and the firm's total number of employees that work to produce output in period t is described by the period-(t+1) job-hiring constraint n,41 = (1-p)n, +9141 and so on for period t+2 and period t+3 and period t+4, etc, The Lagrange function for the firm's profit-maximization problem is AHI f (1, )- Wn - A, . f (n? ) - wn- @ . v. 1+r @ . 142 + ... (1 +r,) . (1+r,+1) + H! (1-p)np, + qFINDv, - np) + Hit ((1-p)mp + 9FIND (FIND V ,HI - 12 ) H142 (1 +r,) (1+[;4) ( ( 1-p)n,41 + 9142 V1+2 - 1742 ) + ..... in which the first two lines contain the infinite-horizon present-value profit function (notice the "..." term at the end of the second line) and the last two lines contain the infinite-horizon present-value job-hiring constraints (notice again the "..." term at the end of the last line). The Lagrange multiplier on the period-t job-hiring constraint is denoted u, , the Lagrange multiplier on the period-(t+1) job- hiring constraint is denoted ,4, the Lagrange multiplier on the period-(t+2) job-hiring constraint is denoted M,42, and so on. a. Based on the Lagrange function presented above, compute the first-order conditions with respect to n,, v, , and V,41 . b. Using the first-order conditions from part a, eliminate the Lagrange multipliers in order to write the period-t job-creation condition - the final solution should be written in a way analogous to the job-creation condition written at the bottom of p. 465 of the textbook (or, equivalently, the job- creation condition that appears at the bottom of slide 83 of the "Search and Matching" Discussion).Convex Costs of Vacancy Posting. A small but growing consensus that has emerged in the past few years is that if the search and matching model were to be used to describe empirically relevant business-cycle fluctuations for unemployment and vacancies in the U.S., one element that needs to be modified is shape of the total vacancy posting function. Using the "long-lasting jobs" framework of Chapter 29, capturing this idea requires a slight modification of the representative firm optimization problem. The modification is that, in any time period t, the total cost of posting vacancies is -.v, , with the exogenous parameter q > 0 (the Greek lowercase letter "varphi") measuring the curvature of the total cost of posting vacancies. Note that the parameter value =1, in hindsight, is the way that we have so far discussed vacancy posting costs. Retaining all of the notation of Chapters 27, 28, and 29, the lifetime profits of the representative firm is A, . f (n? ) - wnp - @ . 12 + 1+r -. V142 - + ... (1 +r,) . (1 +r,) in which (1+r;) is the gross real interest rate between period t and period t +1, (1+r,+1) is the gross real interest rate between period t +1 and period t +2, (1+r(+2) is the gross real interest rate between period t +2 and period t +3, and so on. Exactly like in Chapter 29, the firm's total number of employees that work to produce output in period t is described by the period-t job-hiring constraint n? = (1-p)n + q,"v