Question

Let the period 1 profit of a firm be as follows:
F(K₁) - I₁ - (a/2) [(I₁)^2/ (K₁)]

where a>0 . We omit labor for simplicity.

Profit in future periods is defined analogously. Assume no depreciation for simplicity, so that K₂ = K₁ + I₁ and so on.

a)Assume that the firm maximizes the discounted present value of its current and future profits. Derive the first-order necessary conditions for K₂.

b) Recast this optimality condition in terms of q, where we now define q₁ = 1 + [(aI₁)/K₁]

c)How does this condition differ from the condition we derived in class? Explain your answer in terms of an added future economic benefit from investment in period 1.

d) Interpret the new version of the equation in "asset pricing" terms.
BONUS: What is the "fundamental solution" for q₁?

Answer 1

a. First order condition of K₂ with respect to

F(K₁) - I₁ - (a/2) [(I₁)^2/ (K₁)]

= (a/2) [(I₁)^2/ (K₁)

or = (a) [(I₁)^2/ (K₁)

b Now optimality condition in terms of q₁ put q₁ = 0 we get

0 = 1 + [(aI₁)/K₁]

or -1 = [(aI₁)/K₁]

c. It should differ from the condition because of the putting value and for the future economic benefit for the investment in period 1 the value may differ and it will satisfy the result.

d. The new version of the equation in terms of asset pricing would be the fundamental option when we put q ₁ = 1 so

answer would be [(aI₁)/K₁].