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Economics: Deriving the Factor Demand Curve


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  • Type: Video Tutorial
  • Length: 15:10
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 163 MB
  • Posted: 03/29/2010

This lesson is part of the following series:

Economics: Full Course (269 lessons, $198.00)
Economics: Resource Markets (5 lessons, $9.90)
Economics: The Derived Demand for Labor (3 lessons, $6.93)

In this video lesson, you will learn about deriving the Factor Demand Curve. Taught by Professor Tomlinson, this video lesson was selected from a broader, comprehensive course, Economics. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers economic thinking, markets, consumer choice, household behavior, production, costs, perfect competition, market models, resource markets, market failures, market outcomes, macroeconomics, macroeconomic measurements, economic fluctuations, unemployment, inflation, the aggregate expenditures model, banking, spending, saving, investing, aggregate demand and aggregate supply model, monetary policy, fiscal policy, productivity and growth, and international examples.

Steven Tomlinson teaches economics at the Acton School of Business in Austin, Texas. He graduated with highest honors from the University of Oklahoma and earned a Ph.D. in economics at Stanford University. Prof. Tomlinson's academic awards include the prestigious Texas Excellence Teaching Award given by the University of Texas Alumni Association and being named "Outstanding Core Faculty in the MBA Program" several times. He has developed several instructional guides and computerized educational programs for economics.

