# Factor Demands (and more generally, deriving a cost function)

## Deriving a Cost Function

Outputs are produced from inputs. q = F(x1,x2) is a production function, it gives the quantity of output as a function of the quantities of inputs. It is a physical relationship, not an economic one.

The q*'th isoquant (also the isoquant that makes output q*) is the set of all inputs that exactly make q*. That is all the (x1,x2) pairs for which F(x1,x2) =q*.

An example due to Bressler (1952 Cardboard Model) is y = 20+ 6.67 x2 + 10 x3 - .5 x32 . View the isoquants and production surface.

Lipsey and Courant, Microeconomics, Appendix to Chapter 9 has a similar discussion.

Cost, C(Q,w1,w2) is the least amount of money needed to buy the inputs that produce output Q when the prices of the inputs are w1,w2.

### Graphical Construction of Cost Curves

Assume you have a graph with the quantities of the goods on the axes and an isoquant or isoquants already drawn in.

To find a single point on C(Q*,w1=3,w2=3), proceed as follows: All bundles of inputs that cost the same lie on the same line. The equaton of such a line is c = w1 x1 + w2 x2. Only the w's are known. However all lines of that type (that is all lines with the same w's and different c) have slope -w1/w2. The first step is to draw an isocost line with the slope -w1/w2. The line with vertical and horizontal intercepts of 5 is an isocost line with slope of -w1/w2=-3/3=-1. It nowhere touches the lower or Q* isoquant so no point on this line can actually produce Q*. Now, by eye, draw an isocost line parallel to the first line, and tangent to Q*. Since it is parallel, it has the same prices. Since it is tangent, at the point of tangency, Q* is produced. The cost anywhere on this line is the same: The tangency between the lower isocost line and the q* isoquant is at about the point (3,6). Thus, when w1=w2=3 this way of producing q* is to use 3 units of the first input and 6 units of the second input. The cost of making q* is w1 x1 + w2 x2 = 3 times 3 + 3 times 6 = 27. There is another way, in the diagram to make Q*. Look at the point where the upper isocost line intersects the isoquant Q*. It is the point (10.5,3.5) and it would cost 42 to produce at this point. Now consider the three possible isocost lines. The lower one does not have points that produce Q* and the upper one, while it does have a point that produces Q*, costs more than the middle or tangent line. Thus the point of tangency is the input mix that costs least of all the ways to produce Q*. The inputs that cost the least amount of money and produce Q* are (3,6) and these inputs cost 27. Thus C(Q*,3,3) = 27.

What of C(Q**,3,3), the cost of producing on the upper isoquant? The point of tangency appears to be (4.5,9.5) and so C(Q**,3,3)= 42. There is another way to find what the input bundle (4.5,9.5) costs. Look at the vertical intercept for this isocost line (0,14). This too is a bundle on the isocost line and must cost the same as any other bundle on the line. It too costs 42.

## Conditional Factor Demand

Defintion: The conditional factor demand for x1 is the cost minimizing amount of x1 purchased by a firm as a function of w1, Q and w2 held constant.

To find the quantity of x1 used by firms as a function of the price, w1, draw an isoquant with several tangent isocost lines. The diagram below is such an example..

Based on the graph and the assumption that w2 = 2, find the price w1: On the isocost line with intercept 8, -w1/w2 = -8/11 = -w1/2 or w1 =16/11. The tangency occurs at x1 = 3, so a point on the conditional factor demand function is given by 3 = D(32/39=w1,2=w2,q*), On the isocost line with an intercept of 6, -w1/w2 = -6/11 so w1=12/11. The tangency occurs at x1= 6.5 so 6.5= D(12/11,2,q*). Try using a ruler against the screen and finding another point on the conditional factor demand; put one end of your ruler on 4 on the vertical axis, make your ruler tangent to isoquant q*, read off the point of tangency and the horizontal intercept. Now do the exercise above to find another point on D(w1,2,q*). The chart below is then used to plot the factor demand:

 price of input 1 quantity of input 1 32/39 3 12/11 6.5

For an environmental economist, the interesting question is what happens when one of the inputs is something like air and society does not charge enough for it. State the problem this way: A firm makes use of clean air services to carry away its pollution. It only has to pay for the fans to blow the pollution into the air and does not make any other payments for its waste disposal. Since clean air services are demanded by others for breathing, clean air services should have a price higher than just the cost of running the fans. Thus the price of clean air services, from a social standpoint, is too low. Using the graph above it is easy to see that the quantity of clean air services used per unit of output is too high.