cost minimization and cost curves
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Cost Minimization and Cost Curves. Beattie, Taylor, and Watts Sections: 3.1a, 3.2a-b, 4.1. 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 - PowerPoint PPT PresentationTRANSCRIPT
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Cost Minimization and Cost Curves
Beattie, Taylor, and WattsSections: 3.1a, 3.2a-b, 4.1
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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
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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.
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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
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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
Where C is some level of cost and c(•) is a function
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
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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.
There are two major uses of the cost function: To minimize cost given a certain output
level. To maximize output given a certain cost.
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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
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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
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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.
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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
12
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
yxx
xxy
93
2
946
2
4366
2
))(1(4)6()6(
06
6
2
2
2
13
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
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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)
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Average Variable Cost Average variable cost is defined as the
cost function without the fixed costs divided by the output function.
APP
wAVC
xf
wxAVC
y
wxAVC
)(
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Average Fixed Cost Average fixed cost is defined as the cost
function without the variable costs divided by the output function.
)(xf
TFCAFC
y
TFCAFC
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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.
)()( xf
TFC
xf
wxATC
y
TFC
y
wxATC
y
TFCwxATC
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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.
MPP
wMC
xfyf
yfwyTVC
TFCyfwyTC
dy
ydTVC
dy
ydTCMC
)(y of inverse is )(
)(*)(
)(*)(
)()(
1
1
1
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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.
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Example of Finding Marginal Cost Cont.
ydy
dcMC
ydy
dcMC
ydy
dcMC
y
y
yx
xxy
9
5
)9(10*2
1
)1()9(10*2
10
91030c(y)C
)9310(c(y)C
:givesfunction cost theinto thisPlugging
93
6
2
1
2
1
2
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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.
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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
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Product Curve Relationships When MPP>APP, APP is increasing.
This implies that when MC<AVC, then AVC is decreasing.
When MPP=APP, APP is at a maximum. This implies that when MC=AVC, then AVC
is at a minimum. When MPP<APP, APP is decreasing.
This implies that when MC>AVC, then AVC is increasing.
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Product Curve Relationships Cont.
)(')(
0)(')(
0)(')(
0)]([
)(')(
)(
2
xxfxf
xxfxf
xwxfxwf
xf
xwxfxwf
dx
dAVC
xf
wxAVC
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Product Curve Relationships Cont.
AVCMC
APP
w
MPP
wAPPMPP
MPPAPP
xfx
xf
xxfxf
11
)(')(
)(')(
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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
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Example of Examining the Relationship Between MC and AVC Cont.
0
02
26626
10
6
10
26
10
6
10
6
10
)( 2
x
x
xxxx
MCAVCxMPP
wMC
xxx
x
xf
wxAVC
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Example of Examining the Relationship Between MC and AVC Cont.
x
xx
xxxx
MCAVCxMPP
wMC
xxx
x
xf
wxAVC
0
2
62626
10
6
10
26
10
6
10
6
10
)( 2
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Example of Examining the Relationship Between MC and AVC Cont.
0
26626
10
6
10
26
10
6
10
6
10
)( 2
x
xxxx
MCAVCxMPP
wMC
xxx
x
xf
wxAVC
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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
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Finding the Slope of the Iso-Cost Line
2
1
1
2
2
11
22
2
11
22
22
1122
2211
w
w
dx
dx
w
xw
w
Cx
w
xw
w
C
w
xw
xwCxw
xwxwC
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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.
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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
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Example of Iso-Cost Line Graphically
x2
x1
10
100
X2 = 10 – (1/10)*x1
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Finding the Slope of the Iso-Cost Line
10
110
10
100
10
100
1000
100
100
101000100
100101000
1
2
12
12
12
21
dx
dx
xx
xx
xx
xx
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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.
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Input Use Selection There are two ways of examining how
to select the amount of each input used in production. Maximize output given a certain cost
constraint Minimize cost given a fixed level of
output Both give the same input selection
rule.
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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
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First Order Conditions for the Cost Minimization Problem with Two Inputs
0),(
0
0
0
0
),(),2,1(
21
2
22
2
1
11
1
212211
2
1
xxfy
MPPw
x
fw
x
MPPw
x
fw
x
xxfyTFCxwxwxx
x
x
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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.
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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.
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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
2
1
22
1
1
21,...
10:subject to
100421
xxy
xxMinxxtrw
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Example 1 of Cost Minimization with Two Variable Inputs Cont.
classin doneSolution
010
02
1104
02
1101
1010041),2,1(
2
1
22
1
1
2
1
22
1
12
2
1
22
1
11
2
1
22
1
121
xxy
xxx
xxx
xxyxxxx
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Solving Example 1 Using Ratio of MPP’s Equals Absolute Value of the Slope of the Cost Function
4x
1x20
4
20
xx20
)(x)10(4xy
function production into Plug3.
4
4
1
4
1Set 2.
.1
1
2
12
2
1
22
1
2
2
1
22
1
2
12
1
2
2
1
1
2
2
1
yxand
yx
y
xx
x
x
w
wMRTS
x
x
MPP
MPPMRTS
x
x
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Solving Example 1 Using MRTS from the Isoquant and Setting it Equal to the Slope of the Cost Function
4x
1x
205100
usingfor x Solve .4520
44
1
100
4
1Set3.
100MRTS
MRTS 2.Find
100100
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
w
wMRTS
xy
x
x
xy
x
yx
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Final Note on Input Selection It does not matter whether you are
trying to maximize output given a fixed cost level or minimizing a cost given a fixed output level, 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.