Suppose that we have an infinitely lived representative firm, whose production function is given by yt = Atf(kt,nt). The way in which the total number of employees that it has on its payroll changes from period t to period t + 1 is nt+1 = (1 )nt + vtqt, where nt denotes the number of workers who are working at the firm in period t, is the probability of exogenous separation of the job and the worker, vt denotes the number of vacancies that the firm posts in period t, and qt denotes the exogenous probability of meeting with a potential worker. Let the interest rate be exogenous and constant across time, i.e., rt = r, for all t. The timing of the model is as follows:

• In period t the firm takes nt and kt as given and uses the production function Atf(kt, nt) to produce, in which At denotes total factor productivity (TFP) in period t, kt is the amount of capital , and f(kt,nt) is the firm's production function.
• The firm pays the operating cost, which is wtnt +F, where wt is the real wage, and F is a real fixed cost of operation, paid if the firm is operating in period t.
• The firm decides the investment and hence the next period operating capital, which is evolving according to kt+1 = (1 )kt + It, where It is the investment in period t. Assume that the cost of buying one unit of capital equals to the cost of buying one unit of consumption good (as usual).
• The firm decides how much vacancies to post in period t, i.e. vt, and for each vacancy that the firm posts, the average cost is , so the total cost of posting vacancies is vt.
• The firm discounts the future with a discount factor 1+r .

1. Based on the first-order conditions, explain how the search aspects of labor markets affect the firm's capital demand decisions. (Hint: Compare the FOC for the capital with the RBC model. Is it changing?). Compute the marginal product of capital and explain why it is equal to what you find that it is equal.

For the rest parts of the question assume f(kt,nt) = kt nt .