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)
