Lecture6 Time

Topics in Consumer theory and Intertemporal Choices1Some Topics1.1Elasticities1.Let x i(p;I)be the consumer’s Marshallian demand function forgood i:The leti @x i(p;I)@IIx i(p;I); ij=@x i(p;I)@p jp jx i(p;I);and lets i=p i x i(p;I)Iso that s i 0andnX i=1s i=1:2.We call i the income elasticity of demand for good i; ij the priceelasticity of demand of good i:If j=i; ij is the own-price elasticity of demand of good i:If j=i; ij is called the cross-price elasticity of demand for good i with respect to p j:s i denotes the income share spent on purchase of good i:3.Engel aggregation:X n i=1s i i=1;Cournot aggregation:X n i=1s i ij= s j;j=1:::n:The Engel aggregation says that the share-weighted income elasticities must always sum to one.Cournot aggregation says that the share weighted own-and cross-price elasticities must always sum in a particular way.4.Proof.Recall the budget constraint is I=px(p;I):Di¤erentiateboth sides of it w.r.t.income and get1=nX i=1p i@x i@I:Multiply and divide each element in the summation by x i I;re-arrange,and get1=nX i=1p i x i I@x i@I I x i:Substitute from the de…nitions to getX n i=1s i i=1:To prove the Cournot aggregation,di¤erentiate the budget con-straint w.r.t.p j;we obtain0=[nX i=j p i@x i@p j]+x j+p j@x i@p j: We can combine terms and rearrange to getx j=nX i=1p i@x i@p j:Multiply both sides of the equation by p j=I and getp j x j I =nX i=1p j@x i@p j p i I;multiply and divide each element of th summation by x i and getp j x j I =nX i=1p j x i@x i@p j p i x i I:Substituting from the de…nitions completes the proof.1.2Integrability1.In previous lectures,we have shown that a utility maximizing con-sumer’s demand function must satisfy homogeneity of degree zero, budget balance,symmetry,and negative semide…niteness,along with Engel and Cournot aggregation.2.Both Engel and Cournot aggregation results follow directly frombudget balance.And homogeneity of degree zero is implied by budget balance and symmetry.3.Thus,if x(p;I)is a utility-maximizer’s system of demand function,we may summarize the implications for observable behavior in thefollowing three items alone,and there are no other independent re-strictions imposed on demand behavior by the theory of the utilitymaximizing behavior.(a)Budget Balancedness:px(p;I)=I:(b)Negative Semide…niteness:the associated Slutsky matrix S(p;I)must be negative semide…nite.(c)Symmetry:S(p;I)must be symmetric.4.The problem of recovering a consumer’s utility function from hisdemand function is known as the integrability problem.Integrabil-ity Theorem:A continuous di¤erentiable function x is the demandfunction generated by some increasing,quasiconcave utility func-tion if(and only if,when utility function is continuous,strictlyincreasing,and strictly quasiconcave)it satis…es budget balanced-ness,symmetry,and negative semide…niteness.1.3Revealed preference1.So far,the consumer theory what we have learned is preference-based.In his remarkable Foundations of Economic Analysis,PaulSamuelson(1947)suggested an alternative approach,the choice-based consumer theory.Samuelson showed how virtually everyprediction ordinary consumer theory makes can also be derivedfrom a few simple and sensible assumptions about the consumer’sobservable choices themselves,rather than about his unobservablepreferences.2.The basic idea is simple:if the consumer buys one bundle insteadof another a¤ordable bundle,then the…rst bundle is considered tobe revealed preferred to the second.The presumption is that byactually choosing one bundle over another,the consumer conveysimportant information about her tastes.3.Weak Axiom of Revealed Preference(W ARP).A consumer’s choicebehavior satis…es W ARP if for every distinct pair of bundles x0;x1with x0chosen at price p0and x1chosen at prices p1;p0x1 p0x0=)p1x0>p1x1:In other words,W ARP holds if whenever x0is revealed preferredto x1;x1is never revealed preferred to x0:(illustrate W ARP bygraphs.)4.Now suppose a consumer’s choice behavior satis…es W ARP.Letx(p;y)denote the choice made by this consumer when faced with price p and income y:Note that x(p;y)is not a demand function because we haven’t mentioned utility or utility maximization.We call it the choice function.5.We also assume the budget is balanced,i.e.,px(p;y)=y:We candemonstrate that if a choice function satis…es