cost minimization and cost curves...3 cost function a cost function is a function that maps a set of...
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1
Cost Minimization and Cost Curves
Beattie, Taylor, and WattsSections: 3.1a, 3.2a-b, 4.1
2
Agenda The Cost Function and General Cost
Minimization Cost Minimization with One Variable
Input Deriving the Average Cost and Marginal
Cost for One Input and One Output Cost Minimization with Two Variable
Inputs
3
Cost Function A cost function is a function that maps
a set of inputs into a cost. In the short-run, the cost function
incorporates both fixed and variable costs.
In the long-run, all costs are considered variable.
4
Cost Function Cont. The cost function can be represented as
the following: C = c(x1, x2, …,xn)= w1*x1 + w2*x2 + … +
wn*xn Where wi is the price of input i, xi, for i =
1, 2, …, n Where C is some level of cost and c(•) is a
function
5
Cost Function Cont. When there are fixed costs, the cost function
can be represented as the following: C = c(x1, x2| x3, …,xn)= w1*x1 + w2*x2 + TFC Where wi is the price of the variable input i, xi, for
i = 1 and 2 Where wi is the price of the fixed input i, xi, for i =
3, 4, …, n Where TFC = w3*x3 + … + wn*xn
When inputs are fixed, they can be lumped into one value which we usually denote as TFC
x3 , … , xn are held to some constant values that do not change
Cost Function Cont. Suppose we have the following cost
function: C = c(x1, x2, x3)= 5*x1 + 9*x2 + 14*x3 If x3 was held constant at 4, then the cost
function can be written as: C = c(x1, x2| 4)= 5*x1 + 9*x2 + 56
Where TFC in this case is 56
6
7
Cost Function Cont. The cost function is usually meaningless
unless you have some constraint that bounds it, i.e., minimum costs occur when all the inputs are equal to zero.
8
Standard Cost Minimization Model Assume that the general production function
can be represented as y = f(x1, x2, …, xn).
),...,,(:subject to
...
21
2211,...,,... 21
n
nnxxxtrw
xxxfy
xwxwxwMinn
=
+++
9
Cost Minimization with One Variable Input Assume that we have one variable input (x)
which costs w. Let TFC be the total fixed costs.
Assume that the general production function can be represented as y = f(x).
)(:subject to...
xfy
TFCwxMinxtrw
=
+
Lagrangian Solution for One Variable Input Model Γ 𝑥𝑥 = 𝑤𝑤𝑥𝑥 + 𝑇𝑇𝑇𝑇𝑇𝑇 + λ 𝑦𝑦 − 𝑓𝑓 𝑥𝑥 FOC =>
𝜕𝜕Γ 𝑥𝑥𝜕𝜕𝑥𝑥
= 𝑤𝑤 − λ 𝜕𝜕𝑓𝑓 𝑥𝑥𝜕𝜕𝑥𝑥
= 0
𝜕𝜕Γ 𝑥𝑥𝜕𝜕λ
= 𝑦𝑦 − 𝑓𝑓 𝑥𝑥 = 0
λ = 𝑤𝑤𝜕𝜕𝑓𝑓 𝑥𝑥𝜕𝜕𝑥𝑥
= MC
10
11
Cost Minimization with One Variable Input Cont. In the one input, one output world, the
solution to the minimization problem is trivial. By selecting a particular output y, you are
dictating the level of input x. The key is to choose the most efficient
input to obtain the output.
12
Example of Cost Minimization Suppose that you have the following production
function: y = f(x) = 6x - x2
You also know that the price of the input is $10 and the total fixed cost is 45.
2
...
6)(:subject to
4510
xxxfy
xMinxtrw
−==
+
13
Example of Cost Minimization Cont. Since there is only one input and one output,
the problem can be solved by finding the most efficient input level to obtain the output.
yx
yx
yx
yx
yxxxxy
−±=
−±=
−±=⇒
−−±−−=⇒
=+−⇒
−=
932
9462
43662
))(1(4)6()6(
066
2
2
2
14
Example of Cost Minimization Cont. Given the previous, we must decide whether
to use the positive or negative sign. This is where economic intuition comes in. The one that makes economic sense is the
following:
yx −−= 93
15
Cost Function and Cost Curves There are many tools that can be used
to understand the cost function: Average Variable Cost (AVC) Average Fixed Cost (AFC) Average Cost (ATC) Marginal Cost (MC)
16
Average Variable Cost Average variable cost is defined as the cost
function without the fixed costs divided by the output function.
