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Coldsprings Tavern produces burgers according to the following production function:Q=K^1/3 E^2/3.Coldsprings will produce 400 burgers when...

Coldsprings Tavern produces burgers according to the following production function:Q=K^1/3 E^2/3.Coldsprings will produce 400 burgers when the hourly wage is $10 and the rental price of capital is $20.

a.Find the cost-minimizing bundle of employment and capital.b.Now suppose that the market wage falls to $8 per hour, and at this set of input prices, Coldsprings Tavern produces 450 burgers. Find the cost-minimizing bundleof employment and capital.

c.Solve for the numerical values of the scale and substitution effects for both employment and capital. [Hint: First find the cost-minimizing bundle of employment and capital when the wage rate is $10 per hour and q=450

Solutions

Expert Solution

Q = K1/3E2/3

Total cost (C): C = wN + rK

Cost is minimized when MUE / MUK = w / r

MUE = Q / E = (2/3) x (K / E)1/3

MUK = Q / E = (1/3) x (E / K)2/3

MUE / MUK = 2 x (K / E)

(a) Q = 400 = K1/3E2/3

MUE / MUK = 10/20 = 1/2

2K / E = 1/2

E = 4K

Substituting in production function,

400 = K1/3(4K)2/3

400 = 42/3K1/3K2/3

400 = 42/3K

K = 400 / 42/3 = 158.74

E = 4 x 158.74 = 634.96

(b) Q = 450 = K1/3E2/3

MUE / MUK = 8/20 = 2/5

2K / E = 2/5

2E = 10K

E = 5K

Substituting in production function,

450 = K1/3(5K)2/3

450 = 52/3K1/3K2/3

450 = 52/3K

K = 450 / 52/3 = 153.9

E = 5 x 153.9 = 769.5

(c) w = 10 and Q = 450 = K1/3E2/3

MUE / MUK = 10/20 = 1/2

2K / E = 1/2

E = 4K

Substituting in production function,

450 = K1/3(4K)2/3

450 = 42/3K1/3K2/3

450 = 42/3K

K = 450 / 42/3 = 178.58

E = 4 x 178.58 = 714.32

For employment,

Total effect (TE) = 769.5 - 634.96 = 134.54

Substitution effect (SE) = 769.5 - 714.32 = 55.18

Scale effect = TE - SE = 134.54 - 55.18 = 79.36

For capital,

TE = 153.9 - 158.74 = - 4.84

SE = 153.9 - 178.58 = - 24.68

Scale effect = - 4.84 + 24.68 = 19.84

Rahul Sunny answered 1 year ago
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