Title: Chapter 20 Cost Minimization
1Chapter 20Cost Minimization
2Cost Minimization
- A firm is a cost-minimizer if it produces any
given output level y ³ 0 at smallest possible
total cost. - c(y) denotes the firms smallest possible total
cost for producing y units of output. - c(y) is the firms total cost function.
3Cost Minimization
- When the firm faces given input prices w
(w1,w2,,wn) the total cost function will be
written as c(w1,,wn,y).
4The Cost-Minimization Problem
- Consider a firm using two inputs to make one
output. - The production function is y f(x1,x2).
- Take the output level y ³ 0 as given.
- Given the input prices w1 and w2, the cost of an
input bundle (x1,x2) is w1x1 w2x2.
5The Cost-Minimization Problem
- For given w1, w2 and y, the firms
cost-minimization problem is to solve
s.t.
6The Cost-Minimization Problem
- The levels x1(w1,w2,y) and x2(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
7Conditional Input Demands
- Given w1, w2 and y, how is the least costly input
bundle located? - And how is the total cost function computed?
8Iso-cost Lines
- A curve that contains all of the input bundles
that cost the same amount is an iso-cost curve. - E.g., given w1 and w2, the 100 iso-cost line has
the equation
9Iso-cost Lines
- Generally, given w1 and w2, the equation of the
c iso-cost line isi.e. - Slope is - w1/w2.
10Iso-cost Lines
x2
Slopes -w1/w2.
c º w1x1w2x2
c º w1x1w2x2
c lt c
x1
11The y-Output Unit Isoquant
x2
All input bundles yielding y unitsof output.
Which is the cheapest?
f(x1,x2) º y
x1
12The Cost-Minimization Problem
x2
All input bundles yielding y unitsof output.
Which is the cheapest?
f(x1,x2) º y
x1
13The Cost-Minimization Problem
x2
All input bundles yielding y unitsof output.
Which is the cheapest?
x2
f(x1,x2) º y
x1
x1
14The Cost-Minimization Problem
At an interior cost-min input bundle(a)
and(b) slope of isocost slope
of
isoquant i.e.
x2
x2
f(x1,x2) º y
x1
x1
15A Cobb-Douglas Example of Cost Minimization
- A firms Cobb-Douglas production function is
- Input prices are w1 and w2.
- What are the firms conditional input demand
functions?
16A Cobb-Douglas Example of Cost Minimization
At the input bundle (x1,x2) which minimizes the
cost of producing y output units (a)(b)
and
17A Cobb-Douglas Example of Cost Minimization
(b)
(a)
From (b),
Now substitute into (a) to get
So
is the firms conditionaldemand for input 1.
18A Cobb-Douglas Example of Cost Minimization
and
Since
is the firms conditional demand for input 2.
19A Cobb-Douglas Example of Cost Minimization
So for the production function the cheapest
input bundle yielding y output units is
20A Cobb-Douglas Example of Cost Minimization
So the firms total cost function is
21A Perfect Complements Example of Cost Minimization
- The firms production function is
- Input prices w1 and w2 are given.
- What are the firms conditional demands for
inputs 1 and 2? - What is the firms total cost function?
22A Perfect Complements Example of Cost Minimization
x2
4x1 x2
Where is the least costly input bundle
yielding y output units?
min4x1,x2 º y
x2 y
x1 y/4
x1
23A Perfect Complements Example of Cost Minimization
The firms production function is
and the conditional input demands are
and
So the firms total cost function is
24Average Total Production Costs
- For positive output levels y, a firms average
total cost of producing y units is
25Returns-to-Scale and Average Total Costs
- The returns-to-scale properties of a firms
technology determine how average production costs
change with output level. - Our firm is presently producing y output units.
- How does the firms average production cost
change if it instead produces 2y units of output?
26Constant Returns-to-Scale and Average Total Costs
- If a firms technology exhibits constant
returns-to-scale then doubling its output level
from y to 2y requires doubling all input
levels. - Total production cost doubles.
- Average production cost does not change.
27Decreasing Returns-to-Scale and Average Total
Costs
- If a firms technology exhibits decreasing
returns-to-scale then doubling its output level
from y to 2y requires more than doubling all
input levels. - Total production cost more than doubles.
- Average production cost increases.
28Increasing Returns-to-Scale and Average Total
Costs
- If a firms technology exhibits increasing
returns-to-scale then doubling its output level
from y to 2y requires less than doubling all
input levels. - Total production cost less than doubles.
- Average production cost decreases.
29Returns-to-Scale and Av. Total Costs
/output unit
AC(y)
decreasing r.t.s.
constant r.t.s.
increasing r.t.s.
y
30Returns-to-Scale and Total Costs
- What does this imply for the shapes of total cost
functions?
31Returns-to-Scale and Total Costs
Av. cost increases with y if the
firmstechnology exhibits decreasing r.t.s.
c(y)
c(2y)
Slope c(2y)/2y AC(2y).
Slope c(y)/y AC(y).
c(y)
y
y
2y
32Returns-to-Scale and Total Costs
Av. cost decreases with y if the
firmstechnology exhibits increasing r.t.s.
c(y)
c(2y)
Slope c(2y)/2y AC(2y).
c(y)
Slope c(y)/y AC(y).
y
y
2y
33Returns-to-Scale and Total Costs
Av. cost is constant when the firmstechnology
exhibits constant r.t.s.
c(2y) 2c(y)
c(y)
Slope c(2y)/2y 2c(y)/2y
c(y)/y so AC(y) AC(2y).
c(y)
y
y
2y
34Short-Run Long-Run Total Costs
- In the long-run a firm can vary all of its input
levels. - Consider a firm that cannot change its input 1
level from x1 units. - How does the short-run total cost of producing y
output units compare to the long-run total cost
of producing y units of output?
35Short-Run Long-Run Total Costs
- The long-run cost-minimization problem is
- The short-run cost-minimization problem is
s.t.
s.t.
36Short-Run Long-Run Total Costs
x2
Consider three output levels.
x1
37Short-Run Long-Run Total Costs
In the long-run when the firmis free to choose
both x1 andx2, the least-costly inputbundles
are ...
x2
x1
38Short-Run Long-Run Total Costs
x2
Long-run costs are
Long-runoutputexpansionpath
x1
39Short-Run Long-Run Total Costs
- Now suppose the firm becomes subject to the
short-run constraint that x1 x1.
40Short-Run Long-Run Total Costs
Short-runoutputexpansionpath
Long-run costs are
x2
x1
41Short-Run Long-Run Total Costs
Short-runoutputexpansionpath
Long-run costs are
x2
Short-run costs are
x1
42Short-Run Long-Run Total Costs
- Short-run total cost exceeds long-run total cost
except for the output level where the short-run
input level restriction is the long-run input
level choice. - This says that the long-run total cost curve
always has one point in common with any
particular short-run total cost curve.
43Short-Run Long-Run Total Costs
A short-run total cost curve always hasone point
in common with the long-runtotal cost curve, and
is elsewhere higherthan the long-run total cost
curve.
cs(y)
c(y)
y
44Fixed Costs and Variable Costs
- Fixed costs are associated with fixed factors.
They are independent of the level of output and
must be paid even if the firm produces zero
output. - Variable costs only need to be paid if the firm
produce a positive amount of output.
45Sunk Costs
- Sunk cost an expenditure that has been made and
cannot be recovered. - Once a sunk cost occurs, it should not affect a
firms decision.