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We've seen how a profit-maximizing firm continues to increase its output until the marginal cost is equal to the price of the good. Or, in the case of a firm with market power, the firm continues increasing production until the marginal cost is equal to the marginal revenue. In the short run, remember that a firm can only increase its output by hiring more of the variable input - for instance, labor. Everything else is fixed, so if you want to make more stuff, you've got to get more workers. This means that in the short run, there's an equally good way to describe the condition of profit maximization, and that is that the firm keeps hiring workers until the last worker add just enough revenue to the firm to cover the cost of that worker.
What we're going to do now is consider this alternative way of describing profit maximization. And the payoff will be that we come up with a demand curve for labor, a derived demand - that is, firms hire labor because of the things that you want to buy from the companies that use labor to make them.
Let's start by defining the concept of marginal revenue product. The marginal revenue product is equal to the change in the firm's total revenue that results when it hires an extra worker. The change is what's added to the firm's revenue as a result of the decision to bring another worker into the factory. The marginal revenue product has two components. The first component is the marginal revenue - that is, how much extra revenue is added when the firm produces another unit of output. And the second component is the marginal product - that is, how much output is added when the firm hires an extra worker.
If we flip these over and look at the definitions of the marginal revenue and the marginal product, you can see clearly the relationship between the marginal revenue product and these two concepts that we've learned elsewhere. That is, when a new worker comes into the firm, that worker adds a certain amount of output; and that certain amount of output is going to have a direct effect on the total revenue.
Now consider how this output is going to influence the total revenue. If the firm has market power, then the firm has to lower its price to sell this extra output, and we become very interested in the elasticity of demand and the marginal revenue. But to make this problem easier, let's imagine for the time being that we're dealing with a competitive firm - that's a firm that's able to sell as much output as it likes without influencing the market price of its good. If that's the case, then the change in total revenue that results from a change in output - the extra revenue that's added whenever you sell an extra unit of the good - is simply equal to the price of the good. So we can replace marginal revenue with the goods price. It's always true for a competitive firm that price is equal to marginal revenue.
So now we have a simpler version of the marginal revenue product, and this simpler version has its own name - that is the "value" of the marginal product. The value of the marginal product is defined as the price of the good times the extra output that's produced by the last worker hired. So if you want to know how much a worker adds to the revenue of a competitive firm, the value of the marginal product answers that question; it's simply the physical output of the worker multiplied by the price at which that good sells in the market.
So let's go ahead and use this concept on a firm that we've looked at before, and then see how that firm might use the information in choosing how much labor to hire whenever it's after profit maximization. So here we go to a set of data that we've considered before - the production possibilities of this firm in the short run. We can hire one worker, two workers, three workers, four, and so forth. And here we have the physical output of any given number of workers. Remember we're in the short run with certain fixed inputs, like the size of our factory and the tools the workers have to work with. We're varying only a single input - labor. So one worker can make two television sets a week, two workers can make ten television sets a week, and so forth.
The next thing we want to do as we're trying to figure out how much labor to hire is look at the physical productivity of each additional worker; and we've calculated this information before - it's called the marginal product of labor. So the first worker takes us from zero television sets produced to two television sets produced. The marginal product of that first unit of labor is two television sets.
The second worker brings the total production up to ten units. So ten, minus those two units that we got when we had only a single worker, gives us a marginal product of eight units for the second worker, and so forth. In each case, look at the number of television sets that can be produced when another worker is added, and subtract what was produced without that worker, and you get the marginal product of that last worker. The third worker has a marginal product of 20, the fourth worker has a marginal product of 10, and so forth.
Now, let's suppose that the price of a television set is $100.00. That means that this competitive firm can sell all the television sets it wants to at a price of $100.00 apiece; and that, because the firm has no market power, they don't have to worry about saturating the market. But they will be concerned with how much revenue is added by each additional worker they put in their factory.
So to calculate now the value of the marginal product in this case, we're going to multiply the physical output of each additional worker by the market price of the television - $100.00 - and that's going to give us the contribution of each additional worker to the revenue of the firm, an amount measured in money. And we call that amount the value of the marginal product.
So now I'm ready to add another column to my table, and I add that column by taking the physical output of each worker, multiplying it by the price of a television set, and recording it, and that becomes the value of the marginal product. So the first worker produces two television sets that weren't there before. If we multiply two television sets by the price of $100.00 a television set, we get the value of the marginal product of that first worker is going to be $200.00 - two television sets at $100.00 apiece. Multiply the eight television sets that are added by the second worker by the price of $100.00, and we have the value of the marginal product is $800.00. I can keep going down this list. The third worker adds 20 television sets, so the value of the marginal product of that worker is 20 television sets of market price of $2,000.00. So value of the marginal product for the next worker, worker number four, is $1,000.00. The value of the marginal product for the next worker is $500.00. In each case it's as simple as multiplying the actual physical output of the worker by the price of the television sets. This last worker, by the way, has negative marginal product; the firm would never hire that worker anyway, so we don't even have to calculate that.
