Exercise 1 A firm is faced following production function: q = K5L5 (which is the same as: q = \\/K\\/L). The price of K and L are respectively r= 90 and w = 10. A firm is producing in a good in the long-run using only two inputs, K and L. (a) Represent in the graph the isocosts lines corresponding to two different total costs (C): C=900 and C = 1800. (b) Identify the optimum combination of inputs that maximizes the output of the firm in the long-run, when total costis 900. (c) Find the optimal combination of inputs for total cost equal to 1800. (d) If the firm wants to produce 36 units of its good, what is the optimum combination of inputs for the firm? Exercise 2 Plot the average and marginal product of labour for the following production function: Q= 10L+ 30L2-L3, What is the relationship between the average and marginal product of labour? For what amount of labour, the average product of labour is maximized? Exercise 3 Consider the production function: Q=LK In the short run, K is fixed at K= 100. The wage rate is 25 and the rental rate of capital is 8. Derive the short-run and long-run cost functions. Exercise 4 Consider a firm with production function Q= min{aK, BL}. The wage rate is w, and the rental rate of capital is r. The firm wants to produce Qo units of output. In the short run, K is fixed at K. (a) Calculate the short-run optimal cost as a function of a, 3, input prices w, r and Q. (b) Calculate the long-run optimal cost as a function of , 3, input prices w, r and Q Exercise 5 Determine the returns to scale associated with the following production functions: (a) Q=L2K? (b) Q=L0-25K0.25 (c) Q=3L+2K (d) Q=min (4L, K). Exercise 6 Consider the production functions (a) Q=LK (b) Q=L+2K Where K is the quantity of capital and L is the quantity of labour. Let w denote the wage rate and r denote the price of capital, where r=4w. Derive the demand functions for labour and capital as functions of output (Q). Exercise 7* A firm manufactures quad bikes. Each quad is made using one frame (x1) and four wheels (x2). Each frame cost 10 and each wheel cost 4. The output of the firm can hence be represented by the production function Q=min (x1, 1/4x2). (a) What is an isoquant? Show the isoquants for producing two quads, three quads and seven quads. (b) Derive the input demands for frames and wheels for the firm. How do they depend on input prices? What is the maximum number of quads the firm can produce if the number of wheels is fixed at 12?7 Assume that the firm is producing the maximum number of quads. What is the marginal product of adding an extra frame? (c) Define the firm's cost function and derive it. What is the cost of producing seven quads