# The factor market

The law of (eventually) diminishing marginal product

Economists assume that the process by which a firm uses “inputs” or “factors of production” such as land, labor, and capital to produce final output can be represented by a production function. If q is the level of output, and a,b,c, and cl are the amounts of inputs used by the fimi, we can write the production function as:

which simply means that the amount of output produced by the firm over a given period of tune is a function/of the quantities of a,b,c, and d used in the production process.

If we hold constant the amounts of all inputs but one, input a, we can express the output of the finn as a function of that one input:

The marginal product of a is defined as the increase in total output the finn would obtain by hiring one more unit of a, holding fixed the amounts of all other inputs:

Table 1.3 shows in a hypothetical example how the marginal product of a changes as more units of a are employed by the finn, and Figure 1.6 shows how the level of output of a finn, q, might change as the finn used more units of a. The marginal product of a, Aq/Aa, is in fact the slope of q =f (a).12 Note that in Table 1.3 the marginal product Aq/Aa increases at

Table 1.3 Hypothetical schedule of marginal product of factor a

 Units of factor hired q = Total output Change in output (marginal product, Aq/Aa) 0 0 1 4 4 2 9 5 3 16 7 4 22 6 5 27 5 6 30 3 7 32 2 8 33 1

Figure 1.6 Changes in total product as more of a factor is hired.

first, then subsequently declines from seven to six when the fourth unit of the factor is lin ed. Also in Figure 1.6, the slope of the total product curve q increases at first, but then declines at a. This change occurs because of the famous law of eventually diminishing marginal product. According to this law, as more and more units of a factor a are hired by a firm, holding fixed the amounts of all other factors, the additional output obtained by hiring an additional unit of factor a must eventually decline.

In Figure 1.6 the amounts of all factors other than a are held fixed. Now what would happen if the amounts of the other factors changed? An increase in the amount of another factor, b. might increase the marginal product of a if this is true, we would say that factors a and b are complements. We would expect that factors are complements in many cases. If a law finn increases the amount of online legal materials available to its lawyers (court cases, statutes, regulations, law journal articles, etc.), the productivity of its lawyers will increase. Conversely, the value of those materials to the law finn will increase if the finn hires additional lawyers. Assuming a and b are complements, we can draw’ different total product curves for a: f{a).fi{a),fi{a). corresponding to increasing amounts of b: b, bi. and b}. This is shown in Figure 1.7.

How much of an input will the firm hire?

When we examined the decision a finn makes on how much output it will produce, w-e concluded that to maximize profits, the finn would produce at the level of output w’here marginal revenue equals marginal cost. Another question we might ask is how much of each input (of land, labor, capital, etc.) should a finn hire in order to maximize profits? Of course, the amount of inputs a finn hires will determine the amount of output it produces (through the firm’s production function), so w’e would hope to get the same answer - the same level of output - if w-e pose the question in terms of inputs rather than outputs. And, of course, we do.

When w'e considered w’hat the optimal level of output would be, w'e concluded that an additional unit of output should be produced and sold if. and only if, the additional revenue thereby gained (marginal revenue) was greater than the additional cost of producing that unit (marginal cost). It turns out that when w-e consider how much of each input the finn should hire, w'e should follow’ the same rule. An additional unit of an input should be hired if, and

Figure 1.7 Total product curves for different amounts of a complementary factor.

only if, the additional revenue thereby gained (which is called marginal revenue product) is greater than the additional cost of hiring that unit (called marginal factor cost).

The marginal revenue product of an increase of an input equals the change in the film’s total revenue divided by the change in the amount of the input: ATR/Aa. Note that this fraction can be written as:

where the first term on the right-hand side represents the change in total revenue divided by the change in the amount of the product produced and sold, or marginal revenue, and the second term is the change in total product divided by the change in the amount of the factor used, or marginal product. Thus:

marginal revenue product = marginal revenue x marginal product.

Now let us consider an example of how much of a factor - in this case labor - a firm will hire.

Suppose a rental car agency must decide how many customer agents to hire to staff one of its local offices. The salary of a rental agent is \$100 per day. If the office has many potential customers, the additional revenue to be gained by hiring a second agent may well exceed the revenue obtained from hiring the first agent. If there is only one agent, she would not be able to answer the phone and serve customers who walk in, simultaneously. If the office has only one agent, customers may have to wait a long time to be served, and they may decide to go elsewhere (never to return). In Table 1.4, the marginal revenue product of a second agent exceeds that of the first agent, and the marginal revenue product of a third agent is greater still.

