1. Profit Functions and Optimal Investment

Assume that a firm uses the production function for output Y = F (A, K, L) given capital K, labor L, and productivity A. The function is given by

F (A, K, L) = A(K^ρ + L^ρ)

where 0 < ρ < 1.

(a) If the firm faces the wage rate W per unit of labor L, then write down an expression for the variable profit function Π(K; A, W ) defined by

Π(K; A, W ) = max [F (A, K, L) WL] .
                      L

(b) Assume that the firm with the profit function from above will live for two periods and does not face uncertainty. The firm must choose its period 1 investment level I1 to maximize the present discounted value of dividends

max D1 + D2/1 + r

where D1 = Π(K1; A, W ) -I1, D2 = Π(K2; A, W )+(1 -δ)K2, and K2 = I1 +(1 -δ)K1.

Derive the optimality condition, or intertemporal Euler equation, for investment I1. Interpret this condition, in words.

(c) Given optimal investment choices, what is the elasticity of optimal capital K2 with re- spect to the user cost of capital r + δ?

(d) Given optimal investment choices, what is the elasticity of optimal capital K2 in the second period with respect to productivity A?

2. Unexpected Failure

Consider a firm which lives for two periods t = 1, 2 and faces uncertainty over the capital depreciation rate δ2 in period 2 according to

         1,    with prob. q

δ2 =

          0, with prob 1 - q

where 0 < q < 1. Intuitively q is the probability that, after being used for production in period 2, the capital of the firm will unexpectedly fail and become useless in the capital resale market. Recall that the objective of a firm is to maximize the expected discounted value of dividends, i.e.

max D1 + ED2/1 + r,
I1

s.t. D1 = Π(K1; θ) - I1 D2 = Π(K2; θ) + (1 - δ2)K2

K2 = I1 + K1

Note that in principle, a depreciation rate δ1 for period 1 may also exist which is uncertain in the same manner. However, the equations above implicitly assume that the firm makes its investment choice I1 after observing that δ1 = 0, so this is no longer a random variable.

Also, note that the profit function is written in terms of a profitability index θ, which is constant and certain in both periods.

(a) Write a formula for the expected value of dividends in period 2, ED2, taking as given a level of investment I1.

(b) Derive the optimality condition for investment I1. Interpret this equation, in words.

(c) If q increases, does investment I1 increase or decrease? You may assume that ΠKK ≤ 0. Explain the intuition behind this result.

(d) Assume that the firm faces a financial friction in the form of a borrowing constraint given by D1 ≥ 0. Under what conditions does this borrowing constraint bind?

(e) If the probability of capital failure q in the next period increases, is the firm more or less likely to be borrowing constrained? Explain the intuition behind this result.

3. The "Permanent Profitability Hypothesis"

Consider a firm which solves the infinite horizon value maximization problem subject to uncertainty given by

 max Et s= ⁰Σ_∞ (1 + 1/r)^sD_t+s,

{I_t+s}^∞_s=0

s.t D_t+s, = Π(K_t+s; θ_t+s) - I_t+s

K_t+s+1 = I_t+s + (1 - δ)K_t+s.

Here, capital prices are normalized to be equal to the constant level P_Kt = 1. The firm's profit function takes the Cobb-Douglas form Π(K_t+s; θ_t+s) = θ_t+sK_γ for all s ≥ 0, where γ ≤ 1. The uncertainty that the firm faces involves potential fluctuations in θ, with

log(θ_t+s) = θ¯ + ε_t+s.

Here, θ¯ > 0 is the constant permanent of mean level of log profitability for the firm in all periods, while ε_t+s is a transitory shock to profitability which is independent across periods and normally distributed with ε_t+s ∼ N (0, σ_ε²).

(a) Write the optimality condition for investment It. Explain the intuition behind this equation in words.

(b) Assume that γ < 1. Write an explicit formula for desired capital K_t+1 in terms of θ_t, σ_ε², γ, r, and δ. Hint: If log X ∼ N (µ, σ²), then EX = eµ + σ²/2.

(c) If the transitory profitability level εt increases, the firm's profits in period t increase. Does investment It in period t increase, decrease, or stay the same?

(d) If the permanent profitability level θ¯ increases, does investment in period t increase, decrease, or stay the same? Contrast your answers to this question and part (c).

(e) Assume that γ = 1, and assume that the firm faces quadratic capital adjustment costs AC(I) = φ I2 where φ > 0. Dividends Dt+s can now be written as

D_t+s = Π(K_t+s; θ_t+s) - I_t+s - AC(I_t+s).

Define Tobin's Q to be qt = 1 + φIt. Write the optimality condition for investment It in terms of qt and q_t+1. Explain the intuition behind this equation, in words.

(f) Iterate forwards on this equation, as shown in class, to derive an expression linking qt to the expected present discounted value of profitability θt+s, adjusting for depreciation. Then, write this formula for qt only in terms of the parameters r, δ, θ_t, and σ².

(g) Following the lecture notes, derive a formula for investment It in terms of Tobin's Q qt. Then, write this formula for It only in terms of the parameters r, δ, θ_t, σ2, and φ.

(h) If the transitory profitability level εt increases, the firm's profits in period t increase. Does investment It in period t increase, decrease, or stay the same?

(i) If the permanent profitability level θ_t increases, does investment in period t increase, decrease, or stay the same? Compare your answers to parts (h) and (i) with your an- swers to parts (c) and (d).

(j) Assume that the business in question is a doctor's office, which must continually invest in new equipment. Assume that in period t all of the medical personnel in the office contract a severe but fast-acting stomach illness from a contagious patient. This de- creases their profitability θt in period t. By period t + 1 the staff are expected to make a full recovery. Is this an example of a shift in permanent profitability θ_t for the doctor's office, or a bad realization of ε_t? Based on your answers in this question so far, should the medical firm in period t buy more, less, or the same amount of medical equipment as previously planned?