Chapter 20 Cost
Minimization
20.1 Cost Minimization

Cost minimization: The cheapest way to
produce a given amount of output.
min w1 x1  w2 x2
x1 , x2

s.t.
f ( x1 , x2 )  y
The isocost line: All the combinations of
inputs that yield a given level of cost, C.
w1x1+w2x2=C

To minimizing cost, the firm looks for the
lowest isocost line tangent to the isoquant.
20.1 Cost Minimization
20.1 Cost Minimization
Consider any change in the pattern of
production (△x1, △x2) that keeps output
constant.
 Such a change must satisfy
MP1△x1+ MP2△x2=0
 If we are at the cost minimum, then this
change cannot lower costs, so we have
w1△x1+w2△x2≥0

20.1 Cost Minimization





Now consider the change (-△x1, -△x2).
It also produces a constant level of output.
It too cannot lower costs.
-w1△x1- w2△x2≥0
These implies
w1△x1+ w2△x2=0
It follows that
△x2/△x1 =-w1/w2=-MP1/MP2
20.1 Cost Minimization

The Lagrange multiplier method
L  w1 x1  w2 x2   ( f ( x1 , x2 )  y)

F.O.C.
L
f
 w1  
0
x1
x1
L
f
 w2  
0
x2
x2
w1 f

w2 x1
f
 TRS12
x2
20.1 Cost Minimization

Cost function: c(w1, w2, y)
 The
value function of the cost minimization
problem.

Conditional factor demand functions: x1(w1,
w2, y) and x2(w1, w2, y)
 the
optimal choice of the cost minimization
problem.
EXAMPLE: Minimizing Costs for
Specific Technologies

Perfect complements: f(x1, x2)=min{x1, x2}
 c(w1,

Perfect substitutes: f(x1, x2)= x1+ x2
 c(w1,

w2, y)= w1y+w2y=(w1+w2)y
w2, y)= min{w1, w2}y
Cobb-Douglas: f(x1, x2)= x1ax2b
 c(w1,
w2, y)=Kw1a/a+bw2b/a+by1/a+b
20.2 Revealed Cost Minimization

WACM
 Produce
the same quantity y.
 Choose (x1t, x2t) when the prices are (w1t, w2t).
 Choose (x1s, x2s) when the prices are (w1s, w2s).
20.2 Revealed Cost Minimization

Cost minimization requires that
w1tx1t+w2tx2t ≤w1tx1s+w2tx2s
w1sx1s+w2sx2s ≤w1sx1t+w2sx2t
(w1t-w1s)( x1t-x1s)+(w2t-w2s)(x2t-x2s)≤0
△w1△x1+△w2△x2≤0
20.3 Returns to Scale and the Cost
Function

Unit cost function: c(w1, w2, 1)
 The
minimal cost to produce 1 unit of output.
Average cost function: AC(y)= c(w1, w2, y)/y
 Constant returns to scale:
c(w1, w2, y)=c(w1, w2, 1)y
AC(w1, w2, y )=c(w1, w2, 1)y/y=c(w1, w2, 1)

20.3 Returns to Scale and the Cost
Function

Increasing returns to scale
 The
cost increases less than linearly w/r to output.
 The average cost is declining in output.

Decreasing returns to scale
 The
cost increases more than linearly w/r to output.
 The average cost will rise as output increases.
20.4 Long-Run and Short-Run Costs

Short-run cost function
 The
minimum cost to produce a given level of
output, only adjusting the variable factors.
cs ( y, x2 )  min w1 x1  w2 x2
x1
s.t.
f ( x1 , x2 )  y
20.4 Long-Run and Short-Run Costs

The short-run factor demand function
x1  x1s (w1, w2 , x2 , y)
x2  x2

The short-run cost function can be written as
cs ( y, x2 )  w x ( w1 , w2 , x2 , y )  w2 x2
s
1 1
20.4 Long-Run and Short-Run Costs

Long-run cost function
 The
minimum cost of producing a given level of
output, adjusting all factors.

The long-run cost function can be written as
c( y)  min cs ( y, x2 )
x2

The optimal solution being
x2  x2 ( w1, w2 , y )
20.4 Long-Run and Short-Run Costs
It follows that
x1 (w1, w2 , y)  x1s (w1, w2 , x2 (w1, w2 , y), y)
 The long-run cost is the minimal short-run cost.
 The firm chooses the long-run optimal quantity
of the fixed factor.
 The long-run quantity of the variable factor
equals that of the short-run.

20.5 Fixed and Quasi-Fixed Costs

Fixed costs
 Costs
associated with the fixed factors.
 No fixed costs in the long run.

Quasi-fixed cost
 Costs
need to be paid if the firm produces a
positive amount of output.
 Quasi-fixed costs are possible in the long-run.