Question

The hamburger industry is in its long-run equilibrium with total industry sales based on total market demand of Qd = 4800 − 50P. The production function for hamburgers is described as Q = 6K.5L.5, which all market participants share. Current wages are $8, while rental rate is $72. The state has just decided to raise the minimum wage to $18. How many workers will lose their jobs in the hamburger industry?

Answer 1

The hamburger industry is in its long-run equilibrium with total industry sales based on total market demand of

Qd = 4800 − 50P

or, P = 96 - 0.02.Qd

The production function for hamburgers is described as

Q = 6.K^{​​​​​​0.5}​.L^0.5

Now, Marginal Productivity of Labor is

MPL = dQ/dL = 3.(K/L)^0.5

And, Marginal Productivity of Capital is

MPK = dQ/dK = 3.(L/K)^0.5

Hence, Marginal Rate of Technical Substitution is

MRTS = MPL/MPK

or, MRTS = K/L

We are given that,

wage rate: w = $8

rental rate: r = $72

Now, at equilibrum production,

MRTS = w/r

or, K/L = 8/72

or, L = 9.K........(1)

Putting this in the production function we get,

Q = 6.(K.9K)^0.5

or, Q = 6.(3K)

or, K = Q/18.........(2)

And, L = 9.K

or, L = Q/2..........(3)

Hence, Total Cost is

C = w.L + r.K

or, C = 8.(Q/2) + 72.(K/18)

or, C = 8.Q

Hence, Marginal Cost is

MC = dC/dQ = 8..........(4)

Now, at equilibrum in the market,

P = MC

or, 96 - 0.02.Qd = 8

or, Qd = 4400

Hence, putting Q = 4400 in equation (3) we get thr labor requirement in equilibrum as

L* = Q/2 = 4400/2

or, L* = 2200 units

Now, the wage rate changes from w = $8 to w' = $18.

Hence, at equilibrum,

MRTS = w'/r

or, K/L = 18/72

or, L = 4.K

Putting L = 4.K in the production function we get,

Q = 6.(K.4.K)^0.5

or, Q = 6.(2.K)

or, K = Q/12.........(5)

And, L = 4.K = 4.(Q/12)

or, L = Q/3..........(6)

Now, the Total Cost is

C' = w'.L + r.K

or, C' = 18.(Q/3) + 72.(Q/12)

or, C' = 6.Q + 6.Q

or, C' = 12.Q

Marginal Cost is

MC' = dC'/dQ = 12........(7)

Hence, At equilibrum in the industry,

P = MC'

or, 96 - 0.02.Qd = 12

or, Qd = 4200

Now, putting Q = 4200 in equation (6) we get,

L** = 4200/3

or, L** = 1400

Hence, the labor employment decreased from L* = 2200 to L** = 1400.

Hence, (2200 - 1400) = 800 workers will lose their jobs.

800 workers will lose their jobs in the hamburger industry.

Hope the solution is clear to you my friend.