Macroeconomics II Lecture 5 Chapter 10. A Real Intertemporal ...

Macroeconomics II Lecture 5Krzysztof MakarskiFall 2010Chapter 10. A Real Intertemporal Model with InvestmentIntroNow we build a two period model of the real side of the economy - this will be the base for further analysisand will be used rst for the RBC and the new Keynesian theory.The Representative ConsumerThe Representative Consumer - optimal choiceWe combine the work-leisure choice with the intertemporal consumption choice. The representativeconsumer has h units of time in each period and divides this time between work and leisure in each period.Let w 1 denote the real wage in current period, w 2 the real wage in future period, and r the real interestrate. The consumer pays lump sum taxes to the government of T 1 in the current period and T 2 in the futureperiod. Her choice is to choose the current consumption C 1 , future consumption C 2 , leisure in the currentand future period l 1 and l 2 to make herself as well of as possible given her budget constraint. The consumer'scurrent budget constraint isC 1 + S p 1 = w 1(h − l 1 ) + π 1 − T 1 (10.1)the consumer's future budget constraint isSubstituting for S p from (10.1) into (10.2) and rearranging we getC 2 = w 2 (h − l 2 ) + π 2 − T 2 + (1 + r)S p 1 (10.2)C 1 + C 21 + r = w 1(h − l 1 ) + π 1 − T 1 + w 2(h − l 2 ) + π 2 − T 21 + r(10.3)The representative consumer's problem is to choose (C 1 , C 2 , l 1 , l 2 ) to make herself as well o as possible giventhe budget constraint. Since the consumer's choice is four-dimensional we cannot depict it on a graph, butwe can describe the consumer's choice in terms of three marginal conditions (we have looked at in Chapters4 and 8):• current consumption-leisure choice• intertemporal consumption choice• future consumption-leisure choiceMRS l1,C 1= w 1 (10.4)MRS l2,C 2= w 2 (10.5)MRS C1,C 2= 1 + r (10.6)1

Current Labor Supply• Increases when the current real wage increases. Recall from Chapter 4 that the impact of the realwage on the choice depends on the relative strength on the income and the substitution eect. Herewe assume that the substitution eect is stronger, thus the increase in the real wage will increase laborsupply.• Increases when the real interest rate increases. Consumer not only substitutes consumption intertemporally,but also substitutes intertemporally leisure. We assume that in the case of the increase of thereal interest rate the substitution eect is larger than the income eect, thus when the real interestrate increases the consumer will consume less current leisure and more future leisure. So since currentleisure goes up, current labor supply goes down.• Decreases when the lifetime wealth increases. Since both leisure and consumption goods are normalconsumption of both increases when wealth increases.• Current Labor Supply Schedule Thus we can show the relationship between current real wage and current labor supply, see Figure10.1. On this current labor supply schedule we can also show the eect of the increase in the real wage,see Figure 10.2 and the eect of the increase in the lifetime wealth, see Figure 10.3Current Demand for Consumption Goods• Current income. Recall from Chapter 8 the relationship between current income and current consumption,Figure 10.4.• Real Interest Rate Eects We assume that the substitution eect dominates the income eect, thuswhen the real interest rate increases, current consumption will decrease, Figure 10.5.• Lifetime Wealth Eect. Since both leisure and consumption goods are normal consumption of bothincreases when wealth increases, Figure 10.6. Example of the change in lifetime wealth is a decrease of the present value of taxes.2

Figure 10.1The Representative Consumer’s Current LaborSupply Curve© 2011 Pearson Addison-Wesley. All rights reserved.10-8returnFigure 10.2An Increase in the Real Interest Rate Shifts theCurrent Labor Supply Curve to the Right© 2011 Pearson Addison-Wesley. All rights reserved.10-9return3

Figure 10.3Effects of an Increase in Lifetime Wealth© 2011 Pearson Addison-Wesley. All rights reserved.10-10returnFigure 10.4The Representative Consumer’s Current Demandfor Consumption Goods Increases with Income© 2011 Pearson Addison-Wesley. All rights reserved.10-12return4

Figure 10.5An Increase in the Real Interest Rate from r1to r2Shifts the Demand for Consumption Goods Down© 2011 Pearson Addison-Wesley. All rights reserved.10-13returnFigure 10.6An Increase in Lifetime Wealth Shifts the Demandfor Consumption Goods Up© 2011 Pearson Addison-Wesley. All rights reserved.10-14return5

