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  The Labor Market and MPL used for the Cobb-Douglas production function

The Labor Market:

Economists try to suppress each detail and difference that doesn’t matter to the overall result in order to simplify the analysis and focus it on the key, significant factors that count. For the reason that differences between businesses will not matter, let’s think regarding an economy with K typical--identical--competitive firms, every of which owns one unit of the economy's capital stock. Every of these typical competitive firms hire L workers and every firm pays each worker the same wage W. Every firm sells Y units of its product at a per-unit price P.

The typical firm doesn’t control either the wages it must pay or the prices it receives; those are defined by the market. The firm makes an effort to make as much money as it can. The usual firm’s profits are simply its revenues minus its costs as well as its only costs are the wages it pays to workers. Thus:

Profits = Revenues - Costs
Profits = P x Y - W x L

To determine how many workers to hire, the firm goes with two simple rules:

1. Hire workers to increase output.

2. Stop hiring when the additional revenue from the output produced by the last worker hired just equals his or her wage.

The worth of the output produced by the preceding worker hired is the product price P times the marginal product of labor [MPL]. The cost of hiring the preceding worker is his or her wage W. The firm will remain hiring until:

P x MPL - W = 0

The marginal product of labor is the dissimilarity between what the firm can produce with its current labor force Lfirm, moreover what it would produce if it hired one more worker:

MPL = F(1, Lfirm + 1) - F(1, Lfirm)

The MPL used for the Cobb-Douglas production function is:

1105_mpl cobb douglas production.jpg

Once more we take how much the firm would produce if it hired one more worker as well as subtract how it produces now with its current labor force. Since the firm has merely one unit of capital we can rewrite this equation as:

199_cobb douglas formula.jpg

The term inside the brackets seems like the rate-of-growth of a variable growing by one unit (the firm’s labor force Lfirm) and then increased to a power (the term 1- α). We comprise a standard rule-of-thumb for dealing with such a situation. The comparative growth rate of a variable increased to a power is the proportional growth rate of the variable multiplied by the power to which the variable is increased. This notifies us that the term inside the brackets is:

2476_rate of growth.jpg

Therefore the MPL is:

1805_MPL.jpg

There isn’t anything deep in this math. Certainly the Cobb-Douglas function was cautiously tweaked consequently that it would yield such simple forms for quantities like the MPL. That is why economists use it consequently often. If the Cobb-Douglas production function created more complicated expressions we would not use it.

The firm appoints workers up to the point where the product price times the marginal product of labor equals to the wage:

P x MPL - W = 0

Substituting for the MPL in this equation we obtain:

415_marginal product of labor.jpg

Next we rearrange this equation to observe that the typical firm's demand for workers is:

949_typical firms demand.jpg

For the reason that there are K firms in the whole economy and total economy-wide employment is equal to K times the typical firm’s demand for labor:

1224_typical firms demand for labor.jpg

Define labor supply? The answer is easy: it is the numeral of workers who want to work. The labor market will be in equilibrium when firms' entire demand for workers is equal to the labor force.

Labor market can’t be in equilibrium if wages and prices are flexible? Think about what would take place if supply were not equal to demand. Presume there are more workers than firms wish to hire at current wages and prices. Then a few of the unemployed will underbid their employed fellow workers: offer to take their jobs as well as work for less. Those workers who are employed will answer by offering to accept lower wages to keep their jobs. The wage W will refuse relative to the price level P and the real wage W/P will fall. Since the real wage falls firms will hire more workers.

Presume firms want to hire more workers than there are people in the labor force. A few firms will try to bid workers away from other firms by offering higher wages. The real wage W/P will increase. Since the real wage rises employers will reduce the quantity of labor they demand.

Therefore in equilibrium, labor demand Ld will equal the labor force L:

1056_labor demand.jpg

Labor demand is equivalent to the labor force when the real wage W/P is:

2082_labor demand and real wage.jpg

And each of the K firms in the economy occupies L/K workers. As long as wages as well as prices are flexible enough for this adjustment process to work the economy will stay at full employment.

Note that a full employment economy isn’t necessarily the best or even a good economy. The real incomes of those who do not own chunks of the capital stock are their real wages: W/P = (1-α) x (Y/L). If α is large their real incomes will be small as well as social welfare may be low.

When the labor market is in equilibrium the usual firm produces a level of output equal to:

686_firms level of output.jpg

For the reason that there are K firms total output Y is simply K times the typical firm’s output: Y = K x Yfirm. This is the similar potential output Y* in the Cobb-Douglas form of the production function analysis:

1229_potential output of cobb douglas.jpg

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