Politicians often talk about introducing 'investment tax credits'. If there is investment tax credit, the government refunds

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Question:

Politicians often talk about introducing 'investment tax credits'. If there is investment tax credit, the government refunds a fraction of the investment purchase (or, firms pays discounted price to purchase a capital good). Investment tax credits are like subsidies to the purchase of investment goods. Imagine that instead of price Pt, a firm effectively pays the price Pt (1 − b) to purchase a capital good, where b is the investment tax credit (0 < b < 1).
(a) How would an increase in the investment tax credit affect the investment function? In order to discuss the effect of higher b on the investment function, answer the following questions:
i. A firm borrows money from a bank to purchase 1 unit of capital good (or machine) whose price is Pt at t. Thanks to the investment tax credit, the actual price the rm should pay is (1 − b) Pt. How much should the firm repay at t + 1 when a nominal interest rate is i?
ii. As we learn in class, a capital good is productive at t + 1. If a firm sells the marginal product of 1 unit of machine at the price of Pt+1 at t + 1, what is the firm's gain from the extra production?
iii. Also, a firm re-sells the used machine at Pt+1 at t + 1. Assume that the firm does not have to return the investment tax credit b of the proceeds of the resale. What is the firm's gain from the resale (after depreciation at the rate of δ)?
iv. Find the optimal condition for investment.
v. Draw two curves (one with downward slope and flat one) whose intersection represents optimal K (or optimal investment).
vi. How would an increase in b (investment tax credit) affect the investment?

(b) What would be the effect of an increase in b on output, (real) interest rate, and price level?
From now on, let us assume that the firm HAS TO return a fraction b of the proceed of the resale when it re-sells the used machine at t + 1.
(c) Derive the optimal condition for desired capital stock. The desired capital stock (or investment) is greater compared to (a)?
(d) How would an increase in the investment tax credit affect the investment function? Does investment change more compared to (a)?
(e) What would be the effect of an increase in b on output, (real) interest rate, and price level?