Optimal adjustment of labor demand.7 Consider a firm that has decided to raise the labor input from L0 to an undetermined optimal level L T after encountering a wage shock reduction at t ¼ 0. The adjustment of labor input will imply a cost that varies with L’(t), the rate of change of LðtÞ. The problem is to determine the best speed of adjustment toward L T as well as the level of L T itself. Let the profit function be a function of L only: pðLÞ ðwith p00ðLÞ < 0Þ: The cost of adjusting L is assumed to be: C  L0'  ¼ bL02 þ k ðb > 0; k > 0 and L0 s0Þ

The net profit will be: PðLÞ ¼ pðLÞ  CðL0 Þ The problem of the firm is to maximize the total net profit over time during the process of changing the labor input. The firm has to choose not only the optimal L T but also the optimal T. This is a terminal curve problem, where neither the time nor the state variable is preset. Moreover, the profit will also include the capitalized value of the profit in the post T period, which is affected by the choice of L T and T. The profit rate at time T is pðLTÞ and its present value at T, will be pðLTÞ p and the present value as of t0 will be pðLTÞ p ert . The problem is therefore formalized as maximizing the following discounted stream of profits: maximize PðLÞ ¼ Z T 0  pðLÞ  bL02  k  e rt þ pðLTÞ r e rT (1) s:t: Lð0Þ ¼ L0 and LðTÞ ¼ LT ðwith LT > L0 free; T freeÞ Problem (1) is defined as Problem of Bolza, because of the presence outside the integral of the quantity pðLTÞ p epT,which still varies with LT and T. We can convert Problem (1) (see also Alpha Chiang, op. cit. Section 3.4) into the following standard version: maximize PðLÞ ¼ ZT 0   bL02  k þ 1 r p0 ðLÞL0  e rt þ 1 r pðL0Þ (2) The functional now contains an extra term outside the integral, but it is a constant and therefore it does not affect neither the optimal path L(t) nor the optimal value LT and T.

a. Find the numerical optimal path L*(t), the optimal value L T and the optimal time T* with the following profit function, setting the required boundary conditions pðLÞ ¼ L0:5

b. Find the numerical optimal path L*(t) the optimal value L T and the optimal time T* with the following profit function, setting the required boundary conditions: pðLÞ ¼ 2mL  nL2 where m ¼ 1 and n ¼ 0.05. Enumerate and plot the results. In both cases, set Lð0Þ ¼ 10, p¼0.05, k ¼ 0:9 and b ¼ 1.