The Representative FirmFirm's choiceFirm Choices:• Current Production In the current period the representative rm produces output according to thefollowing production functionY 1 = z 1 F (K 1 , N 1 ) (10.7)• Future Production In the future period output is produced according toY 2 = z 2 F (K 2 , N 2 ) (10.8)• Investment and Capital As in the Solow model we assume that the future capital stock is given byK 2 = (1 − d)K 1 + I 1 (10.9)where d− depreciation rate and I 1 − investment Since at the end of the future period there is (1−d)K 2capital left, for simplicity, we assume that it is converted into consumption goods.Prots and Current Labor Demand• Current Prots Current prots areπ 1 = Y 1 − w 1 N 1 − I 1 (10.10)• Future Prots Future prots areπ 2 = Y 2 − w 2 N 2 + (1 − d)K 2 (9.11)• The Present Value of Prots Prots are paid out to the shareholders of the rm as dividend incomein each period. There is one shareholder in this economy - representative consumer, and the rmacts in the interest of this shareholder. This implies that the rm maximizes the present value of theconsumers dividend income (which serves to maximize the lifetime wealth of the consumer), the rmthen maximizesV = π 1 + π 2(9.12)1 + rby choosing current labor demand N 1 , future labor demand N 2 and current investment I 1 .• Current Employment Choice As in Chapter 4 the rm hires current labor until the current marginalproduct of labor equals the current real wageMP N = wand the demand curve for labor in the current period is identical to the marginal product of labor,Figure 10.7 As in Chapter 4 the labor demand increases with total factor productivity z or with theinitial capital stock K, Figure 10.8.6

Figure 10.7The Demand Curve for Current Labor Is theRepresentative Firm’s Marginal Product ofLabor Schedule© 2011 Pearson Addison-Wesley. All rights reserved.10-21returnFigure 10.8The Current Demand Curve for Labor Shifts Dueto Changes in Current Total Factor Productivity zand in the Current Capital Stock K© 2011 Pearson Addison-Wesley. All rights reserved.10-22return7

The Investment DecisionThe choice of investment by the representative rm involves equating the marginal cost of investmentwith the marginal benet of investment.• The Marginal Cost of Investment Let MC(I) denote marginal cost of investment, thenMC(I) = 1 (10.13)The marginal cost of investment is for the rm is what it gives up, in terms of the present value ofprots, V . By investing one unit of capital in the current period it cannot sell, thus it decreases thepresent value of prots by 1.• The Marginal Benet of Investment The marginal benet of investment, denoted by MB(I) is whatone extra unit of investment adds to the present value of prots. Notice all the benets come in termsof future prots π 2 . One unit of investment, will increase capital tomorrow by one unit. And theadditional output in the future period is equal to the future's marginal product of capital MP K2 plusan additional (1 − d) units of remaining at the end of the future period. Thus one unit of investmentincreases future prots by MP K2 + 1 − d, in calculating the marginal benet of investment we have todiscount the future gains, thusMB(I) = MP K 2+ 1 − d1 + r• The Optimal Investment Rule The rm invests until MB(I) = MC(I)or after rearrangingMP K2 + 1 − d1 + r(10.14)= 1 (10.15)MP K2 − d = r (10.16)which says that the future net marginal product of capital is equal to the interest rate. Thus the realinterest rate represents the opportunity cost of investing in the representative rm.• The Optimal Investment Schedule The optimal investment schedule is the rm's future period netmarginal product of capital, Figure 10.9 Also, the optimal investment schedule changes due to thefollowing factors increases when future total factor productivity increases increases when current capital stock decreasesThose changes are presented in Figure 10.10 Example: r = 5%, see Table 10.1.Investment with Asymmetric Information and the Financial Crisis• What if a bank is not able to distinguish good and bad rms (as discussed in Chapter 9), than therewill be spread. Denote r L as a lending rate and if x is a default premium we will haver L = r + xand thusMP K2 − d − x = r• An increase in default premium would (if banks perceive that bad borrowers have become more prevalent)lower investments, which can be represented as a shift of an optimal investment schedule, seeFigure 10.118

Figure 10.9Optimal Investment Schedule for theRepresentative Firm© 2011 Pearson Addison-Wesley. All rights reserved.10-27returnFigure 10.10The Optimal Investment Schedule Shifts to theRight if Current Capital Decreases or FutureTotal Factor Productivity Is Expected to Increase© 2011 Pearson Addison-Wesley. All rights reserved.10-28return9

Table 10.1Data for Christine’s Orchard© 2011 Pearson Addison-Wesley. All rights reserved.10-29returnFigure 10.11The Effect of an Increased Default Premium on aFirm’s Optimal Investment Schedule© 2011 Pearson Addison-Wesley. All rights reserved.10-33return10

