Labor Demand

The functional relationship between the quantities of outputs and inputs is called the “production function”. The functional relationship between output and labor can be defined as follows: q = f(E) where q and E are quantities of output and labor, respectively. f(.) is a function that defines the relationship between output and labor.

If the demand for a firm’s product increases, the firm needs to employ more labor. In other words, the demand for labor is a derived demand, i.e., it depends on the demand for the product produced by the firm. Changes in the demand for the product induce changes in the demand for labor. The effect of changes in product demand on labor demand is mediated through the production technology used by the firm.

The average product is the ratio between output and labor and shows how much one worker can produce (the output per worker). It is also defined as “labor productivity”. The marginal product is the increase in the quantity of output that results from employing one more unit of labor.

The average product increases when the marginal product is higher than the average product and declines otherwise so that the marginal product curve intersects with the average product curve at its maximum (p: 61, Figure 3.2). Therefore, at the peak of the average product curve, the marginal product should be equal to the average product.

The response of the firm to a shift in product demand can be analyzed in two time spans, the short run and the long run. The short run is the period in which the firm can change its labor input but not its capital input. The long run is the period during which the firm can change both the quantities of labor and capital.

When the capital input is fixed, the marginal product of labor, MP_E, is expected to decline after a certain level of employment because of diminishing returns to labor. When the firm employs more labor without changing the quantity of capital, the workers will start to share the capital equipment, and this will reduce each worker’s productivity. This is called the law of diminishing returns, and it is assumed to operate over some level of employment.

The change in industry level labor demand in response to changes in the wage rate is defined as the short run elasticity of labor demand. Algebraically, the short run elasticity of labor demand is equal to the ratio between the percent change in short run employment and the percent change in the wage rate:  δ_SR = (ΔE_SR/E_SR) /(Δw/w) where δ_SR is the short run elasticity of labor demand, E_SR is the short run employment in the industry, and w is the wage rate. Δ denotes the change in the variable.

Since there is a negative relationship between the demand for labor and the wage rate, the short run elasticity of labor demand is negative.

When the short run labor demand function is steeper (more vertical), the elasticity gets smaller in absolute value. This type of labor demand function is called inelastic because wage rate changes will not significantly affect labor demand. On the contrary, if the labor demand function is flatter (more horizontal), the absolute value of the elasticity is higher, and a slight change in the wage rate may cause substantial changes in the labor demand. This type of labor demand function is called elastic.

The first order conditions for profit maximization state that the firm should employ workers at the point where the wage rate is equal to the value of the marginal product of labor, and the capital stock at the point where the rental price of capital is equal to the value of the marginal product of capital. Moreover, these two conditions imply that

w/r = MP_E /MP_K

In other words, at the profit maximizing levels of labor and capital, the wage-rental ratio should be equal to the ratio between the marginal products of labor and capital.

The second order conditions (f_EE < 0, f_KK < 0, and f_EE f _KK – f² _KE > 0) state that there should be diminishing returns to labor and capital, and the labor demand function should be downward sloping in the long run.

The MP_E /MP_K ratio is called the marginal rate of technical substitution. The marginal rate of technical substitution of labor for capital is the rate at which capital can be reduced for every one unit increase in labor while keeping output constant. It equals the absolute value of slope of the isoquant.

The derivation of the long run labor demand functions at the firm level reveals that there are two factors that determine the link between the wage rate and the demand for labor: a change in the quantity of output and the substitution between inputs.

The scale and substitution effects have the same impact on the demand for labor. If the wage rate declines, the quantity of labor demanded increases, so that the long run demand function at the firm level is downward sloping. The scale and substitution effects work in the opposite direction for the capital input. If the wage rate declines, the firm expands its output and uses more capital. The scale effect implies an increase in the capital input. Since labor is now cheaper and capital is more expensive, the firm will substitute labor for capital, and the demand for capital will decrease due to the substitution effect.

Labor and capital are perfect substitutes in linear production functions. Substitution is not possible in the Leontief production function. The Cobb-Douglas production function allows some degree of substitution between capital and labor.

Marshall’s Laws of Derived Demand defines four determinants of the elasticity of labor demand:

1. Labor demand is more elastic the greater the elasticity of substitution.
2. Labor demand is more elastic the greater the elasticity of demand for the output.
3. Labor demand is more elastic the greater labor’s share in total costs.
4. Labor demand is more elastic the greater the supply elasticity of capital.

In all sectors, the demand elasticity is higher for production workers than for administrative employees. Demand elasticities are lower in capital goods industries, which employ relatively more skilled workers, and higher in consumer goods industries, which are more labor intensive.

The sign of the cross-price elasticity shows if these two inputs are gross complements or gross substitutes. If the inputs i and j are gross complements, a decline in the price of the j^th input will increase the demand for the i^th input, and the cross-price elasticity will be negative. If inputs are gross complements, when one input gets cheaper, the complementary input will be used more because these two inputs are used “together” (like computers and printers). If the inputs i and j are gross substitutes, a decline in the price of the j^th input will reduce the demand for the i^th input, and the cross-price elasticity will be positive.

As with the own-price elasticity of demand, the cross-price elasticity of demand is also determined by scale and substitution effects. If two inputs are substitutes in production, the substitution effect is negative, but if the scale effect is larger than the substitution effect, these inputs will be gross complements.

“Informal labor” or “informal employment” refers to those employees who are not registered in any social security organization. The most obvious cost of informality is the cost of punishment if the firm is detected. Therefore, the firm will employ informal labor as long as the cost of informal employment (the probability of detection times the penalty) is less than the benefit of formal employment (the difference between the labor cost and the net wage).

We can define at least three types of wages in the Turkish context:

• “net wage” is the income (in cash or inkind) received by the worker,
• “gross wage” is the money paid by the firm for the worker,
• “labor cost” is the employer’s total cost of employing a worker.

The firm nominally “pays” the gross wage to the worker, but the employee’s social security contributions (for pensions, health insurance, etc.) and income tax are withheld and paid to the government by the employer. Therefore, the net wage is equal to the gross wage minus social security contributions and the income tax to be paid by the worker. The employer also contributes to the social security for each employee in proportion to her gross wage. Therefore, the labor cost for the employer is equal to the gross wage plus the employer’s social security contributions.

Adjustment costs are important in understanding many policy-relevant labor market dynamics. For example, labor productivity rises during expansion and declines in recession. Adjustment costs can explain changes in labor productivity during a business cycle, and governments can take measures to improve economic welfare by considering these effects.

Adjustment costs have two components, variable and fixed costs. Variable adjustment costs change with the number of workers hired or fired. In contrast, fixed adjustment costs are independent of the number of workers involved.  Variable firing and hiring costs are likely to be different in magnitude. In many cases, firing costs are much higher than hiring costs because of labor market laws and regulations. If firing costs are higher than hiring costs, the downward adjustment in labor demand (firing workers) will be much slower than the upward adjustment (hiring workers). Fixed costs make hiring and firing a lumpy process. For example, in the process of hiring workers, the firm will compare the benefits and costs of hiring and will only hire workers if the benefits are higher. Therefore, if the firm needs to hire only a few workers and the fixed costs of hiring are high, it will postpone hiring those workers and will hire new workers only if the number of workers to be hired is sufficiently high.