W ARP and budget balancedness,then it must satisfy two other properties implied by utility maximization,namely,homogeneity of degree zero and negative semide…niteness of the Slutsky matrix.6.It turns out that in the two-good case,budget balacedness togetherwith homogeneity imply that the Slutsky matrix must be symmet-ric.Therefore,for two goods,the integrability theorem tells us that the choice function must be ulitility generated.For more than two goods,W ARP and budget balancedness are not equivalent to the utility-maximization hypothesis.7.Strong Axiom of Revealed Preferences(SARP)is satis…ed if forevery sequence of distinct bundles x0;x1;:::x n;where x0is revealed preferred to x1;x1is revealed preferred to x2;:::;and x n 1is re-vealed preferred to x n;but it is not the case that x n is revealed preferred to x0:8.A theory of demand built only on SARP,a restriction on ob-servable choice,is essentially equivalent to the theory of demand built on utility maximization.Under both SARP and the utility-maximization hypothesis,consumer demand will be homogeneous and the Slutsky matrix will be negative semide…nite and symmet-ric.2Intertemporal Choices1.We discuss some topics dealing with the behavior of a consumerand an economy over time.The objects of choice will now be streams of consumption over time.2.If it is the discrete time,U(c t)=TX t=0 t u(c t)(1)where u(:)is instantaneous utility function: is the discount factor. and =11+r;where r is the discount rate.Most of the time,weregard the real interest rate as the discount rate.Here is we assume the utility function is additive over time.3.If it is the continuous time,U(c t)=Z T0u t(c t)e rt dt:2.1Two periods1.We…rst review the two-period model.Denote consumption ineach of the two periods by(c1;c2).The consumer has an initial endowment of w1in period1,and can invest his wealth in two assets.One asset pays a certain return R0;the other asset pays a random return of e R1:2.Suppose that the consumer decides to consume c1in the…rst periodand to invest a fraction s of his wealth in the risky asset and a fraction of1 s in the certain asset.Therefore his second period wealth(consumption)ise w2=e c2=(w1 c1)(e R1s+R0(1 s))=(w1 c1)e R where e R=e R1s+R0(1 s)is the consumer’s portfolio return.3.Since the portfolio return is uncertain,the consumer’s second-period consumption is uncertain.We suppose that consumer hasa utility function of the formU(c1;e c2)=u(c1)+ Eu(e c2)4.Let V1(w1)be the maximum utility the consumer can achieve if hehas wealth w1in period1u(c1)+ Eu[(w1 c1)e R](2)V1(w1)=maxc1;sAs we learn before,V1(w1)is essentially an indirect utility:it gives maximized utility as a function of wealth.5.FOCs of(2)areu0(c1)= Eu0(e c2)e R(3)Eu0(e c2)(e R1 R0)=0(4) Equation(3)is the intertemporal optimization condition,the Euler equation.It says that the marginal utility of consumption in period1must be equal to the discounted expected marginal utility of consumption in period2.Equation(4)is a portfolio optimization condition:it says that the expected marginal utility of moving a small amount of money from the safe to the risky asset should be0.By equation(3)and(4),we can solve for the optimal consumption and the portfolio choice.2.2The life-cycle/permanent-income hypothesis1.Consider an individual who lives for T+1periods whose lifetimeutility isU(c t)=TX t=0u(c t)(1+r)t(5)Assume the initial wealth is w0and the income for period t is I t.Thus the individual’s budget constraint isTX t=0c t(1+r)t w0+T X t=0I t(1+r)t2.The Lagrangian for his or her maximization problem is thereforeL=TX t=0u(c t)(1+r)t+ w0+T X t=0I t(1+r)t T X t=0c t(1+r)t!(6)3.The FOC for c t isu0(c t)= (7) Since(7)holds in every period,the marginal utility of consump-tion is constant.And since the level of consumption uniquely de-termines its marginal utility,this means that consumption must be constant.Thus c0=c1=:::=c T:4.This analysis implies that the individual’s consumption in a givenperiod is determined not by income that period,but by income over his or her entire lifetime.This is the key idea of the life-cycle/permanent-income hypothesis.2.3The Ramsey problem1.We now extend this analysis to examine aggregate capital accu-mulation in an economy.A representative agent who lives forever and has utility functionU(c t)=1X t=0 t u(c t)(8)At time t