[PoFC] Investment and the Interest Rate Spread• Negatively correlated.• Particularly during recent nancial crisis, see Figure 10.12 and Figure 10.13GovernmentGovernmentThe government must satisfy its present-value budget constraintG 1 + G 21 + r = T 1 + T 21 + r(9.17)Competitive EquilibriumThe Current Labor Market and the Output Supply Curve• Equilibrium in the labor market for the given real interest rate. In the labor market the real wage andemployment is determined. Employment in equilibrium determines output (given the capital stock),from the production function, see Figure 10.14• Slope of Output SupplyReal Interest Rate Eects Next suppose that the real interest rate increasesfrom r 1 to r 2 , this increases labor supply, labor supply curve shifts right, this increases equilibriumemployment from N 1 to N 2 , and output from Y 1 to Y 2 . Thus output increases when the real interestrate increases, see Figure 10.15• Shifts in Output Supply Government spending Decrease in lifetime wealth increases labor supply, thus labor demand curveshifts to the right. It leads to the shift of output supply curve to the right, see Figure 10.16. Anexample of a decrease of lifetime wealth is the increase in government spending (which will increasetaxes today or in the future). Current Total Factor Productivity (or current capital stock) An increase in total factor productivityincreases labor demand, thus labor demand curve shifts, it leads nally to the shift of theoutput supply curve, see Figure 10.17.11

Figure 10.12Investment and the Interest Rate Spread© 2011 Pearson Addison-Wesley. All rights reserved.10-34returnFigure 10.13Scatter Plot: Investment vs. Interest Rate Spread© 2011 Pearson Addison-Wesley. All rights reserved.10-35return12

Figure 10.14Determination of Equilibrium in the Labor MarketGiven the Real Interest Rate r© 2011 Pearson Addison-Wesley. All rights reserved.10-37returnFigure 10.15Construction of the Output Supply Curve© 2011 Pearson Addison-Wesley. All rights reserved.10-38return13

Figure 10.16An Increase in Current or Future GovernmentSpending Shifts the Y s Curve© 2011 Pearson Addison-Wesley. All rights reserved.10-39returnFigure 10.17An Increase in Current Total Factor ProductivityShifts the Y s Curve© 2011 Pearson Addison-Wesley. All rights reserved.10-40return14

The Current Goods Market and the Output Demand Curve• Now we turn to the goods market. The equilibrium condition in the goods market isY 1 = C d 1 (r) + I d 1 (r) + G 1 (10.19)In Figure 10.18 we show the total demand for goods (the right hand side of the equation (10.19))as a function of current income. The demand for current consumption goods is determined by theintersection of the C d 1 (r) + I d 1 (r) + G 1 curve with the 45 o line.• Construction of Output DemandReal Interest Rate Eects The next step is to derive the outputdemand curve. Suppose that the real interest rate goes up from r 1 to r 2 , this increases output demand,the C d 1 (r) + I d 1 (r) + G 1 curve shifts down, thus quantity of goods demanded falls, see Figure 10.19• Shifts in Output Demand Curve Increase in government expenditure from G 1 to G 2 , then the C d 1 (r) +I d 1 (r) + G 1 curve shifts up. Since the interest rate does not change this means that the output demandcurve shifts to the right, see Figure 10.20. Also the following factors increase demand, and thus shiftthe C d 1 (r) + I d 1 (r) + G 1 curve up and thus shift the output demand curve to the right. decrease in the present value of taxes (increases consumption) increase in future income (increases consumption) increase in future total factor productivity (then investment increase) decrease in the current capital stock (then investment increase)The Complete Real Intertemporal Model• We have all building blocks for our real intertemporal model. Model is presented in Figure 10.21 whichpresents market clearing in the labor market (panel (a)), and in the goods market (panel (b)).• Next we perform experiments to see how changes in exogenous variables changes the equilibrium inour economy.A Temporary Increase in Government PurchasesDirect eect of an increase in G• See Figure 10.22, AB is equal to BF (45 o line).• The vertical distance DF between lines is equal to ∆G−MP C∆G = (1 − MP C) ∆G, where ∆G is anincrease in demand due to higher gov. expenditure and MP C∆G is a decline in current consumptionfrom a fall in current wealth by ∆G due to higher future taxes (Ricardian equivalence must hold)• Increase in income by AB increases consumption by DB thus DBMP C is constant). Substituting∆Y −(1−MP C)∆G∆YAB−DFAB= MP C orAB= MP C (assume= MP C and solving gives ∆Y = ∆G. return15

Figure 10.18The Demand for Current Goods© 2011 Pearson Addison-Wesley. All rights reserved.10-41returnFigure 10.19Construction of the Output Demand Curve© 2011 Pearson Addison-Wesley. All rights reserved.10-42return16

