Macroeconomics Coursework Assignment
Consider a discrete-time infinite-horizon real business cycle (RBC) model aug- mented for investment shocks. The economy features two agents: i) a rep- resentative household who consumes, supplies physical capital to the firm (i.e., invests), supplies labor to the firm, and receives the firm's profits (if positive); ii) a representative firm that rents capital and demands labor from the household and pays the householdís wage and capital incomes.
The representative household derives utility from consumption (Ct ) and ex- periences disutility associated with labor (Nt ) according to the following sepa- rable lifetime utility function:
where E0 represents the expectations operator, 0 < β < 1 is the householdís discount factor, and b > 0 is the constant weight attached to the disutility from labor.
The household supplies labor Nt to the firm and receives its wage bill WtNt ; with Wt representing the wage rate. Denoting Kt as the existing capital stock, rt as the rental rate on capital, and It as the total capital investment spending, the householdís budget constraint is:
Ct + It ≤ WtNt + rt Kt : (2)
Physical capital evolves according to:
Kt+1 = (1 — δ) Kt + μt It ; (3)
where 0 < δ < 1 is the capital depreciation rate. The shock μt to the marginal e¢ ciency of investment represents an exogenous disturbance to the process by which investment goods are transformed into installed capital to be used in pro- duction (e.g., Greenwood, Hercowitz, and Hu§man 1988; Justiniano, Primiceri, and Tambalotti 2010, 2011). The shock μt follows an auto-regressive AR(1) process:
μt = (μt - 1 )Pμ exp (sdμ ."t(μ)) ; (4)
with 0 < Pμ < 1 denoting a persistence (auto-correlation) parameter, and "t(μ) are random mean-zero, serially-uncorrelated white-noise shocks with a constant standard deviation sdμ > 0.
Finally, the firm produces output according to a standard constant-returns- to-scale (CRS) production function:
Yt = At Kt(α)Nt1 -α: (5)
where At is a total factor productivity (TFP) shock that follows an AR (1) process and satisfies A = 1 in steady state.
Please Answer the Following Questions:
1. Taking the investment tax rate, the wage rate, and the rental rate of capital as given, formulate the householdís dynamic optimization problem and calculate the first-order conditions with respect to Ct ; Kt+1 ; and Nt : Define λt as the Lagrange multiplier on the householdísbudget constraint (2) in the optimization problem and substitute (3) in (2) for It. What is the marginal utility of consumption and how is it related to the interest rate rt and the investment shock μt? Explain the Euler equation with respect to the physical capital stock.
2. The firm maximizes its profit function Πt = At Kt(α)Nt1 -α — WtNt — rt Kt :
Explain the profit function and calculate the firm's first-order conditions with respect to Nt and Kt taking input prices Wt and rt as given. Explain the intuition behind the optimality conditions and why the firm earns zero profits in equilibrium. In particular, show that WtNt + rt Kt = Yt :
3. Combining (2), (3), (5), and the firm's zero profit condition derived in question (2), derive and explain the economyís market clearing condition.
4. Write down and explain the general equilibrium conditions of an economy with TFP and investment shocks. .
5. Derive the steady state equilibrium conditions of the model. In the steady state you may assume that there are no investment shocks, i.e., μt = μ = 1. Explain your answer.
6. Assume that the parameters of the investment shock are set to Pμ = 0:95 and sdμ = 0:01. Simulate the model following a 1% drop in μt : On the same graph, compare the model-implied dynamics of this investment shock with those resulting from a 1% fall in At : Set the TFP persistence parameter and standard deviation to PA = 0:95 and sdA = 0:01, re- spectively. All other structural parameters should remain consistent with those used in the baseline RBC simulations from the lecture slides and the RBC_Basic.mod file. Discuss the general equilibrium e§ects of each shock, and explain the transmission channels involved. For the purpose of displaying your results, please simulate the log-linear annualized dynam-ics of Output, Consumption, Investment, Capital, Labour, MPK, Interest
Rate, Wage Rate, and the Investment / TFP Shock (a 3x3 display in the display matlab file) . Read Greenwood et al. (1988); Justiniano et al. (2010, 2011); Brinca et al. (2016) to build further intuition.
7. Compare the modelís standard deviations of (log) output, (log) invest- ment, and (log) consumption following a TFP shock and a combination of TFP and investment shocks to their U.S. data counterparts from 1970:Q1 to 2020:Q1, obtained from Federal Reserve Economic Data (FRED) pro- vided by the St. Louis Fed. Ensure that the empirical data series are converted to quarterly percentage changes to align with the logarithmic model variables. Create a simple table showing the standard deviations for the data, TFP shock, and TFP + investment shocks. Which model better explains the variability of the key variables in the data?