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3.4 Factor prices question Imagine Bavis is producing economics degrees following production function q(L, K) =...

3.4 Factor prices question

Imagine Bavis is producing economics degrees following production function q(L, K) = L^0.5 K^2. In the short run, capital is fixed at ̄K= 5. Bavis faces price p = 150 and can hire as many workers as it would like at a constant wage w = 75.

A. Find equilibrium labor (L∗) and wages.

B. What are Bavis’s profits at this equilibrium?

C. Prove that this profit level is a global maximum.

- Thank you!

Expert Solution

a)

q = L0.5K2

K = 5

q = L0.5(5)2

Short run production function

q = 25L0.5

profit = pq - wL

= 150(25)L0.5 - 75L

= 3750L0.5 - 75L

d/dL = 3750(0.5)L0.5 - 1 - 75

d/dL = 1875 L- 0.5 - 75

put d/dL = 0

1875 L- 0.5 - 75 = 0

1875 L- 0.5 = 75

1875 = 75L0.5

L0.5 = 25

L* = 625

q = 25L0.5

= 25(625)0.5 = 625

b) Profit = pq - wL

= 625625 - 75625

= 390,625 - 46,875 = 343,750

c) Now

d/dL = 1875 L- 0.5 - 75

d2/dL2 = 1875(- 0.5)L- 0.5 - 1

= - 937.5 L- 1.5 < 0

Therefore Profit is maximum at L = 625

Rahul Sunny answered 2 years ago

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