APPwAVC
xfwxAVC
ywxAVC
=
=
=
)(
17
Average Fixed Cost Average fixed cost is defined as the cost
function without the variable costs divided by the output function.
)(xfTFCAFC
yTFCAFC
=
=
18
Average Total Cost Average total cost is defined as the cost
function divided by the output function. It is also the summation of the average fixed
cost and average variable cost.
)()( xfTFC
xfwxATC
yTFC
ywxATC
yTFCwxATC
+=
+=
+=
19
Marginal Cost Marginal cost is defined as the derivative of the cost
function with respect to the output. To obtain MC, you must substitute the production
function into the cost function and differentiate with respect to output.
MPPwMC
xfyfyfwyTVC
TFCyfwyTCdy
ydTVCdy
ydTCMC
=
=
=
+=
==
−
−
−
)(y of inverse is )()(*)(
)(*)(
)()(
1
1
1
20
Example of Finding Marginal Cost Using the production function y = f(x)
= 6x - x2, and a price of 10, find the MC by differentiating with respect to y.
To solve this problem, you need to solve the production function for x and plug it into the cost function. This gives you a cost function that is a
function of y.
21
Example of Finding Marginal Cost Cont.
−==
−==
−−−==
−−==⇒
−−==
−−=⇒
−=
−
−
ydydcMC
ydydcMC
ydydcMC
y
y
yx
xxy
95
)9(10*21
)1()9(10*210
91030c(y)C
)9310(c(y)C
:givesfunction cost theinto thisPlugging93
6
21
21
2
22
Notes on Costs MC will meet AVC and ATC from below
at the corresponding minimum point of each. Why?
As output increases AFC goes to zero. As output increases, AVC and ATC get
closer to each other.
23
Production and Cost Relationships Summary Cost curves are derived from the
physical production process. The two major relationships between
the cost curves and the production curves: AVC = w/APP MC = w/MPP
24
Product Curve Relationships Cont.
)(')(
0)(')(
0)(')(
0)]([
)(')()(
2
xxfxf
xxfxf
xwxfxwf
xfxwxfxwf
dxdAVC
xfwxAVC
>
<
>
<
>
<
>
<
=⇒
=−⇒
=−⇒
=−
=
=
25
Product Curve Relationships Cont.
AVCMC
APPw
MPPw
APPMPP
MPPAPP
xfxxf
xxfxf
>
<
>
<
>
<
>
<
>
<
>
<
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
11
)(')(
)(')(
26
Product Curve Relationships When MPP>APP, APP is increasing.
=> MC<AVC, then AVC is decreasing. When MPP=APP, APP is at a maximum.
=> MC=AVC, then AVC is at a minimum. When MPP<APP, APP is decreasing.
=> MC>AVC, then AVC is increasing.
27
Example of Examining the Relationship Between MC and AVC
Given that the production function y = f(x) = 6x - x2, and a price of 10, find the input(s) where AVC is greater than, equal to, and less than MC.
To solve this, examine the following situations: AVC = MC AVC > MC AVC < MC
28
Example of Examining the Relationship Between MC and AVC Cont.
002
26626
10610
2610
610
610
)( 2
=⇒=⇒
−=−⇒−
=−
⇒
=−
==
−=
−==
xx
xxxx
MCAVCxMPP
wMC
xxxx
xfwxAVC
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Example of Examining the Relationship Between MC and AVC Cont.
xxx
xxxx
MCAVCxMPP
wMC
xxxx
xfwxAVC
>⇒−>−⇒
−>−⇒−
>−
⇒
>−
==
−=
−==
02
62626
10610
2610
610
610
)( 2
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Example of Examining the Relationship Between MC and AVC Cont.
026626
10610
2610
610
610
)( 2
>⇒−>−⇒−
<−
⇒
<−
==
−=
−==
xxxxx
MCAVCxMPP
wMC
xxxx
xfwxAVC
31
Review of the Iso-Cost Line The iso-cost line is a graphical
representation of the cost function with two inputs where the total cost C is held to some fixed level. C = c(x1,x2)=w1x1 + w2x2
32
Finding the Slope of the Iso-Cost Line
2
1
1
2
2
11
22
2
11
22
22
1122
2211
ww
dxdx
wxw
wCx
wxw
wC
wxw
xwCxwxwxwC
−=
−=⇒
−=⇒
−=⇒+=
33
Example of Iso-Cost Line Suppose you had $1000 to spend on
the production of lettuce. To produce lettuce, you need two
inputs labor and machinery. Labor costs you $10 per unit, while
machinery costs $100 per unit.
