Intermediate Microeconomics questions

1.True or False (give the explanation)
a. A firm has two variable factors and a production function f(x1, x2) =sqrt (2×1+4×2). The technical rate of substitution between x1 and x2 is constant.

b. The Marginal Product of a factor is just the derivative of the production function with respect to the amount of this factor, holding the amounts of other factor inputs constant.

c. A fixed factor is a factor of production that is used in fixed proportion to the level of output.

2. Multiple Choices (give the explanation)
A) Which of the following production functions exhibit constant returns to scale? In each case y is output and K and L are inputs. (1) y=K1/2L2/3 ; (2) y=3K1/2L1/2; (3) y=K1/2+L1/2; (4) 2K +3L.
(a) 1,2, and 4
(b) 2, 3, and 4
(c) 1, 3, and 4
(d) 2 , and 3
(e) 2 , and 4

B) A competitive firm produces output using three fixed factors and one variable factor. The firm’s short run production function is q=163x-2×2, where x is the amount of variable factor used. The price of output is $3 per unit and the price of the variable factor is $9 per unit. In the short run, how many units of x should the firm use.
(a) 20.
(b) 80.
(c) 19.
(d) 40.
(e) None of the above.

C) A profit maximizing competitive firm uses just one input, x. Its production function is q=8×1/2. The price of output is 16 and the factor price is 8. The amount of the factor that the firm demand is.
(a) 10
(b) 22.63
(c) 64
(d) 48
(e) None of the above
3. Consider the production function f(x) =x11/4×26/8. Answer the following: (give the explanation)
(a) What kind of production function is this? Can you draw the isoquant map?

(b) Derive the marginal product with respect to the first input, and compute it for x2=8. Derive the marginal product with respect to the second input.

(c) What kind of returns to scale does this technology exhibit? If a technological break-through changes the exponents to 1/3 and 4/5, what kind of returns to scale would this technology exhibit?

(d) Derive the Technical Rate of Substitution if the exponents of the original production function are both 1.