Macroeconomic Theory: Steady State Analysis and House Prices, Study notes of Introduction to Macroeconomics

Download Macroeconomic Theory: Steady State Analysis and House Prices and more Study notes Introduction to Macroeconomics in PDF only on Docsity! Econ 712 Macroeconomic Theory- First Exam. University of Wisconsin Instructor: Rody Manuelli October 16, 2004 1 Instructions • Please answer all questions. If you get stuck in one section move to the next one. Do not waste time on questions that you find hard to solve. • Partial credit will be awarded if it is clear that you were approaching the prob- lem in an essentially correct manner. • This is a closed book exam. Students may bring one page (both sides) or two pages (single sides) of notes. • Please hand in the exam promptly at 12 Noon • Each question is worth 50 points. The point total for each section is indicated at the beginning of the section. Look at these “prices” when deciding how to allocate your time!! • If you believe that a question is wrong or poorly worded, please make the “minimal” necessary changes to make it “beautiful” and well posed. Of course, unnecessary changes will result in a lower grade. • Please use one blue book for each question and write only on the “right” page. (The odd numbered page in a newspaper.) • Please remember to put your name in each blue book. • Good luck ! 1 2 Questions Problem 1 (Habit Persistence) Consider an economy populated by a large num- ber of households with utility functions given by ∞X t=0 βtu(ct − φzt), 0 < β < 1, φ ≥ 0 where u : R+ → R, is twice differentiable, increasing and strictly concave (if necessary you may assume that it satisfies the Inada condition). The variable ct is individual consumption, and zt is a measure of lagged consumption. To be precise, zt ≥ ∞X j=0 (1− δc)jct−1−j, 0 ≤ δc ≤ 1. It follows that, alternatively, it is possible to describe the law of motion for zt as zt+1 ≥ (1− δc)zt + ct. In this setting, zt is a measure of ‘habit persistence,’ as it implies that the marginal utility of any given level of consumption decreases the higher the level of past con- sumption. The technology in this economy is standard and given by ct + xt ≤ f(kt), kt+1 ≤ (1− δk)kt + xt, where the functions f is strictly concave, increasing and satisfies Inada conditions. Assume δk ∈ (0, 1) 1. (15 points) Let the planner maximize the utility of the representative agent subject to all the feasibility constraints. Argue that, under some condition on (φ, δc) an interior steady state exists and is unique. Describe the condition that (φ, δc) has to satisfy. 2. (10 points) What does the model say about the impact of cross-country differ- ences in how much people care about past consumption –as measured by φ– on the steady state output per worker. 3. (10 points) Define a competitive equilibrium in which, in each period, households trade (at least) one period bonds, capital, consumption and investment goods. Assume that consumption is taxed at the rate τ , that is, the cost of purchasing c units of consumption is (1 + τ)c. The revenue produced by this tax is rebated in a lump-sum fashion to the households. 2 state, habit persistence does not play any role in determining consumption since the marginal rate of substitution is given by the discount factor. Moreover, with inelastic labor supply, consumption taxes are not distortionary. For this to be the case, they would have to create a wedge between current and future consumption. However, a constant tax rate does not distort the choice between present and future consumption, as both are taxed at the same rate. Problem 3 (Housing Prices) Consider an economy populated by a large number of identical households with utility function given by E0{ ∞X t=0 βtu(ct, ht)}, 0 < β < 1, where ct is consumption of a nondurable good, and ht is consumption of housing. The function u is twice differentiable, increasing and strictly concave (if necessary you may assume that it satisfies the Inada condition). In this economy, there are I trees per household, with each of them ‘producing’ dit units of nondurable consumption at t. Assume that ct = PI i=1 dit. Each household trades in shares to all the trees, bonds of all maturities, a full set of Arrow-Debreu state contingent securities and housing. One unit of housing costs qt units of non-durable good. Thus, the total cost of a house of ‘size’ ht is qtht. In equilibrium, the supply of housing per household is given by a stochastic process {ht}. 