Figure 10.20The Output Demand Curve Shifts to the Right ifCurrent Government Spending Increases© 2011 Pearson Addison-Wesley. All rights reserved.10-44returnFigure 10.21The Complete Real Intertemporal Model© 2011 Pearson Addison-Wesley. All rights reserved.10-45return17

A Temporary Increase in Government Purchases• Impact Eects As government spending go up, present value of taxes goes up, thus consumers decreaseconsumption of both leisure and consumption goods, thus labor supply goes up (N s curve shifts right).Furthermore we know that a decrease in consumption of consumption goods (recall MP C < 1) thedemand for current consumption goods falls by less than government purchases rise. Therefore thetotal demand for goods must rise (so Y d curve shifts right). See Figure 10.22 and Figure 10.23 Labor Supply goes up (G goes up, thus taxes will go up, thus wealth goes down). Output Supply goes up (since Labor Supply shifts). Output Demand goes up (see above).• Equilibrium Eects Goods Market: Y ↗, r ↗ Labor Market: N ↗, w ↘ The Composition of Output: C ↘, I ↘∗ Government multiplier: Since - as shown ealier - Y d shifts by exactly ∆G nally income -due to increase in the interest rate - increses by less then the initial increase in G, thus thegov. multplier is smaller than 1.∗ Notice that an initial decline in consumption equal to MP C∆G - as shown ealier - is notoset by an increase in consumption due to an increse in income, since income goes up by lessthan ∆G and thus consumption - due to increase in income goes up by less than MP C∆G,so overall consumption falls.∗ Notice that in equilibrium the interest rate increased, which decreased investment.Gov Multiplier• Its value is debated. New classicals (Barro) argue it's less than 1, while Keynesians (Ch. Romer) claimit's larger than one.• Romer's estimate is based on models not immune to Lucas critique (not based on micro foundations)• In our model multiplier generally is smaller than one, but can get higher with: government spending nanced with borrowing when Ricardian equivalence does not hold (e.g.consumers are credit constrained). decline in lifetime wealth (due to higher future taxes) increases labor supply atter Y s curve (high response of labor supply to changes in the interest rate)First two points are more likely during a credit crunch.• To have a fully developed new Keynesian argument we need the fully developed new Keynesian model(soon).18

Figure 10.22The Effect of an Increase in G on theTotal Demand for Goods© 2011 Pearson Addison-Wesley. All rights reserved.10-47return1return2Figure 10.23A Temporary Increase in Government Purchases© 2011 Pearson Addison-Wesley. All rights reserved.10-49return19

An Increase in Current Total Factor ProductivityAn Increase in Current Total Factor Productivity• Impact Eects, See Figure 10.25 Labor Demand (since MP N goes up) Output Supply (since labor demand shifts)• Equilibrium Eects Note: as the interest rate goes up (see panel b. on Figure 10.25) the labor supplycurve shifts left (see panel a. on Figure 10.25) Goods Market: Y ↗, r ↘ Labor Market: N? (likely increases),w ↗ The Composition of Output: C ↗, I ↗ (since output goes up and the interest rate goes down)An Increase in Future Total Factor ProductivityAn Increase in Future Total Factor Productivity• Impact Eects, see Figure 10.26 Output Demand (since investments go up)• Equilibrium Eects Goods Market: Y ↗, r ↗ Labor Market: N ↗, w ↘ The Composition of Output: C ↘, I ↗An Increase in Credit Market Uncertainty: Asymmetric Information and theFinancial CrisisAn Increase in Credit Market Uncertainty• Source: Asymmetric Information and the Financial Crisis; focus on rms.• An increase in credit market uncertainty leads to an increase in credit rate spreads, thus given anyinitial lending rate (or central bank interest rate) borrowing rates for rms go up, thus the investmentsgo down, Y d curve shifts left. The rest is like for the negative supply shock.• Impact Eects, see Figure 10.28 Labor Supply Output Demand• Equilibrium Eects Goods Market: Y ↘, r ↘ Labor Market: N ↘, w ↗ The Composition of Output: C ↗, I ↘20

Figure 10.25The Equilibrium Effects of an Increase inCurrent Total Factor Productivity© 2011 Pearson Addison-Wesley. All rights reserved.10-55returnFigure 10.26The Equilibrium Effects of an Increase inFuture Total Factor Productivity© 2011 Pearson Addison-Wesley. All rights reserved.10-57return21

Summary• Derived a Complete Real Model of the Economy labor market goods market• Performed experiments with eects of dierent shocks. government expenditure (gov. multiplier) current TFP shock future TFP shockReadingWilliamson, Chapter 10.22

Figure 10.28The Effect of an Increase in Credit Market Risk© 2011 Pearson Addison-Wesley. All rights reserved.10-60return23