There will be some number of agents that is optimal for the rental office, holding fixed the amount of its other factors, e.g.. the size of the rental office, its location, inventory of vehicles, advertising expenditures, telephone lines, computers, fax machines, and the like. Table 1.4 indicates that for this particular office the optimal number of agents is three. The

Table 1.4 Hypothetical schedule of marginal revenue product of labor for car rental agents

 No. of rental agents Total rental rev enue per day (S) Marginal revenue product per day (\$) 0 0 1 90 90 2 200 110 3 340 140 4 430 90 5 500 70 6 550 50 7 580 30

marginal revenue product of a third agent, \$140, exceeds her marginal cost of \$100, while the marginal revenue product of a fourth agent, \$90, would be less than her marginal cost. If there are already three agents in the office, who can handle most of the customer traffic, a fourth agent may not have much to do for most of the day. With the fourth agent, this rental office would be subject to the law of eventually diminishing marginal product. Note also that the marginal revenue product of the fourth agent would surely increase if the firm were to increase the amount of complementary factors at the rental office, e.g., by adding more vehicles and telephone lines, or by doing more advertising.

Next we consider how a change in the price of a factor affects the firm’s decision of how much of that factor, or other factors, it will line.

Substitution and scale effects of a change in factor price

The substitution effect

A change in the price of a factor has both a substitution effect and a scale effect. The substitution effect arises because at a given level of output, a film will be inclined to hire more of a factor whose price has fallen, and less of a factor whose price has increased. This obviously has implications for the amount of other factors that the firm hires. If, for example, the price of capital falls, or the productivity of capital increases, which effectively reduces the price of the services of capital, one would expect the film to substitute toward capital and away from labor. With the advent of bar code technology, stores do not need as many workers to mark prices on goods, or to add up the prices of goods at the cash register. The introduction of automatic teller machines greatly reduced the need for bank tellers.

Since the objective of a firm is to maximize profits, the firm will seek to minimize costs at every level of output. Therefore if the price of an input increases, the firm will tend to substitute toward the use of other inputs. If, for example, the cost of labor increases, a firm may substitute toward capital and hire fewer workers. Laws that were intended to improve working conditions for migrant agricultural workers in the United States required the owners of farms to provide them with housing and other amenities. As a consequence, many farms decided to buy machinery to harvest their crop rather than hire migrant workers.

Another example is the salad bar. In many restaurants in the United States, a customer who wants a salad goes to the salad bar to prepare it himself. In restaurants that do not have a salad bar, a restaurant employee prepares the salad in a bowl, and serves it to the customer at his table. Salad bars are generally found in countries where wages are high, like the United

States. The use of a salad bar enables the restaurant to reduce its use of labor, but allows the customer to obtain a larger salad than he would get otherwise, hi high-wage countries it costs less to give the customer more lettuce, tomatoes, croutons, etc., than it would to line the labor to prepare a salad. In countries in which the cost of labor is low, like China, restaurants will not use a salad bar.

The same analysis applies to the automated telephone sendee (called automated interactive voice-response systems), which can be used either to receive calls (“If you wish to refill a prescription, press 1; if you wish to talk to a pharmacist, press 2 ...”) or to make calls. In high-wage countries, firms will substitute away from using then employees to make calls toward these services, or will hire individuals hi low-wage countries to talk on the telephone. In low-wage countries films can minimize costs by having then own employees talk to the film's customers.

A similar obseivation applies to the web sites of many companies. Often the web site encourages a customer to contact the company with an email, rather than by a phone call, by making the phone number difficult for the customer to find. Customer requests made by email can be handled with fewer employees than such requests made by phone.

Web sites of companies with employees in low-wage countries are more likely to invite customers to contact them by phone.

The scale effect

That is the substitution effect. What about the scale effect of a change in factor price? This arises because a change in the price of a factor affects the film’s desired level of output. A supermarket that adopts bar code technology (an effective decline in the price of capital) will be able to lower its prices and thus attract more customers and expand its operations. It will then employ more managers, truck drivers, workers to stock its shelves, etc. Similarly, a bank that installed automatic teller machines was able to, in effect, reduce the price of its banking services by improving their quality: a customer could now make withdrawals at any time of the day or week, rather than just during regular banking hours. A bank that effectively reduces its price will be able to increase its output. Thus a bank would attract more customers, and would need to hire more loan officers, secretaries, security guards, etc.

An increase in wages increases the costs of a firm, which means the firm must raise its price, in which case its sales and output will decline. Because its output has declined, the finn will hire fewer amounts of factors such as labor and capital. Conversely, a decline in wages leads to an increase in sales and output. A hotel chain that hires immigrant workers at a low wage will be able to offer lower room prices to guests, and thus will obtain more business than its competitors. The hotel chain will then need to increase its use of other types of labor - security guards, plumbers, electricians, and others.

Another way to understand the scale effect is through the effect of a change in factor price on the amounts of other factors employed by the finn. To understand this, suppose a film received an addition to its capital from out of the blue, as a gift. If capital and labor are complementary factors - the usual case - this increase in capital would increase the marginal product of the labor employed by the film, which would induce the film to line more labor. When the finn hires more labor, the marginal product of its capital will increase, leading the finn to acquire more capital, which in turn increases the marginal product of labor. Of course this sequence of events does not continue indefinitely; the film will converge to a new equilibrium with larger amounts of capital, labor, and other factors.