the capital is K t:The production function is q t=f(K t): The consumer must decide how much to invest and how much to consumec t+i t=q tWe assume that u(:)and f(:)are strictly concave,strictly increas-ing and twice continuously di¤erentiable.And the initial capital stock is K0:2.Capital depreciates at rate ;thus,with gross investment i t;thecapital stock next period isK t+1=(1 )K t+i t=(1 )K t+f(K t) c t We will write this more simply asK t+1=F(K t) c t(9) The Lagrangian isL=1X t=0 t u(c t)+ t(F(K t) c t K t+1)3.Ignoring the possibility of a boundary solution,the…rst order con-ditions are@L@c t= t u0(c t) t=0(10)@L@K t+1= t+1F0(K t+1) t=0(11)@L@ t=F(K t) c t K t+1=0(12) Combining(10)and(11)yields the necessary conditionMRS(c t;c t+1)=u0(c t)u0(c t+1)=F0(K t+1)(13)From(12)again we get the constraint(9).4.From(9)and(13)we can get the steady-state equilibrium.F0(K )=1=1+r(14)c =F(K ) K (15) Since F(:)is strictly concave,K is unique.5.Now we attempt to…gure out the path that the economy convergesto the steady state equilibrium.From the equation(13),u0(c t)>:Since u(:)is strictly concave,it follows u0(c t+1)i¤F0(K t+1)>1that c t+1>c t i¤F0(K t+1)>1:Hencec t+1>c t i¤K t+1<Kc t+1<c t i¤K t+1>K :(16)6.Next consider capital accumulation.The accumulation constraint,(9)can be rewritten as followsK t+1 K t=F(K t) c t K t(17) Thus the capital stock is constant ifc t=F(K t) K t=F(K t+1) K t+1ThereforeK t+1 K t>0i¤c t<F(K t+1) K t+1K t+1 K t<0i¤c t>F(K t+1) K t+1(18)7.With(16)and(18)as well as the steady-state equilibrium,we candraw the phase diagram.Starting from a low capital stock,there isa sequence{c t;K t+1}1t=1in phase III that approach the stationaryequilibrium point E:8.Since the consumption path approaches c ;it follows directly from(10)that the sequence of shadow prices approaches0.Also thelimiting capital stock is…nite.Thus the value of the capital stockt K t+1approaches0so the Transversality Condition holds.Giventhe concavity of the production function and the utility function,it follows that the sequence{c t;K t+1}1t=0is optimal.9.Transversality Condition(TC):let{c t;K t+1}1t=0satisfying the FOCs.The TC holds if the associated sequence of shadow prices{ t}1t=0yields a sequence of{ t K t+1}1t=0that approaches0as T growslarge.2.4Dynamic programming reformulation1.Another(probably better)way to solve the intertemporal(or dy-namic)optimization problem is the method of dynamic program-ming,which breaks the above sequence problem into a two-periodoptimization problem.2.For the Ramsey problem above,we can de…ne the indirect utilityfunction at the period0asV(K0)=maxK1f u[F(K0) K1]+ V(K1)g whereV(K1)=maxK2f u[F(K1) K2]+ V(K2)g: By the same token we can de…ne V(K2);V(K3);:::3.Notice those indirect utility functions share the same recursivestructure,the time subscribes have become a nuisance:we do not care what the date is.We can write the Ramsey problem with current capital stock K asf u[F(K) K0]+ V(K0)g(19)V(K)=maxK0The above equation is also called the value function,or the Bellman equation.And K here is so-called the"state variable".4.For the sequence problem discussed in section1.3,the choice vari-ables are the sequence{c t;K t+1}1t=0,this seems to be daunting in the sense that it contains in…nity numbers of choice variables and constraints.However,the maximizers for the recursive problem are obtained as functions which are called as"policy functions".Solving the Ramsey problem in the recursive form,we can obtain the following:c(K):Policy function for current period’s consumptionK0(K):Policy function for next period’s capital The goal of the problem in the recursive formulation is to obtain these functions.This is the reason why the recursive formulation is called as a"functional equation".The policy and the value functions are all functions of the state variable.5.The…rst-order and the envelope condition for(19)would beu0[F(K) K0]= V0(K0)(20) andV0(K)=F0(K)u0[F(K) K0](21) respectively.(20)equates the marginal utility of consuming cur-rent output to the marginal utility of allocating it to capital and enjoying augmented consumption next period.(21)states that the marginal value of current capital,in terms of total discounted util-ity,is given by the marginal utility of using the capital in current production and allocating its return to current consumption.。