1. (10 points) Go as far as you can describing how to price trees and houses as a function of the exogenous stochastic processes [{dit}, {ht}]. 2. (15 points) Assume that u(c, h) = [cαh1−α]1−θ 1− θ , 0 < α < 1, θ > 0, and that ht = h̄ > 0. Go as far as you can deriving the implications of the model for house prices. How is qt related to the value of the ‘stock market.’ Note: define the value of the stock market as pt = PI i=1 pit, where pit is the price of tree i. 3. (15 points) Assume that u(c, h) = [cαh1−α]1−θ 1− θ , 0 < α < 1, θ > 0, and that ht = µct, µ > 0. Go as far as you can deriving the implications of the model for house prices. How is qt related to {R−1jt }∞j=1 –the collection of prices of j-period bonds? 5 4. (10 points) Let the function u be arbitrary, and let the stochastic processes [{dit}, {ht}] be given. Assume that households do not own the houses they live in; they rent houses from developers. That is, in every period, they pay a certain amount of rent, rt. Go as far as you can to derive the equilibrium per period rent and the price of a house. Does the stochastic process for qt in this environment differ from the one you computed in 1? [Assume that the utility function and the stochastic processes are the same] Solution 4 (Sketch) 1. The household solves maxE0{ ∞X t=0 βtu(ct, ht)} subject to ct + qtht+1 + ∞X j=1 R−1jt bjt + IX i=1 pitsit+1 + Z X q(xt, x 0)zt(xt, x 0)dx0 ≤ ∞X j=1 R−1j−1tbjt−1 + IX i=1 sit+1(pit + dit) + zt−1(xt−1, xt) + qtht. This budget constraint is standard, and we follow the convention that, in period t, the household sells its house (and receives qtht), and then it purchases the house that it will live in next period. The first order conditions include (among others) uc(ct, ht)qt = βEt{uc(ct+1, ht+1)qt+1 + uh(ct+1, ht+1)}, uc(ct, ht)pit = βEt{uc(ct+1, ht+1)[pit+1 + dit+1]}, R−1jt = β jEt ½ uc(ct+j, ht+j) uc(ct, ht) ¾ . Repeated substitution in the first equation implies that the price of the house is given by qt = ∞X j=1 βjEt ½ uh(ct+j, ht+j) uc(ct, ht) ¾ . This expression can also be written as qt = ∞X j=1 βjEt ½ uc(ct+j, ht+j) uc(ct, ht) uh(ct+j, ht+j) uc(ct+j, ht+j) ¾ . Since the price of a tree satisfies pit = ∞X j=1 βjEt ½ uc(ct+j, ht+j) uc(ct, ht) dit+j ¾ , 6 it follows that houses are priced like stocks (trees) with a dividend equal to the marginal rate of substitution between consumption of nondurables and housing. 2. Using the specific utility function and the assumption that the stock of housing is constant, it follows that qt = ∞X j=1 βjEt ½ uc(ct+j, h̄) uc(ct, h̄) (1− α)ct+j αh̄ ¾ . Since the value of the stock market is pt = ∞X j=1 βjEt ½ uc(ct+j, h̄) uc(ct, h̄) ct+j ¾ , it follows that the value of housing is a constant proportion of the value of the stock market. More precisely, the model implies that qth̄ = 1− α α pt. 3. In this case, simple calculations show that qt = ∞X j=1 βjEt ( c−θt+j c−θt (1− α) αµ ) , or qt = (1− α) αµ ∞X j=1 R−1jt . 4. If the household rents the house, then its budget constraint is given by ct + rtht + ∞X j=1 R−1jt bjt + IX i=1 pitsit+1 + Z X q(xt, x 0)zt(xt, x 0)dx0 ≤ ∞X j=1 R−1j−1tbjt−1 + IX i=1 sit+1(pit + dit) + zt−1(xt−1, xt), where rt is the per period rent. It is immediate to verify that, in an interior solution, uh(ct, ht) = rtuc(ct, ht) Now, since a house is identical to a tree that drops dividends given by {rt}, the standard pricing formula for trees (as applied to houses) is qt = ∞X j=1 βjEt ½ uc(ct+j, ht+j) uc(ct, ht) rt+j ¾ , 7