Question

NEED ANSWERS OF PART (f,g,h,j)

Problem 2 [21 marks] Consider a firm that uses two inputs. The quantity used of input 1 is denoted by x, and the quantity used of input 2 is denoted by x2. The firm produces and sells one good using the production function f(x1, x2)-4x053x25. The final good is sold at price P $10. The prices of inputs 1 and 2 are w$2 and w2 $3, respectively. The markets for the final good and both input goods are treated as competitive markets by the firm, that is, it takes prices as given a) Show whether the production function has increasing, decreasing, or constant returns to scale. [2 marks] b) Find the marginal product of each input. Show whether the production technology obeys the law of diminishing marginal products. [2 marks] c) Draw the isoquant for an output level of 12. Clearly label the axes and the curve and show any two input bundles on the curve by indicating their coordinates. [2 marks] d) Does the firm have convex production technology? Explain. [2 marks] e) Find the technical rate of substitution. Does the technology show diminishing technical rate of substitution? Explain. [2 marks] Assume in the short run that x2 is fixed at x2 100 f) Write down the firm's profit function and the firm's short run profit maximisation problem. Find the firm's optimal use of input 1, the associated optimal quantity of the output good, and the firm's profit level. [4 marks] Now consider the long run, where the quantity of input 2 can be varied. g) According to your answer in part a), does the firm have a profit maximising plan in the long run? If no, explain why. If yes, is the plan unique? [2 marks] h) Write down the firm's profit function and the firm's long run profit maximisation problem. Find the firm's optimal use of input 1, input 2, the associated optimal quantity of the output good, and the firm's profit level. [4 marks] j) Explain why the profit level in the long run must be at least as high as the profit level in the short run. [1 mark]

Answer 1

a) Let the quantity of $x_1$ and $x_2$ increase $\alpha$ times to
X-
and $\alpha x_2$ respectively. The firm's production function is:

0.5 05 lax1,ax) - 4(aX1 +3(aX2)0 a5 (4x95 +3x25) a05f(xt, x2)

Hence, increasing both inputs by $\alpha$ increased production by less than $\alpha$ . Therefore, this production function exhibits decreasing returns to scale.

b) Marginal product of each input can be computed by differentiating the production function with respect to the two inputs individually:

2 05, MP MP2X05, MP 1.5 1.5x05, or MP 05 X

It is clear that as quantities of $x_1$ and $x_2$ are increased, their Marginal Products fall. Hence, the production function obeys the law of Diminishing Marginal Product.

c) The following image shows an isoquant for a level of production of 12.

X1 -15 0,15 10 Isoquant for output level of 12 5 9,0 10 0 15 X2 LO LO LO

d) Yes, since the production function is convex to the origin, the production technology is convex.

e) The firm's MRTS (Marginal Rate of Technical Substitution) is the ratio of the two inputs' marginal products.

4x05 2 MRTS 3x0.5

As the quantity of $x_1$ is increased, the MRTS falls. Hence, the MRTS is diminishing.