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21. Marginal Product of Labor
Questions:
- What is the law of diminishing returns?
- Given a short run production function, find the marginal product of labor function.
3. What amount of labor will a firm hire? Why? Why does the law of demand apply to labor?
Marginal Product
The marginal product of an input is the change in output resulting from a one unit change in the use of an input. All the other inputs are assumed to be given. For example, the marginal product of labor is the change in output resulting from a one unit change in the amount of labor used. Since it is usually true that increases in labor increase output, the marginal product of labor is the increase in output resulting from a one unit increase in the use of labor, or the decrease in output due to a one unit decrease in the amount of labor.
For example, suppose you couldn’t change the number of machines you had in the short run, but could use more labor to operate the machines. If some machines were idle, then increasing the labor utilized would allow the idle machines to be used. The additional output produced is the marginal product of labor. On the other hand, if all the machines are being used, getting more output is a little tough. Rather than letting the machines stand idle when workers are on breaks, you might hire another worker to spell workers when they take breaks. While that worker might end up spending some idle time and some machines might stand idle anyway (given the nature of breaks,) more output could be produced. The extra output might be small compared to what could be produced if the worker was working with another machine, but whatever extra output you can get is the marginal product of labor. Of course, goods typically use materials as well. If some materials are unused, then more can be used. But suppose the firm is making shirts. It might be possible to get more shirts out of a given amount of cloth by using more labor in careful cutting. The amount of scrap cloth would be reduced.
Marginal Product of Labor Function
The marginal product function is found by taking the derivative of the production function with respect to an input. For example:
Q = 5L.5
Use the power rule:
MPL = 5* 100L.5-1
= 2.5L-.5
That means 2.5 times 1 divided by the square root of the amount of labor used.
Suppose a firm used 10,000 man hours of labor. Then the marginal product of labor would be:
MPl = 2.5 (10,000)-.5
= 2.5 * 1/10000.5
= 2.5/100
= .025
This says that if one more man/hour of labor was used, then output would increase by .025 units.
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The “Law” of Diminishing Marginal Returns
The “Law” diminishing marginal returns is that while it is possible that the marginal product of an input could increase when only a little is used, as more and more of an input is used, marginal product will decrease. In other words, if the amount of all other inputs are given, as more and more units of any one input are used, output will increase, but the increases in output will get progressively smaller.
The Cobb-Douglas production function exhibits diminishing marginal returns. But is it a good reflection of reality? Why is this law true?
Well, maybe it isn’t even true. One possibility is that inputs need to be combined in fixed combinations. Like one worker per machine and three yards of cloth per shirt. With one worker on a machine producing one shirt every 12 minutes. If that is true, then marginal product will be constant if one input is increased and the other inputs are idle, but when one runs out of any input, then no increase in output is possible. For example, an extra worker with no machine produces nothing.
On the other hand, careful cutting or allowing one worker to spell others, or having workers carry things between machines or the like might increase output. But not much! That means that additional amounts of labor increase output only a small amount. That is diminishing marginal returns. (Of course, the Cobb Douglas production function shows marginal product falling continuously rather than being pretty much constant and then dropping like a stone.)
Marginal Revenue Product
A firm purchases (or hires) an input because it helps increase output. The point, of course, of producing more output is to sell that output for money. The marginal revenue product of an input is the change in revenue due to a one unit change in the use of an input, other things being equal. For example, the marginal revenue product of labor is the change in labor resulting from a one unit increase the amount of labor a firm uses. That would be the increase in revenue due to a one unit increase in the amount of labor used or the decrease in revenue due to a one unit decrease in the amount of labor used.
The marginal revenue product of an input is equal to the marginal revenue times the marginal product of the input.
MRPl = MR * MP
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Suppose you used 10,000 units of labor (and 2500 units of capital.) From the last topic, we know that we can produce 500 units of output. If your marginal revenue function is 3253-.1Q and marginal revenue is $3203.
If your marginal product of labor is .025 then your marginal revenue product of labor is:
MRPl = MR * MPl
= .025 * 3203
= 80.075
This means that if you used one more man/hour of labor per year, your firm would expand production at a rate that would earn an extra $80. (Well, a bit more.)
Demand for an Input
The benefit from buying (or hiring) another unit of an input is the extra revenue that can be earned from the extra output produced with it--its marginal revenue product. But that must be compared to the additional cost. What must be paid for the input? The firm will choose to buy (or hire) an amount of an input such that the marginal revenue product of the input is equal to its price (or rental rate.) That is the firm’s demand for the input.
For example, a firm will hire an amount of labor such that the marginal revenue product of labor is equal to the wage rate. Suppose the marginal revenue product of labor is greater than the wage rate. Then if the firm hires more labor, the extra revenue earned from the extra output that is produced with the labor will be greater than what must be paid for the labor. That adds to profit. What if the wage rate is greater than the marginal revenue product of labor? Then if the firm uses less labor, the decrease costs because less labor is hired is greater than the decrease in revenue because less output is produced and sold. Since costs fall by more than revenue, profits rise.
The amount of labor a firm will hire can be found using the marginal revenue product of labor schedule and a given wage rate. Just find the amount of labor where the marginal revenue product of labor is equal to that wage rate.
If the wage rate were $5 and the marginal revenue produce of labor is a bit of $80, the using more labor would add to profit.
Law of Demand for Labor
Why does the law of demand apply to labor? Well, suppose a firm is hiring the amount of labor that maximizes profit. That is the amount of labor at which the marginal revenue product is equal to the wage rate. Now, suppose the wage rate falls. The wage is less than the marginal revenue product of labor. A firm can make more profit by hiring more labor. So a lower wage rate results in an increase in the quantity of labor demanded by the firm.
What if the wage rate rises? It will be above the marginal revenue product of labor. A firm can make more profit by hiring less labor. The lost revenue will be smaller than the savings in wages. So a higher wage rate results in a decrease in the equantity of labor demanded by the firm.