Question

Suppose that the firm has a production function described by

q=0 if L≤6

q = −(L^3)/6+K(L^2)+26K if L>6

Further suppose that we are concerned only in the short run and that the units of capital employed are currently fixed at 8. Also, suppose that labor units are integer. That is, they can only be in terms of whole numbers 1, 2, 3, ... and so on. Suppose that the price of each unit produced by the firm is 2.

1. Graph the labor demand curve. (Any nine evenly distributed evaluation points will suffice)

2. Graph the labor demand curve, but in a scenario where the price of each unit produced by the firm equal to 3.

3. How many more workers can the firm afford after the price increase of the produced good (from 2 to 3) if w= 290/3

4. Calculate the elasticity of labor demand in the short run when price of each unit produced by the firm is 2 and w changed from 290/3 to 198/3. Round to the nearest hundreth.

Answer 1

1.) labor demand function and curve at price = 2

The labor demand curve is where the marginal revenue product of labor equals the wage rate.

And the marginal revenue product of labor is the additional revenue brought by adding an additional labor.

Marginal revenue product of labor = Marginal product of labor x Price of the good

Since,

The marginal product of labor is the additional output produced by an additional labor.

That is, it is the derivate of output with respect to labor units.

Putting the value of units of capital that is,

K = 8

Now the labor demand is is the marginal revenue product of labor equals wage rate.

The price of the good is given to be P = 2

Putting the marginal revenue product of labor equal to the wage rate.

Therefore this is the labor demand function where w is the wage rate and L is the labor units.

Plotting this we will get the labor demand curve. Labor demand curve is the downward part of the marginal revenue product where the marginal product is decreasing. Therefore we will plot from 14 to 22 labor quantity. We can see that the labor demand curve is downward sloping, that is as the wage rate increases, quantity of labor demanded decreases.

2.) Labor demand at price = 3

Now the price of the good has increased to 3.

Therefore labor demand is:

Wage rate equals to the marginal revenue product of labor.

Since marginal revenue product is the price times marginal product of labor.

Therefore,

Therefore labor demand equation is,

Plotting nearly nine points, the labor demand curve is the downward sloping part of the marginal revenue product of labor.

3.) When the price of the good increases to 3 and wage rate is 290/3

Labor that the firm demand at price 2 when the wage rate is 290/3

Putting the wage rate into the demand function of labor when the price was 2

When the price of the good is 2 labor demand function is;

Putting wage rate = 290/3, we get the quantity of worker that the firm could afford.

Therefore at wage rate of 290/3, the firms demand labor units of approx 29.

Putting wage rate to be 290/3 and in the labor demand function when the price is 3

Labor firm can demand at a price of 3 is approx 30 labor units

Therefore the firm can afford one more labor after the price increases to 3

4.) The elasticity of labor demand as the price wage rate decreases from 290/3 to 198/3

The elasticity of demand is the responsiveness of the firm to change its quantity of labor demand as the wage changes.

Wage rate changes from 290/3 to 198/3

As we have found earlier that:

At wage rate 290/3, labor demand is = 28.6

Quantity of labor demanded at wage rate = 198/3

At wage rate 198/3, labor demand is 29.8

The elasticity of labor demand is -0.13 when the wage rate decreases from 290/3 to 198/3.

Thereofre the elasticity of labor demand is inelastic. As the modulus of elastiicty is less than one.