Econ 510a (second half)Yale UniversityFall 2005Prof. Tony SmithHOMEWORK #1This homework assignment is due at the beginning of class on Wednesday, October 26.1. (a) Consider a version of the static model that we discussed in class on October 19 inwhich a typical consumer’s preferences are given by: u(c, `) = log(c) + A log(`),where A > 0, and the firm’s production function is y = zkαn1−α, where 0 < α < 1.(Let both the number of consumers and the number of firms be equal to 1.) Findan algebraic expression for the competitive equilibrium amount of labor supplyin terms of primitives (i.e., in terms of the parameters describing technology andpreferences). How do changes in z and A affect equilibrium output, consumption,and labor supply? Explain (i.e., try to give some economic intuition for youranswers).(b) Now introduce a government that taxes labor income (i.e., income from wages) at aproportional rate τ and returns the pro cee ds in a lump-sum fashion to consumers.A typical consumer’s budget constraint reads:c = rk0+ (1 − τ)w(1 − `) + T,where τ > 0 and T > 0 is a lump-sum transfer to consumers. Each consumer takesthe prices r and w and the transfer T as given when solving his decision problem.The government must satisfy its budget constraint: T = τwn, i.e., total transfersto consumers must be equal to total revenues from the tax on labor income. Findan algebraic expression for the competitive equilibrium amount of labor supply.How do changes in z, A, and τ affect equilibrium output, consumption, and laborsupply? Explain.(c) Is the competitive equilibrium allocation in part (b) identical to the Pareto opti-mal allocation? Explain why or why not.(d) Suppose instead that the transfer is financed by a proportional tax on capitalincome. Do the competitive equilibrium and Pareto optimal allocations coincidein this case? Explain why or why not.2. Consider a version of the static model in the first problem (without taxes) in which atypical consumer’s preferences are given by:u(c, `) =c1−σ− 11 − σ+ A`1−γ− 11 − γ,where both σ and γ are positive. (Note that, as σ and γ both approach 1, thisutility function converges to the one in the first problem.) Find an equation thatdetermines the competitive equilibrium amount of labor supply. (You will not able tosolve explicitly for labor supply as a function of primitives.) How do increases in zand A affect equilibrium output, consumption, and labor supply? Explain. (Hint: Todetermine the effect of changes in z on labor supply, totally differentiate the expressiondetermining labor supply with respect to n and z, compute the derivative dn/dz, andthen try to “sign” the derivative.)3. Consider an economy with two time periods (labelled 0 and 1) in which a typicalconsumer has preferences given by: log(c0) + β log(c1), where ctis consumption inperiod t. Each consumer is endowed with k0units of capital at the beginning of period0 and with one unit of time in each period.In each period, there is a price-taking, profit-maximizing firm that produces goodsusing capital and labor. These goods can be either consumed or saved in the form ofcapital that can be used in production in the next period. Let the firm’s productionfunction be: y = zkαn1−α, where k is the amount of capital rented by the firm and nis the amount of labor rented by the firm in a given period.Because leisure is not valued (leisure does not appear in the utility function), eachconsumer supplies labor inelastically, i.e., he supplies one unit of labor in each timeperiod. The only interesting decision that a consumer makes, then, is how much tosave in period 0. Let k1be the amount of capital that a typical consumer saves inperiod 0. Then each consumer faces a pair of budget constraints:c0= r0k0+ w0− k1andc1= r1k1+ w1,where rtis the rental price of capital in period t (expressed in terms of period-t con-sumption goods) and wtis the wage rate in period t (again expressed in terms ofperiod-t consumption goods). Each consumer takes these prices as given when decid-ing how much to save. Because period 1 is the last period of his life, each consumerconsumes all of his resources in period 1. (Note that, as in the first two problems, thereis an implicit assumption that capital is use d up entirely in the process of production,i.e., the depreciation rate of capital is equal to one.)In equilibrium, the markets for goods, labor, and capital must clear in both timeperiods.(a) Find an explicit expression (in terms of primitives) for the competitive equilibriumcapital stock in period 1. Use your answer to determine the equilibrium rate ofreturn on savings between p eriods 0 and 1. In addition, determine the equilibriumallocation of consumption across the two time periods. How do changes in z affectthe equilibrium allocation? Explain.(b) Formulate a social planning problem for this economy and show that the allo-cation chosen by the social planner is identical to the competitive equilibriumallocation that you determined in part (a). Illustrate this result using an appro-priate