Compare equation N&S 10.6 (last dollar rule) to equations 11.45 and 11.46.Prove that equations 11.45 and 11.46 imply that the last dollar rule is satisfied when the firm has made the profit-maximizing input decisions. just think about combining 11.45 and 11.46 into one condition

b)Let's demonstrate the preceding point with an example.For w = $5, r =$15, a production function q = 4 L^0.5 + 12 K^0.5, and a constant price of $50 per unit of q, solve for L and K based on the MRP rules at 11.45 and 11.46, showing your work.( should get 400 for each.)Then, using the values you found, demonstrate that the last dollar rule holds.Finally, show that q = 320, given the L and K that you found.

c)For an output q = 320 and the above production function, solve for L, K and the minimum expenditures using the production function and the RTS = w/r condition.

d)Suppose now that a firm has output q = 100 with the above production function, input prices, and output price.Calculate L and K , and show that the firm is not complying with the MRP conditions.How should it change its behavior?

e)Equation 11.47 is satisfied for this production function; for example,

f_LL = -L^-1.5

The remainder of this problem examines how L and K respond to a change in w.

Fix K at 400; it cannot vary.Let the wage rate w fall from $5 to $4.That is, given fixed w and P, and in the background fixed K.

f)Solve for the new level of L at w = $4.

g)Now let K and L both vary and use the facts of part (b) to solve for K and L, but with the sole change that w = $4.Compare your results to (f).and briefly explain why the concerns it raises are not a problem for this production function.