Consider an economy with N inï¬ntely-lived identical agents, who are endowed with one unit of time and whose preferences can be written as
U0 =â X t=0βt logCt.
There are two goods in the economy, a consumption good (c) and an investment good (x), produced by competitive ï¬rms with following technology
Fit (Kit,Nit) = Kα it (AitNit)1âα , i â{c,x},
where Ait+1 Ait= 1 + γi, i â{c,x}.
Let δ denote the depreciation rate, so that
Kt+1 = (1âδ)Kt + Xt.
- Solve the optimization problem of the ï¬rms (denote pit the price of good i in period t, wt the wage rate, and Rt the rental rate of capital) and ï¬nd the
optimal capital-labor ratio and the relative price. Assume that there is free mobility of labor and capital across sectors.
- Solve now the optimization problem of the consumer and ï¬nd the Euler equation. You can normalize the price of the investment good to 1.
- Deï¬ne aggregate ouptut as Yt = ptCt + Xt. Rewrite it as the one-sector output using the clearing conditions for capital, labor, consumption good, and
investment good.
- Suppose that Rt is constant over time. Show that this would imply a growth rate of γx for the capital stock, the aggregate output, and the investment good
production.
- Compute now the growth rate of the consumption expenditure when the capital rental rate of capital is constant, and ï¬nd the value for R that is consistent
with a Generalized Balanced Growth Path. 6. [Extra credit] How would your answers change if the preferences becamePâ t=0 βt log(Ct âc)?
Sample Solution
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