Macroeconomic Model Building

Fall 2017

Problem Set 5

Due: 20. Nov. 17:00

Note: If we ask for an equation, do not substitute the value of exogenous variables!

Discussion is encouraged, however, you have to solve the problem set individually. In line

with the Code of Conduct of Corvinus University of Budapest, accepting solutions from

others is an act of academic misconduct and will be penalized by a grade of zero to the

assignment. If your logic is correct but you miscalculate the result, you still get half of the

points.

1. (10 points) The representative household aims to maximize

U =X∞t=1βt−1(Xκt + θ ln(1 − Lt))

with respect to its budget constraint, where

Xt = Cψt G1−ψu,t

and the useful government expenditure is

Gu,t = γGt

The goal of the government is to maximize household utility. The firm is profit maximizing, and its production function is

Yt = atKαt L1−αt

We know that government expenditures are financed by lump sum taxes in each period

(Gt = Tt).

(a) Write down the Lagrangian of the household (0.2 points)

(b) Derive all the FOCs of the household (0.8 points)

(c) Derive the Euler equation of the household in general, and in steady state (0.4

points)

(d) Derive the labor supply of the household in general, and in steady state (0.4 points)

(e) Derive the capital supply of the household in general, and in steady state (0.4

points)

(f) Derive the steady state investment function (0.2 point)

(g) Derive the labor demand of the firm in general, and in steady state (0.4 points)

(h) Derive the capital demand of the firm in general, and in steady state (0.4 points)

(i) What is the steady state production function? (0.2 point)

Macroeconomic Model Building – Problem Set 5 – Page 2 of 2 Due: 20. Nov. 17:00

(j) State the goods market clearing condition in general and in steady state (0.4 points)

(k) State the assets market clearing condition in general and in steady state (0.4 points)

(l) State the labor market clearing condition in general and in steady state (0.4 points)

(m) State the capital market clearing condition in general and in steady state (0.4

points)

(n) Write down the Lagrangian of the government (0.2 points)

(o) Derive the FOC of the government (0.4 points)

(p) Derive a decision rule for the optimal government expenditure in general and in

steady state (0.4 points)

(q) Derive a way to solve this model in MATLAB for any plausible parameter values

(only in steady state) (2 points)

(r) Create a plot with the steady state government expenditure on the Y axis, and

different values (between 0.01 and 0.99) for the parameter ψ on the X axis. Attach

this plot, and explain what is shown in it! (Note: you can set all other parameters

of the model to any plausible value) (0.5 points)

(s) Create a plot with the steady state consumption on the Y axis, and different values

(between 0.01 and 0.99) for the parameter ψ on the X axis. Attach this plot, and

explain what is shown in it! (Note: you can set all other parameters of the model

to any plausible value) (0.5 points)

(t) Create a plot with the steady state government expenditure on the Y axis, and

different values (between 0.01 and 0.99) for the parameter γ on the X axis. Attach

this plot, and explain what is shown in it! (Note: you can set all other parameters

of the model to any plausible value) (0.5 points)

(u) Create a plot with the steady state consumption on the Y axis, and different values

(between 0.01 and 0.99) for the parameter γ on the X axis. Attach this plot, and

explain what is shown in it! (Note: you can set all other parameters of the model

to any plausible value) (0.5 points)