The Firm [15 marks]
Consider the following modified version of the model of the firm studied in class. The firm lives for two periods, producing output Y in the first period, and output Y 0 in the second period by operating the following technologies:
Y = zF(K,Nd), and Y 0 = z0H(K 0),
where K (K 0) denote physical capital in the first (second) period, Nd is labour demanded in the first period, and z (z0) denotes total factor productivity (henceforth TFP) in the first (second) period. The function F is increasing in both arguments, it is concave, and it satisfies constant returns to scale. The function H is increasing and concave.
The firm hires labour in a competitive market at the wage w. Capital in the first period K is fixed, and physical capital depreciates at a rate of 100%, so that K 0 = I, where I denotes investment.
Assume that the level of future total factor productivity is uncertain. Specifically, z0 = zH with probability φH ∈ (0,1), and z0 = zL with probability φL = 1−φH, where zL < zH.
1. Suppose that the goal of the firm is to maximise the expected discounted value of its profits, and that the firm discounts future payoffs using the interest rate r. Write down the optimisation problem of the firm.
2. Using the optimisation problem of the firm, find the optimality condition which pins down the firm’s investment demand. How does the expected value of z0 affect optimal investment demand? Provide an intuitive line of reasoning.
3. Suppose that the government confiscates production in the second period (i.e., it taxes it at 100%) if future TFP is zH. On the other hand, if future TFP is zL, the government subsidises production in the second period at the rate s per unit of output produced. Write down the new optimisation problem of the firm, and derive an optimality condition that pins down optimal investment demand.
4. Assume the subsidy rate s is such that s <φφHLzzLH . How would the introduction of this policy affect the investment demand of the firm? Justify your answer.

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