Question 7: Hiring, Job Openings, and Matching. Consider the representative firm's profit maximization framework in a few ways. First, let's think in terms of one period, rather than the two-period analysis. Second, suppose that the physical capital stock is k = 1. (Note that no time subscripts are needed because the analysis that follows is entirely within "one period" of time). Third, despite the seeming lack of importance of physical capital, there is still diminishing marginal product of labour in the representative firm's production function.

The production function is: f(n) = An^1- where A > 0 represents "total factor productivity," belongs to (0, 1), and the job-hiring constraint is n = qv.

Here v denotes the quantity of job opening (aka job vacancies) and q belongs to (0, 1) denotes the probability that any job opening finds a suitable employee. Hence, the probability of not filling a job is (1 - q).

w: (the Greek lowercase letter "omega") denotes the cost incurred by the firm to "advertise" each job opening. Think of this as the time and expenses that managers and the human resources (HR) department would have to spend in order to attempt to hire a new employee.

1. Set up the firm's profit function and the Lagrangian function and compute the FOC with respect to labour demand n job openings v.
2. Based on the two FOC you computed above, construct this framework's "labour demand function"
3. Starting from the "labour demand condition" you obtained in part b, suppose for this part only that q = 1. How does your solution in part b compare to the "labour demand function" obtained in the classical model? Describe briefly in both mathematical terms and in terms of economics.
4. Now suppose the HR-related costs of advertising job opening is w = 0. With q < 1 and w = 0, how does your solution in part b compare to the "labour demand function" in the classical model? Describe briefly in both mathematical terms and in terms of economics.