8.Consider a production function Q = f(L,K) where L and K are, respectively, the number of units of homogeneous labour and homogeneous capital and Q denotes the number of widgets. Let w and r denote, respectively, the wage rate and the rental rate per machine per hour.

a.Graphically depict an isocost line if a firm can only spend a total of E dollars to hire homogeneous labour (L) at a wage rate of w or to rent a machine (K) at a price of r per hour of rental time, or to hire both inputs, labeling both the vertical and horizontal intercepts in your diagram in terms of E, r and w.

b.Suppose that, by spending E on inputs, the maximum quantity of widgets that a firm could produce is Q*. Suppose that the production function for the Q* isoquant indicates that Q*= f(L*, K*) and that E=wL* + rK*. Assume that (L*, K*) is an efficient allocation of inputs. Depict this situation graphically in a fully labeled diagram.

c. The owner of the firm finds that if she were to spend $1 less on hiring labour than she spends when she hires L*, and instead were to spend $1 more on renting a machine than she spends when she rents K*, or alternatively, if she were to spends $1 more on hiring workers than she does when she hires L* and instead, were to spend $1 less on renting a machine than she spends on K*, she would find that she would not gain anything and she would still produce Q* at the lowest possible cost of E. Why must this condition be satisfied whenever a firm is allocating its inputs efficiently?

d.Now suppose that the workers belong to a strong labour union which negotiates a new wage contract, raising the wage rate from w to 2w. Show why the firm could no longer produce Q* if its expenditure cannot exceed E.

e.If the firm could borrow money to fill the order for Q* widgets after the wage rate doubles to 2w, graphically depict the new efficient (i.e., cost-minimizing) allocation of inputs, denoting these as (L, K).

PLEASE EXPLAIN