The Cost-Minimization Problem We said a firm maximizes profit. Part of this involves minimizing costs for any given level of output. For given w 1, w 2 and y, the firm’s cost- minimization problem is to solve subject to

The Cost-Minimization Problem The levels x 1 *(w 1,w 2,y) and x 1 *(w 1,w 2,y) in the least-costly input bundle are the firm’s conditional demands for inputs 1 and 2. The (smallest possible) total cost for producing y output units is therefore

How do we solve this problem? We can draw isocost lines. That is, all input bundles that have the same cost. We can draw the isoquant curve going through that level of output. The cost associated with the lowest isocost curve that has a point on the isoquant is the cost. With smooth preferences, TRS=-w1/w2

Cobb-Douglas example Cobb-Douglas function. What is the TRS? Use TRS=-w1/w2 and the production function. What is demand and the demand expansion path? What is the cost function c(y)?

Other technologies What is the cost function c(y) of fixed proportions y=min{x1,x2}? What is the cost function of perfect substitutes y=x1+x2?

Returns to Scale. CRS would have c(2y)=2c(y). IRS would have c(2y)<2c(y). DRS would have c(2y)>2c(y). What does this say about average cost? Hint: average cost=c(y)/y.

Average cost. What does average cost imply about the shape of the cost curve? Average cost is just the slope of the line from the origin to the point on the cost curve (y,c(y)). DRS implies this is increasing, IRS decreasing, CRS constant. When does a function switch from IRS to DRS?