Question

Consider an economy with the following Cobb-Douglas production function: Y = K1/3L 2/3 . The economy has 1,000 units of capital and a labor force of 1,000 workers. 1

a. Derive an equation describing labor demand as a function of the real wage and the capital stock. (Hint: this is a review from what we did in Chapter 3)

b. If the real wage can adjust to equilibrate labor supply and labor demand, what is the resulting equilibrium real wage? In this equilibrium, what are employment, output, and the total amount earned by workers? (you may assume labor is supplied inelastically, i.e. due to perfectly competitive markets, workers will work at any price will work at any price)

c. Now suppose Congress, concerned about the welfare of the working class, passes a law requiring firms to pay workers a real wage of one unit of output. How does this wage compare to the equilibrium wage?

d. Congress cannot dictate how many workers firms hire at the mandated wage. Given this fact, what are the effects of this law? Specifically, what happens to total employment, output, and the total amount earned by workers?

Answer 1

Production function:

a) Profit maximizing condition:

where r is the rent for capital K and w is the wage rate for labor L. ........ (i)

differentiating profit equation w.r.t K and L gives

................. (ii)

............. (iii)

solving (iii) and setting it equal to zero gives demand for labor

........ (iv)

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b) Labor demand = Labor supply

and K =1000 as given

w/p = $2/3 per unit of output.

total employment is 1000 workers,

total output = (1000)^1/3 * (1000)^2/3 = 1000

Total amount paid = total employment * real wage rate = 1000 * 2/3 = $666.66 or $ 667.

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c) The mandated real wage rate of $1 per unit of output is greater than the current real wage rate of $2/3 per unit of output.

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d) Total employment will remain same at 1000 workers. (ideally with increase in wage rate, the number of workers may increase but due to perfect competition, the workers will remain constant)

Total output will also be 1000 units

and amount earned will increase from $667 to $1000.

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