4-1. Suppose there are two inputs in the production function, labor and capital, and these two
inputs are perfect substitutes. The existing technology permits 1 machine to do the work of 3
persons. The firm wants to produce 100 units of output. Suppose the price of capital is $750 per
machine per week. What combination of inputs will the firm use if the weekly salary of each worker
is $300? What combination of inputs will the firm use if the weekly salary of each worker is $225?
What is the elasticity of labor demand as the wage falls from $300 to $225?
Because labor and capital are perfect substitutes, the isoquants (in bold) are linear and the firm will use
only labor or only capital, depending on which is cheaper in producing 100 units of output.
The (absolute value of the) slope of the isoquant (MPE/ MP )Kis 1/3 because 1 machine does the work of
3 men. When the wage is $900 (left panel), the slope of the isocost is 300/750. The isocost curve,
therefore, is steeper than the isoquant, and the firm only hires capital (at point A). When the weekly wage
is $225 (right panel), the isoquant is steeper than the isocost and the firm hires only labor (at point B).
Weekly Salary = $300 Weekly Salary = $225
slope =MP EMP K
slope=MP /EP K =1/3
slope = w/r slope = w/r
The elasticity of labor demand is defined as the percentage change in labor divided by the percentage
change in the wage. Because the demand for labor goes from 0 to a positive quantity when the wage
dropped to $225, the (absolute value of the) elasticity of labor demand is infinity.
22 4-2. (a) What happens to the long-run demand curve for labor if the demand for the firm’s output
The labor demand curve is given by VMP E= MR x MP . Es demand for the firm’s output increases, its
marginal revenue also increases. Thus, an increase in demand for the firm’s output shifts the labor
demand curve to the right.
(b) What happens to the long-run demand curve for labor if the price of capital increases?
To determine how an increase in the price of capital changes the demand for labor, suppose initially that
the firm is producing 200 units of output at point P in the figure. The increase in the price of capital
(assuming capital is a normal input) increases the marginal costs of the firm and will reduce the profit-
maximizing level of output to say 100 units. The increase in the price of capital also flattens the isocost
curve, moving the firm to point R. The move from point P to point R can be decomposed into a
substitution effect (P to Q) which reduces the demand for capital, but increases the demand for labor, and
a scale effect (Q to R) which reduces the demand for both labor and capital. The direction of the shift in
the demand curve for labor, therefore, will depend on which effect is stronger: the scale effect or the
23 4-3. Union A wants to represent workers in a firm that hires 20,000 person workers when the wage
rate is $4 and hires 10,000 workers when the wage rate is $5. Union B wants to represent workers in
a firm that hires 30,000 workers when the wage is $6 and hires 33,000 workers when the wage is $5.
Which union would be more successful in an organizing drive?
The union will be more likely to attract the workers’ support when the elasticity of labor demand (in
absolute value) is small. The elasticity of labor demand facing union A is given by:
η = percent ∆ L / percent ∆ w = (20,000–10,000)/20,000 ÷ (4–5)/4 = –2.
The elasticity of labor demand facing union B equals (33,000–30,000)/33,000 ÷ (5–6)/5 = –5/11 ≈ –.45.
Union B, therefore, is likely to have a more successful organizing drive as 0.45 < 2.
4-4. Consider a firm for which production depends on two normal inputs, labor and capital, with
prices w and r , respectively. Initially the firm faces market prices of w = 6 and r = 4. These prices
then shift to w = 4 and r = 2.
(a) In which direction will the substitution effect change the firm’s employment and capital stock?
Prior to the price shift, the absolute value of the slope of the isocost line (w/r) was 1.5. After the price
shift, the slope is 2. In other words, labor has become relatively more expensive than capital. As a result,
there will be a substitution away from labor and towards capital (the substitution effect).
(b) In which direction will the scale effect change the firm’s employment and capital stock?
Because both prices fall, the marginal cost of production falls, and the firm will want to expand. The scale
effect, therefore, increases the demand for both labor and capital (as both are normal inputs).
(c) Can we say conclusively whether the firm will use more or less labor? More or less capital?
The firm will certainly use more capital as the substitution and scale effects reinforce each other in that
direction, but the change in labor employed will depend on whether the substitution or the scale effect for
4-5. What happens to employment in a competitive firm that experiences a technology shock such
that at every level of employment its output is 200 units/hour greater than before?
Because output increases by the same amount at every level of employment, the marginal product of
labor, and hence the value of the marginal product of labor, does not change. Therefore, as the value of
the marginal product of labor will equal the wage rate at the same level of employment as before, the
level of employment will not change.
24 4-6. Suppose the market for labor is competitive and the supply curve for labor is backward
bending over part of its range. The government now imposes a minimum wage in this labor market.
What is the effect of the minimum wage on employment? Does the answer depend on which of the
two curves (supply or demand) is steeper? Why?
Equilibrium is attained where the supply curve intersects the demand curve, and the equilibrium
employment and wage levels are E* and w*, respectively. When the minimum wage is MIN thew
firm wants to hirD E workers bSt E workers are looking for work. As long as the downward-sloping
portion of the supply curve is to the right of the demand curve, the fact that the supply curve is downward
sloping creates no problems beyond those encountered in the typical competitive model. An interesting