The Cost-Minimization Problem

1
The Cost-Minimization Problem

• We said a firm maximizes profit. Part of this
  involves minimizing costs for any given level of
  output.
• For given w1, w2 and y, the firms
  cost-minimization problem is to solve

subject to
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The Cost-Minimization Problem

• The levels x1(w1,w2,y) and x1(w1,w2,y) in the
  least-costly input bundle are the firms
  conditional demands for inputs 1 and 2.
• The (smallest possible) total cost for producing
  y output units is therefore

3
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

4
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)?

5
Other technologies

• What is the cost function c(y) of fixed
  proportions yminx1,x2?
• What is the cost function of perfect substitutes
  yx1x2?

6
Returns to Scale.

• CRS would have c(2y)2c(y).
• IRS would have c(2y)lt2c(y).
• DRS would have c(2y)gt2c(y).
• What does this say about average cost? Hint
  average costc(y)/y.

7
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?