Question

Suppose that the production technology described above can be captured by the production function Q = 40S0.25 R0.25, where Q is your exam grade, S is the number of hours spent studying, and R is the number of hours spent resting. MPs = 105-0.75 R0.25 and MPR = 10S0.25 R-0.25 (d) Prove that this production process indeed have decreasing returns to scale. (e) What is the equation of isoquant? (f) What is the equation for a slope of an isoquant? What is this called? What does it indicate? (g) Set up the cost minimization problem and solve for the conditional studying and resting demands as functions of ps(what you are willing to pay to get back an hour of studying), PR (what you are willing to pay to get back an hour of sleep), and Q (your exam performance). (h) Discuss the demand functions you derived in g). Is rest and studying normal inputs? What happens to optimal amount rest as pr increases? What happens to optimal amount of studying as ps decreases? (i) Now, assume that you are always willing to pay $5 to get back an hour of sleep and $20 to get back an hour of studying. What is your "optimal production plan”? (j) Derive the cost function and simplify the function as much as you can. (k) Continue with total cost function derived in part j) and derive the av- erage cost. Are marginal and average cost curves for this problem upward or downward sloping? Explain. What is the relationship be- tween MC and AC? (1) What is your "optimal production plan” if you wish to reach the exam performance of 100? (m) What is the cost of this 100-score performance?

Answer 1

Q = 40S^0.25R^0.25

(d)

When both inputs are doubled, new production function is

Q* = 40 x (2S)^0.25(2R)^0.25 = 40 x 2^0.25 x 2^0.25 x S^0.25R^0.25 = 2^0.5 x 40S^0.25R^0.25 = 2^0.5 x Q

Q*/Q = 2^0.5 < 2

Since doubling both inputs less than doubles output, there is decreasing returns to scale.

(e)

40S^0.25R^0.25 = Q

R^0.25 = (Q/40) / (S^0.25)

R = {[(Q/40)⁴] / S} [equation of isoquant]

(f)

Equation for slope of isoquant is called MRTS.

MRTS = MPS/MPR = R / S

It indicates the number of hours of rest than must be given up to increase one additional unit of studying.

(g)

Utility is maximized when MRTS = pS/pR

R/S = pS/pR

R = S x (pS/pR)

Substituting in production function,

40S^0.25R^0.25 = Q

S^0.25[S x (pS/pR)]^0.25 = Q/40

S^0.25 x S^0.25 x (pS/pR)]^0.25 = Q/40

S^0.5 x (pS/pR)]^0.25 = Q/40

S^0.5 = (Q/40) x [(pR/pS)]^0.25]

Squaring,

S = (Q² / 1600) x [(pR/pS)]^0.5]

R = (Q² / 1600) x [(pR/pS)]^0.5] x (pS/pR) = (Q² / 1600) x [(pS/pR)]^0.5]

NOTE: As HOMEWORKLIB Answering Policy, 1st 4 parts are answered.