A paper company generates a large amount of waste byproducts (Q) in its paper pr

Question

A paper company generates a large amount of waste byproducts (Q) in its paper production process daily. The company can either use its incinerator technology (K) to eliminate the waste from its production site or send byproducts to an adjacent, government-approved, waterway to remove the waste off site (in effect, water (W) becomes an input to its disposal process). The resulting paper waste removal function is Q=6K1/2W2/3. The marginal product of K is: . The marginal product of water is: . Currently, the company is using 4 hours of incinerator time per day (so K=4) and 8 cubic tons of water to dispose of waste byproducts (So W=8). The per hour cost of incinerator time is $5.00 and cost of using the waterway is $2.50 per cubic ton.

a.         Does this production function exhibit increasing, decreasing, or constant returns to scale? Illustrate your answer mathematically.

b.         At these levels of input use, what are the numerical values for MPK, and MPW?

c.         It the company employing the cost-minimizing amount of K and W?

If your answer is yes, explain why.

If your answer is no, explain how you think the company should K and W to improve profitability.

Explanation / Answer

Q = 6K1/2W2/3

So, MPK = dQ / dK = 6 x (1/2) x W2/3 / K1/2 =3W2/3 / K1/2

MPW = 6 x (2/3) x K1/2 / W1/3 = 4K1/2 / W1/3

(a)

To check returns to scale, let us increase for K & W to double their values, that is: K* = 2K, W* = 2W

Q* = 6 x (2K)1/2(2W)2/3

= 6 x K1/2W2/3 x (2)1/2(2)2/3

= Q x 21.17

Q* / Q = 2.25 > 2

So, doubling of both inputs have more than double the output, so this exhibits increasing returns to scale.

(b) Given, K = 4 & W = 8

Q = 6 x (4)1/2 x (8)2/3 = 6 x 2 x 4 = 48

MPK = 3W2/3 / K1/2 = 3 x (8)2/3 / (4)1/2 = 3 x 4 / 2 = 6

MPW = 4K1/2 / W1/3 = 4 x (4)1/2 / (8)1/3 = 4 x 2 / 2 = 4

(c) Cost minimizing input combination is when

MPK / MPW = PK / PW

Here, MPK / MPW = 6/4 = 1.5

PK / PW = 5 / 2.5 = 2

(MPK / MPW) < (PK / PW)

Or, (MPK / PK) < (MPW / PW)

So, company should use more of K (and/or less of W) to achieve least cost equilibrium.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.