Econ 510a (second half)Yale UniversityFall 2006Prof. Tony SmithFINAL EXAMINATIONThis is a closed-book and closed-notes exam. You have three (3) hours to complete the exam.There are four (4) questions on the exam for a total of 100 points. The points allocated toeach part of each question are indicated below. Please put the answer to each question ina different blue book. To receive full credit, you must provide convincing explanations tosupport your answers. Please write as neatly as possible.1. A farmer in Italy produces bread and wine for his own consumption. The farmeris endowe d with T hours of time in each period. He allocates his time between twoactivities: baking bread and pressing grapes to make grape juice. The farmer doesnot value leisure. The only input required to produce bread and grape juice is labor.Moreover, the production technology is linear: each unit of time devoted to bakingbread produces one unit of bread and each unit of time devoted to pressing grapesproduces one unit of grape juice. The farmer does not consume grape juice, but eachunit of grape juice becomes (via the process of fermentation) one unit of wine (whichhe does consume) in the next period. Both bread and wine are perishable (i.e., non-storable) goods.The farmer allocates his time between baking bread and pressing grapes in order tomaximize the lifetime utility of his consumption of bread and wine; this utility is givenbyP∞t=0βtu(bt, wt), where btis the amount of bread consumed in period t, wtis theamount of wine consumed in period t, and β ∈ (0, 1). The farmer is endowed withw0> 0 units of wine in period 0.(a) [8 points] Formulate the farmer’s optimization problem as a dynamic program-ming problem (i.e., display the Bellman equation for the farmer’s problem). Iden-tify clearly the state and control (or choice) variables.(b) [6 points] Find the farmer’s Euler equation.(c) [6 points] Explain how to compute the slop e of the farmer’s decision rule at thesteady state. You do not need to calculate the slope, but you need to describe aprocedure for doing so in sufficient detail so that someone with no knowledge ofeconomics could implement it.12. Consider an exchange economy with two types of infinitely-lived consumers, each ofwhich comprises one-half of the economy’s population. The two types of consumershave identical preferences over consumption streams given by:E ∞Xt=0βtlog(ct)!,where β ∈ (0, 1). The two types of consumers differ in their endowments of the (non-storable) consumption good. Type-1 consumers have a constant endowment of 1 inevery period. Typ e -2 consumers have a stochastic endowment: in each period, it is1 with probability π and 0 with probability 1 − π, where the probability π does notdepend on time or on the previous realization shocks. Markets are complete.(a) [9 points] Carefully define a competitive equilibrium with date-0 trading for thiseconomy.(b) [9 points] Find the prices of the Arrow securities in this economy. Show yourwork.(c) [9 points] A two-period risk-free bond is a sure claim to one unit of the consump-tion good two periods from now. Express the price of a two-period risk-free bond(relative to the price of the current consumption good) as an explicit function ofthe Arrow security prices. Your answer should depend on the current state of theeconomy. (Note: You can answer this question even if you were unable to answerthe question in part (b).)3. Consider an exchange economy with two types of infinitely-lived consumers, each ofwhich represents half of the economy’s population. Each consumer has the same endow-ment stream {ωt}∞t=0. A consumer of type i, i = A, B, has preferences over consumptionstreams of the form∞Xt=0βtc1−σit− 11 − σi,where β ∈ (0, 1). Assume that σ−1A> σ−1B> 0: the elasticity of intertemporal substi-tution of type-A consumers is larger than that of type-B consumers. Consumers tradea risk-free bond in each period (i.e., a sure claim to one unit of the consumption goodin the next period). There is no restriction on borrowing except for the no-Ponzi-gamerestriction. Each consumer begins with zero asset holdings.(a) [10 points] Carefully define a sequential competitive equilibrium for this econ-omy.2(b) [10 points] Suppose that ωt= ¯ω for all t. Show that, in this case, the economyhas a steady state. Express the steady-state interest rate in terms of primitives.(c) [7 points] Now suppose that the endowment grows over time: ωt+1= (1 + g) ωt,where g > 0. In this case, does there exist a steady state, that is, an equilibriumin which the consumption of a type-A consumer grows at the same rate as theconsumption of a type-B consumer? Explain why or why not.4. Consider a ne oclassical growth model in which the government has gove rnment expen-ditures equal to g in every period. The government finances these expenditures bytaxing labor income at a proportional rate τtin period t. The government balances itsbudget in every period: for all t, the tax rate τtis chosen so that period-t expenditures(i.e., g) are exactly equal to the revenues raised by the lab or income tax.The economy is populated by a continuum (of measure one) of identical consumers.Each consumer has preferencesP∞t=0βt(log(ct) + A log(`t)), where ctis period-t con-sumption, `tis period-t leisure, and β ∈ (0, 1).Consumers do not value government expenditures, so that government expendituresare a pure drain on output in this economy: the aggregate resource constraint is¯yt= ¯ct+ ¯xt+ g,where ¯ytis aggregate output in period t, ¯ctis aggregate consumption in period t, and¯xtis aggregate investment in period t.Each consumer is endowed with k0units of capital in period 0 and with one unit oftime in each period. In each period, consumers rent the services of capital and labor tofirms in competitive markets. Firms maximize profits and produce output accordingto: y = kαn1−α, where y is output, k is capital, and n is labor supply; the parameterα ∈ (0, 1). Finally, capital accumulates according to kt+1= (1 − δ)kt+ xt, whereδ ∈ (0, 1].(a) [10 points] Carefully define a recursive competitive equilibrium for this economy.(b) [10 points] Show that the steady-state capital-to-labor ratio does not depend ongovernment expenditures g.(c) [6 points] How do changes in g affect aggregate labor supply in the steady state?If you c annot obtain a complete answer, then display an equation that relatessteady-state labor supply to g and