ECON 5011, Macroeconomics (MA), Fall 2023
Assignment #2
Due: Friday, December 15, 2023 5pm
Stochastic Growth Model
In this assignment you will analyze the stochastic growth model (which is a real business cycle (RBC) model without endogenous labor supply). In particular, let’s assume that production technology takes the Cobb-Douglas form:
F(z t;kt; lt ) = z tkt(α)lt1-α
where 0 < α < 1. kt is the capital stock at time t and z t is a technology shock to the economy. Without loss of generality we assume that lt = 1. log(z t) follows an AR(1) process log(z t) = ρ log(zt - 1 ) + εt , εt N(0; σ2 ). Economy’s resource constraint is given by:
ct +it = F(z t;kt; 1) = f (z t;kt ):
The capital stock of the economy evolves according to
kt +1 = (1 - δ )kt +it :
Let the preference of the representative household be
E0
β t log(ct ):
Let’s set β = 0:987, α = 0:40, δ = 0:012, ρ = 0:95 and σ = 0:007.
1. [5 points] Write down the Bellman equation for social planner’s problem and derive the Euler equa- tion. In this economy are social planner’s allocations same with competitive (decentralized market) equilibrium allocations (i.e., are competitive (decentralized market) equilibrium allocations pareto optimal?)? Why or why not?
2. [5 points] Solve for the deterministic steady state value of capital,i.e., capital’s steady state level in the deterministic neo-classical model when z t = 1 At .
3. [50 points] I provide you the Markov process that approximates zt with 11 states (i.e., Z = fz1 ; z2 ; : : : ; z11g).1 Using the Markov process and the parameter values above, solve the value function, V(z; k) by em-
ploying value function iteration method and obtain optimal decision rules, g k(z; k) (policy function for next period’s capital), gc(z; k) (policy function for consumption), etc.2 For each ziplot V(zi ; k)
along the k dimension on the same figure. Namely, the x-axis on your graph will bek and the y-axis will be the value function. There will be 11 lines, one for each zi on your figure.3 Plot another figure for policy function of capital, g k(z; k) in the same manner.
4. [10 points] Bonus Question: For δ = 1 we may have an analytical solution. Guess and verify that
the value function takes the following form when δ = 1:4
V(z; k) = A+ Blogk +Clog(z):
5. [10 points] In the previous question the analytical solution should bek( = g k(z; k) = aβ zka . Com- pare the accuracy of your numerical solution with the above analytical solution when δ = 1. Namely, solve the model numerically by setting δ = 1 (using value function iteration method) and obtain the policy function for next period’s capital and then compare this policy function with k( = g k(z; k) = aβ zka by plotting these two together on the same figure.
6. [10 points] You now show how a shock to the current capital stock affect the optimal choices of consumption (ct ), investment (It), capital (kt +1),and output (yt) by plotting the impulse response functions. For this purpose, suppose the economy is at its deterministic steady state and all of a sudden the current capital stock increases by 1% in period 0. In particular, you’ll assume that k — 1 = k* , z — 1 = 1 and at time t = 0 there is a 1% shock to capital, k0 = 1:01k* , z0 = 1. Then you’ll find kt +1 = g(kt;zt) for t = 0; 1; 2::::5 Once you have the series offkt +1gAt, it is easy to findfytgAt , fct gAt, and fIt gAt .
7. [20 points] In this part, you are required to compute the amplitude of fluctuations (volatility mea- sured by standard deviation) in aggregate variables and their correlation with aggregate output. For this purpose, let δ = 0:012 and simulate the economy for a long period of time.starting from some initial level of z0 and k0 (probably steady state value). Namely, once you have the policy functions you can easily simulate the model. Start with an initial condition for the capi- tal stock, say the deterministic steady state level (k0=k* ), and with an initial condition for z; say z0 = 1. Then draw a sequence of technology shocks {zt400 using the Markov transition ma- trix for 400 periods. Use the policy functions, g k(z; k) and gc(z; k) to determine the time series {kt +1 ; ct
{kt +1 ; ct 0(4)1(0)0 ), compute long run averages for consumption, investment, and output, and percentage deviations of these variables from their long-run averages.
For example, for capital long run average is given by k(¯) = 0(4)1(0)0 kt +1,/200. Then percentage deviation from the long run average is defined by k(ˆ)t = kt k(¯). Then, compute standard deviation of these deviations (e.g., standard deviation of {k(ˆ)t +10(4)1(0)0 ) and their correlations with deviations in output (e.g., correlation between {k(ˆ)t
US data below:
Table 1: Fluctuations of the output components
|
|
St. Dev. (σ) of % dev |
Correlation with output |
|
Output |
1.72 |
1.00 |
|
Consumption |
1.27 |
0.83 |
|
Investment |
8.24 |
0.91 |
|
Hours |
1.6 |
0.86 |
Source: Cooley (1995)
8. [50 points] Bonus question: Now solve the stochastic neo-classical growth model with endogenous labor supply (the standard RBC model). Assume the following preferences:
E0
β t [log(ct ) + blog(1 -lt )]
where ct and lt are consumption and labor supply, respectively. Let’s set b = 2. All other features of the model are same with the exogenous labor supply case (with δ = 0:012). Solve this model using value function iteration and compare your results from the model to the post-war US data and to the results of the exogenous labor supply model.
Instructions
1. You can form a group of two. The program name should include the names of the group members in the following format “Project1 FirstLastName1 FirstLastName2.m”
2. Write comments in the program before each major step so that an outsider who reads your program can understand what you are doing and you can understand what you did when you later check your program.
3. Email your programs to Brian Prescott ([email protected]) or upload to the Canvas.