So what do we do with this information? What does the firm do with this information? The firm asks itself: "How many workers should we hire?" This is essentially the same question as: "How many television sets should we produce?" because, in the short run, the only way you can change your output is by hiring more or less of the variable input. By choosing the amount of labor to have on the factory floor, the firm is, in the short run, choosing how many TV sets to produce. The choice of having five workers is the choice to produce 45 television sets a week.
So how does the firm make that decision? Suppose the firm is considering going from four workers to five workers. You can see that that fifth worker adds $500.00 to the revenue of the firm. Is that enough? Well, it depends completely on how much you have to pay that worker. If the worker's wage is $300.00, by all means, hire that worker, because the worker is adding $500.00 to your revenue, but only $300.00 to your costs. On the other hand, if that worker costs $600.00, it wouldn't pay to make that extra hire, because the $500.00 in extra revenue wouldn't cover the costs of bringing that worker into your factory.
So now you can see how the firm is making its decision. Keep hiring workers up to the point at which that last worker add just enough to your revenue to cover the cost of bringing that worker onboard. This is the decision to continue hiring until the wage is equal to the value of the marginal product. Keep hiring your workers until congestion of your fixed inputs pushes productivity low enough that that last worker adds just enough output to allow you to cover the wage of hiring that worker.
So now we're ready to draw a demand curve for labor. The demand curve for labor is a graphical representation of this rule - that the firm continues to hire workers until the wage is equal to the value of the marginal product. If I graph the numbers that I just derived, the value of the marginal product turns into this nice green curve. The value of the marginal product is increasing when the marginal product is increasing, and then is decreasing when the marginal product is decreasing. That means this region over here is the region where teamwork and specialization are dominating the production process and this region over here is the region where the fixed inputs are becoming increasingly congested.
Now the firm that wants to maximize its profits is going to keep hiring labor as long as the value of the marginal product is greater than the wage. That is, as long as each additional worker is adding more to the firm's revenues than it's adding to the firm's costs.
You wouldn't stop back here with less labor than L*, because those extra workers are adding more to the firm's revenue than they're adding to the cost; the VMP is greater than the wage. You wouldn't proceed to a labor output that's greater than L*, because the workers are still costing the same wage, but they're adding now not enough to the firm's revenue to justify the expense. This is the point at which the firm has maximized its profits with respect to labor hires. Keep hiring labor until the last worker that you hire has just added enough to the firm's revenue to cover the wage.
This is a derived demand. "Derived" means that it's demand that comes from another demand. It is because customers want to buy the output of this firm that this firm has a demand for labor. And anything that changes - either the technology of production or the demand for the final good - will shift this green curve. Consider, for example, what would happen if this good were to get very popular, and its price to rise. In that case, the green curve would shift upwards, as a higher price for each unit of output would mean that each additional unit is adding more to the firm's revenue. So an increase in price would shift the VMP curve upwards.
Technology could also do the same thing. If labor became more productive - perhaps because the firm acquires more capital in the long run - or if labor just learns how to do things better, then the value of the marginal product curve would shift upwards, because each additional worker hired adds more output, and, therefore, more to the firm's bottom line.
We've been calling this the value of the marginal product because we've maintained the assumption that the price was constant. That is, the firm was competitive, it had no market power. But you get exactly the same shape of a curve if the firm has market power; you just have to be careful about calculating marginal revenue, which depends on the elasticity of demand. In the end though, you get a very similar looking curve called the marginal revenue product curve. It has the same properties of the curve that we've been studying.
So let me conclude by making one observation then about this new rule for profit maximization of firms in the short run, this new rule that wage is equal to the value of the marginal product, or a wage is equal to the marginal revenue product. And that is that if you look closely at this rule, it's exactly the same as the old rule that we use for profit maximization, which was price equals marginal cost.
Let me go through this argument carefully with you. The rule that we've just derived says that the firm will keep hiring labor until the wage is equal to the value of the marginal product. Now the value of the marginal product is going to be the price of the good multiplied by the marginal product of the variable input. So price times marginal product is equal to the wage for the profit-maximizing firm. Now I'm going to move these symbols around just to get a little bit more room on the board. But watch what happens when we carefully consider what the marginal product is. The marginal product of labor is the change in output that you get when you hire an additional worker.
Well, the reciprocal of this marginal product is going to be the amount of labor that you need to hire to produce one additional unit of output. Follow my logic - if one additional worker can produce four television sets, then you need only 1/4th of a worker, or 1/4th of a worker's time, to produce an additional television set. So if we multiplied both sides of this equation by the reciprocal - that is, the labor requirement, how much labor do you need to produce an extra unit of output - we get a very familiar equation.
Look over here on this side. Here we have the changing labor necessary to produce another unit of output. Multiply by the wage - that is, what you have to pay to get that extra worker. Well, that's nothing more than the marginal cost of producing another television set. The labor you need multiplied by the wage it takes to hire that labor. So let's go ahead and re-label that expression "marginal cost." Now look at the other side of the equation. Here we have change in labor, change in labor, change in output, change in output. This term simply cancels; multiplication leaves us something equal to 1.
And here you have it, a familiar rule. The firm will continue adding workers, continue expanding output in search of higher profits, until marginal cost is equal to price. And that's the familiar rule for profit maximization. So whether you look at it as a matter of choosing your output, or whether you look at the firm's demand for labor, wage equals value of the marginal product. It all boils down to the same intuition, the same decision.
Resource Markets
The Derived Demand for Labor
Deriving the Factor Demand Curve Page [3 of 3]

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