March 25, 2004 Kjetil Storesletten. Macroeconomic Topics Homework 1 Due: April 23 1 Theory 1.1 Aggregation Consider an economy consisting of a continuum of agents of measure 1 who solve max P t=0 βt c 1 γ it / (1 γ). Half the population have efficiency units e 1 =0.5, and the other half have efficiency units e 1 =1.5, so the aggregate labor input is N =1. Agentsderiveincomefromlabor,w t e i, and capital, r t k it,wherew t and r t are aggregate wage rate and interest rate in period t, respectively, and k it is capital holdings of agent i in period t. The aggregate production function is Y = KN,whereK and N are aggregate capital and labor input, respectively. Capital does not depreciate. 1. Define a competitive equilibrium, assuming that prices correspond to marginal productivities. 2. Compute steady-state capital stock K and interest rate r. 3. What is the distribution of wealth in steady-state? 4. Suppose an extreme politician holds power for a brief period of time (Giske) and redistributes all wealth evenly. How does this redistribution of wealth affect future accumulation of capital? Describe the transition towards steady-state from an initial capital stock level of K 1 <K. 1 Make sure you do all the exercises. If you have problems, please do not hesitate to contact me by email. Good luck! 1
1.2 A stochastic endowment economy Consider a stochastic pure-exchange economy. There are two people, i =1, 2, in the economy who live forever. The stochastic endowment process for each household is given by e 1 t = s t for all t 0 e 2 t = 1 for all t 0, where s t is a random variable evolving according to a two-state time-invariant Markovchainwithvaluess t = s =0and s t = s =1. Each consumer values sequences of state-contingent consumption {c i t(s t )} t=0,s t S according to t+1 u i (c i )= X X t=0 s t S t+1 β t π t (s t )ln(c i t(s t )), where π t (s t ) is the probability of the history s t =(s 0,s 1,..., s t ). 1. Define an Arrow-Debreu equilibrium for this economy. Be careful to specify all the elements of the equilibrium. 2. Compute the Arrow-Debreu prices in this economy. 3. Define and solve the social planner s problem for this model. Obviously the solution will be indexed by the Pareto weight α (0, 1), the weight on the utility of the first consumer. 4. Describe the relationship between the solutions (indexed by α) of the social planner s problem and the Arrow-Debreu equilibrium. 5. Define a sequential markets equilibrium for this economy. Be careful to specify all the elements of the equilibrium. Define the natural borrowing limit for each consumer. 6. Compute the price of one-period state-contingent bonds for the following transition matrix:.6.4 P =.1.9 and with β =.96. How many of these bonds are there? 2
2 Robinson-Crusoe Consider a finite-horizon Robinson-Crusoe economy. The single household has a utility function over consumption of fish, c t,andlabor,l t,givenby TX β t [ln(c t )+γln(1 l t )], t=0 where 0 <β<1. Crusoe has a technology which transforms labor into output of fish given by y t = A t lt α, where 0 < α < 1, and{a t } T t=0 is a sequence of numbers. The fish can only be used for consumption. (Leftover fish rots. Crusoe does not have a refrigerator). 1. A social planner wishes to maximize Crusoe s utility subject to appropriate technology constraints. Set up the social planner s problem. 2. One interpretation of the production technology is that there is a fixed supply of one boat (actually a catamaran, designated by the symbol k) so that the technology is y t = A t k 1 α t l α t. This technology has constant returns to scale. Assume that k t is owned by Crusoe and that the technology is operated by a firm which rents capital, k t (catamarans), and labor, l t,inacompetitivemarketatfactor prices r t and w t.letp t denote the Arrow-Debreu price of one unit of consumption at date t. (a) Set up Crusoe s decision problem as a household problem. This household is endowed with k units of capital in each period and must choose how much to consume and how much to work in each period. (b) Set up the firm s problem. (c) What are the resource constraints? (d) Define a competitive equilibrium. 3
(e) Compare a competitive equilibrium to the planner s problem above. (f) Solve for the competitive equilibrium allocations. (g) How does the competitive equilibrium change when A 0 rises? When A 1 rises? 3 Numerical analysis 3.1 Warm-up The program solve_cd.m on the course web (in the file http://folk.uio.no/kjstore/teaching/top solves equation (3) in Floden s notes (Lecture March 18). 1. Write an equation solver that takes as input a one-dimensional function f and uses Newton s method to find a root x such that f (x) =0. 2. Replace fzero.m in solve_cd.m with your own equation solver and check that you obtain similar results. 3. Experiment with different values for µ and σ in solve_cd.m. Whatis the relationship between precautionary savings s and these parameter values? 4. Use qnwnorm.m in CompEcon to implement Gaussian quadrature instead of Monte-Carlo integration. Start with a small number of nodes (for example 5). Increase the number of nodes until the results are stable. How many nodes do you need? 5. Assume that the utility function is u (c, h) =logc + b (1 h)1 1/γ 1, 1 1/γ and assume γ =0.3 and b =(1 α) 1/γ /α. Solve for c 1, h 1,ands 1 using analytical tools as far as you can. What equations do you have to take to the computer? 6. Solve the model in (5) numerically. 4
3.2 Deaton/Aiyagari 1. Assume agents solve max X t=0 β t c 1 γ t 1 γ subject to the following constraints holding for all t: c t + a t+1 = we t +(1+r) a t a t+1 0 c t 0, where e t is i.i.d with e t {0.2, 1.8} and P (e =1.8) = 0.5. Assume β =0.95, r =4%,andthatw =1. (a) Solve for the decision-rules using the collocation method. Use a 20- degree approximant. [Hint: the program deaton_iid_shell.m onthecoursewebcontainssomeofthecodeyoumaywantto use.] (b) Replicate figures 1-2 in Deaton (1991). Does your solution seem OK? (c) What is the aggregate wealth in this economy? 2. The economy consists of a continuum (of measure 1) of agents like the ones above, where e it denotes agent i 0 s effective hours of work in period t. Using the law of large numbers, the average labor input is (in steady-state) N =1. Assume that the economy is closed and that the constant return to scale production function is given by Y = K θ N 1 θ = K θ, with θ =0.4. The capital stock evolves according to K t+1 =(1 δ) K t + INV t, where INV t is investments in period t and δ =10%is a depreciation rate. 5
(a) Define a competitive equilibrium for this economy. In particular, explain why one equilibrium condition should be that prices equal marginal productivities in this economy, i.e. that w t = (1 θ) K θ r t = θk θ 1 δ. (b) Compute the equilibrium values of r and w (obviously, in equilibrium those values differ from the values assumed in exercise 1 above). 6