Now imagine that the initial change is a decline in the price of capital, hi this case there is both a substitution effect, which induces the film to substitute toward capital and away from other factors such as labor, and a scale effect, which will affect the amounts of other factors desired by the film. Since the price of capital falls, the film will respond by buying more capital. Assuming capital and labor are complementary factors, the marginal product of labor will increase, and the film will wish to hire more labor. This effect will work in the opposite direction from the substitution effect.

hi general, the scale effect can be greater or less than the substitution effect. If the overall result of a decline in the price of capital is that the film employs less labor, we say that capital and labor are gross substitutes. If the overall result is that the firm hires more labor, we say that capital and labor are gross complements.

We could also ask a more theoretical question, namely what happens if there is no scale effect, so that the firm’s output remains the same. Suppose again there is a decline in the price of capital. If the firm would then wish to line less labor, we would say that capital and labor are net substitutes. If the firm would wish to line more labor, we say that capital and labor are net complements.

Isoquants

We can analyze the question of how much of each factor of production the film should hire with the use of curves called isoquants. The word “isoquant” is derived from Lathi words meaning “equal or same quantity.” Isoquants are curves that show all the different combinations of inputs that can be used to produce a given level of output. In Figure 1.8, there are two inputs: labor per year is the input on the horizontal axis and capital per year is on the vertical axis. The diagram shows three isoquants representing three different levels of output. The isoquant for Q = 100 units of output shows that this quantity of output can be produced with

Figure 1.8 Isoquants for three levels of output.

five units of capital and two units of labor, or alternatively with two units of capital and four units of labor.

The problem facing the firm is how to maximize the output it produces for a given total expenditure on factors of production, or (stating the same problem differently) how to minimize the total cost of factors of production that will yield a given level of output. Suppose we have only two factors of production, capital, К and labor, L. The cost of a unit of capital per unit time is r, and the cost of a unit of labor per unit time is w. The total amount spent by the firm on factors of production, which we will call C, is rK + wL. We can represent all the possible combinations of factors that can be purchased for a given total expenditure C by something called an isocost line, meaning a line showing the same or equal total cost at all points on the line in Figure 1.9, at point A the firm would be spending all of C on capital, and at point В it would be spending it all on labor. Since we know that C = rK + wL, and at point A we know that L = 0, the amount of capital purchased at point A is C/r. Similarly, since we know that C = rK + wL, and at point В we know that К = 0, the amount of labor purchased at point В is C/w. Also, since rK = C-wL, we know that:

Thus the slope of the isocost line is:

Now what would happen if the wage increased from w to a higher wage, wf! In this case the point where the isocost line crosses the horizontal axis would shift inward from C/w to C/w2, shown in Figure 1.9. The isocost line now points more steeply toward labor, since labor has now become more expensive. Similarly, suppose the cost of capital increased from r to a higher rent, n. In this case the point where the isocost line crosses the vertical axis

Figure 1.9 Effect of changes in the price of a factor on the isocost line.

Figure 1.10 Maximizing output for a given level of expenditure on factors.

would decline from C/r to C/n, which is shown in the diagram. The isocost line now points more steeply toward capital, since capital is now more expensive.

For a given total expenditure C spent on factors of production, the firm would want to produce as much output as possible. Note in Figure 1.10 that the firm could produce eighty units of output by operating at either ofpomts F or G; however it could produce 100 units of output by operating at point E*. Since in order to maximize profit it wants to maximize the output it produces for a given level of expenditure on factors, it will produce at point E*, lining Ke units of capital and Le units of labor. Note that it would like to produce 120 units of output for the level of expenditure C, but this higher isoquant cannot be attained from this isocost line.

Finally, Figure 1.11 shows how the firm that wants to produce 100 units of output would respond to different relative prices of capital and labor, in choosing how much of each factor to buy. If the relative wage of labor is high, as on isocost line IC, the film will choose point E, and hue K units of capital and L units of labor. If, however, the relative wage of labor is lower, as on isocost line 1C the firm will choose pomt Ej, and line K} units of capital and I2 units of labor. Recall from the discussion earlier in this chapter that the substitution effect of a change in factor price shows how the firm would change its purchases of factors if it stayed at the same level of output where it was operating before the change in factor price. Here the film's response to a reduction in the relative price of labor, gomg from isocost line ICi to /Ст, would be to increase its purchases of labor fromli to I2, and reduce its purchases of capital from K to Ki.

These represent the pure substitution effects of the change in relative factor price. There will also be scale effects. If, for example, the wage of labor falls, the costs of the film fall, and it will choose a higher level of output. This means a shift to a higher isoquant.