Problem 1

Consider a problem of choosing the level of capital stock by two firms: A and B. For simplicity we will assume that production requires capital only. We will follow the convention established at the lecture that in time period t the level of capital Kt is predetermined, but the firm can choose its future level of capital, Kt+1. Firm A does not own its capital stock, but instead rents it at price r K, the rental cost of capital. The problem of maximizing the value of firm A is given by: max {Kt+1} t=0 X t=0 1 (1 + r) t F (Kt) r KKt Firm B owns its capital stock, and can adjust its level via investment. The problem of maximizing the value of firm B is given by: max {Kt+1} t=0 X t=0 1 (1 + r) t [F (Kt) It] subject to Kt+1 = (1 ) Kt + It

(a) Derive the first order condition of firm A.

(b) Derive the first order condition of firm B.

(c) What condition has to be satisfied for both firms to choose the same level of Kt+1?

(d) Imagine you are the owner of firm C, which rents capital goods to firm A. What would be the maximal level of r K that you could charge this firm?

(e) What would happen if you demanded higher rental rate than one found in (d)?

Problem 2

Consider the following problem of a manager maximizing the value of the firm: max {Lt, In t , Kt+1} t=0 X t=0 1 (1 + r) t " (1 ) K t L 1 t wtLt Kt I n t 2 (I n t ) 2 Kt # subject to Kt+1 = I n t + Kt t = 0, 1, . . . , where is a tax levied on firm's profits, K is firm's capital stock, L are firm's employees, (0, 1) is output elasticity w.r.t. capital, I n is net investment and (0, 1) stands for capital depreciation. Parameter describes the magnitude of capital installation costs. Note that the tax code does not treat installation costs as tax deductible.

(a) Write down the problem in the Lagrangian form and derive the first order conditions.

(b) Find the steady state level of q (the Lagrange multiplier). Is it equal to 1? Hint: by definition in the steady state firm's capital stock is constant and net investment is 0. (c) Find the desired level of firm's capital stock per employee, K/L, treating interest rate r as given.

(d) Suppose the tax on firm's profits is reduced. What happens with the firm's investment if its level of capital stock per employee was at the level from (c) prior to the tax change? (e) What happens with the firm's investment if its level of capital stock per employee was lower than the level from (c) prior to the tax change?

Problem 3

Let's examine the role of taxes in the Solow-Swan model. Imagine that the behavior of an economy may be summarized by the following three equations: Kt+1 = It Kt It = s (1 ) Yt Yt = K t (AtLt) 1 Assume that population grows at rate n and technology at rate g, so that Lt+1/Lt = Nt+1/Nt = 1 + n and At+1/At = 1 + g, respectively. Income in this economy is taxed with rate and the tax revenues are used for government consumption which is useless from the point of view of households.

(a) Transform the three equations into per effective labor form, i.e. divide them by AtLt. Make use of notational convention xt Xt/ (AtLt).

(b) Find the balanced growth path level of capital per effective labor k in this economy.

(c) Discuss the effects of changes in parameters , n, g, s, on the economy's balanced growth path level of capital per effective labor k .

(d) Discuss the effects of changes in parameters , n, g, s, on the economy's balanced growth path level of consumption per effective labor c .

(e) Households care about the level of consumption per capita, i.e. ct. This variable grows at rate g once the economy reaches its balanced growth path. Discuss whether low g or high g is better from the point of view of households.