, the model is consistentwith technological innovations having been key to changes inthese variables



Using the Model


Increase in z: Short-Run Predictions of the Model

In the model, an increase inTFP:

◮ Increases output◮ Increases consumption

(pro-cyclical)◮ Increases the real wage

rate (pro-cyclical)◮ Has an ambiguous effect

on hours worked (?)

Key business cycles co-movements (from Chap. 3):

◮ Employment is pro-cyclical◮ Consumption is

pro-cyclical◮ Real wage rate is

‘pro-cyclical’

We conclude that fluctuations in TFP may be the primary causeof business cycles if in the short run the substitution effectdominates the income effect (Real Business Cycle Theory)


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