34
Example of Iso-Cost Line Cont. Given the information above we have
the following cost function: C = c(labor, machinery) = $10*labor +
$100*machinery 1000 = 10*x1 + 100*x2
Where C = 1000, x1 = labor, x2 = machinery
35
Example of Iso-Cost Line Graphically
x2
x1
10
100
X2 = 10 – (1/10)*x1
36
Finding the Slope of the Iso-Cost Line
101
1010
10010
1001000
100100
101000100100101000
1
2
12
12
12
21
−=
−=⇒
−=⇒
−=⇒+=
dxdx
xx
xxxx
xx
37
Notes on Iso-Cost Line As you increase C, you shift the iso-cost
line parallel out. As you change one of the costs of an
input, the iso-cost line rotates.
38
Cost Minimization with Two Variable Inputs Assume that we have two variable inputs (x1
and x2) which cost respectively w1 and w2. We have a total fixed cost of TFC.
Assume that the general production function can be represented as y = f(x1,x2).
),(:subject to 21
2211,... 21
xxfy
TFCxwxwMinxxtrw
=
++
39
First Order Conditions for the Cost Minimization Problem with Two Inputs
( )
0),(
0
0
0
0
),(),,(
21
2
22
2
1
11
1
21221121
2
1
=−=∂Γ∂
=−⇒
=∂∂
−=∂Γ∂
=−⇒
=∂∂
−=∂Γ∂
−+++=Γ
xxfy
MPPwxfw
x
MPPwxfw
x
xxfyTFCxwxwxx
x
x
λ
λ
λ
λ
λ
λλ
40
Implication of MRTS = Slope of Iso-Cost Line Slope of iso-cost line = -w1/w2, where w2 is
the cost of input 2 and w1 is cost of input 1. MRTS = -MPPx1/MPPx2 This implies MPPx1/MPPx2 = w1/w2 Which implies MPPx1 /w1 = MPPx2/w2 This means that the MPP of input 1 per dollar
spent on input 1 should equal MPP of input 2 per dollar spent on input 2.
41
Example 1 of Cost Minimization with Two Variable Inputs
Suppose you have the following production function: y = f(x1,x2) = 10x1
½ x2½
Suppose the price of input 1 is $1 and the price of input 2 is $4. Also suppose that TFC = 100.
What is the optimal amount of input 1 and 2 if you want to produce 20 units.
42
Example 1 of Cost Minimization with Two Variable Inputs Cont. Summary of what is known:
w1 = 1, w2 = 4, TFC = 100 y = 10x1
½ x2½
y = 20
21
221
1
21,...
10:subject to
100421
xxy
xxMinxxtrw
=
++
43
Example 1 of Cost Minimization with Two Variable Inputs Cont.
classin doneSolution
010
021104
021101
1010041),2,1(
21
221
1
21
221
12
21
221
11
21
221
121
=−=∂Γ∂
=
−=
∂Γ∂
=
−=
∂Γ∂
−+++=Γ
−
−
xxy
xxx
xxx
xxyxxxx
λ
λ
λ
λλ
44
Solving Example 1 Using Ratio of MPP’s Equals Absolute Value of the Slope of the Cost Function
4x1x
204
20
xx20
)(x)10(4xy
function production into Plug3.
441
41Set 2.
.1
1
2
12
21
221
2
21
221
2
12
1
2
2
1
1
2
2
1
=⇒=⇒
==
=⇒
=
=⇒
=⇒
==
==
yxandyx
y
xxxx
wwMRTS
xx
MPPMPP
MRTSx
x
45
Solving Example 1 Using MRTS from the Isoquant and Setting it Equal to the Slope of the Cost Function
4x1x
205100
usingfor x Solve .4520
441
100
41Set3.
100MRTS
MRTS 2.Find100100
Isoquant Find .1
1
2
12
2
12
1
21
2
2
1
21
2
1
2
11
2
1
2
2
=⇒=⇒
=
=
==⇒
−=−⇒
−=−=
−=∂∂
=
==
−
−
−
−
yyyx
x
yyx
xy
wwMRTS
xyxx
xyx
yx
46
Final Note on Input Selection You want to have the iso-cost line
tangent to the isoquant. This implies that you will set the absolute
value of MRTS equal to the absolute value of the slope of the iso-cost line.