11. Extended Cost Function Example 2

8/12/2019 11. Extended Cost Function Example 2

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Extended Cost Example 2


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Marginal and average product curves

The marginal product of input 1 is the partial derivative of the productionfunction with respect to the first input.

The average product of input 1 is the ratio of output to input 1.

Given the short-run production function y = f(z1, 16) = 2(z1)0.25

the marginal and average product functions are

1 0.75

1 2 1( ; 16) (0.5)( )MP z z z

1 0.75

1 2 1( ; 16) 2( )AP z z z


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Examples of graphs of marginal and average product curves

MP1

Z1

Units of

output

AP1

MP1

Z1

Units of

output

AP1


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Given the fixed level of z2, let Z*(y) denote the amount of input 1needed to produce output y (i.e., f(Z*(y), z2) = y). Then

AP1(Z*(y))Z*(y) = y,

the short-run variable cost to produce y is w1Z*(y), and the short-runaverage variable cost to produce y is

Thus there is an inverse relationship between SAVC and AP1

.Combining this with the relationship between any marginal andcorresponding average functions, we see that SMC >SAVC if andonly if MP1


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Graphs of short-run cost and product curves.

Note output y for the cost curve corresponds to input level Z*(y) for theproduct curves.

Z*(y)

MP1

Z1

Units of

output

AP1

y

SAVC

output

$

SMC


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Relationship between long-run and short-run cost curves

Fixthe input prices, w1and w2.We will use the notation C(y), MC(y), AC(y), z1(y), and z2(y) to denote the

long-run cost, long-run marginal cost, long-run average cost, and thetwo conditional factor demands, respectively.

We will use the notation STC(y; z2), SMC(y; z2), and SATC(y; z2) to denotethe short-run total cost, short-run marginal cost, and short-run averagetotal cost, respectively, where each short-run function depends on boththe output level, y, and the level of the fixed factor, z2.

Some simple observations provide the insights necessary to understand

the relationship between the long-run and short-run cost curves.


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1) In the long-run, it is always possible to mimic what was done in the

short-run, so it cannot be cheaper to produce y in the short-run than inthe long-run, i.e.,

STC(y; z2) C(y) for every z2 and y.

Since the average cost functions are obtained by dividing the total cost

functions by the same output level, a similar condition holds for averagecost functions:

SATC(y; z2) AC(y) for every z2 and y.


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(2) If, in the short-run, the fixed factor is at the optimal (long-run) levelfor producing a certain output level, then the short-run cost of producingthat output level is no more than the long-run cost (since the variablefactor may also be set at the (long-run) optimal level for producing thatoutput).

Combined with observation (1), this yields

STC(y; z2(y)) = C(y) for every y.

Once again a similar condition holds for the average cost functions:

SATC(y; z2(y)) = AC(y) for every y.


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Together, these two observations show that the long-run cost curve is whatis called the lower envelope of the short-run cost curves. This isillustrated for an example in which there are only three possible valuesfor z2, 1, 2, and 3. For each output level, the long-run cost is theminimum of the three possible short-run costs for producing that output.

C(y)

STC(y;1)

output

$ STC(y;2) STC(y;3)


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A similar description applies when z2 is not constrained to only threepossible values.

[The shape shown for the long-run cost function is not general. The long-run cost function must satisfy two properties: in the long-run the firm canget out of business, so C(0) = 0; and long-run cost must increase asoutput increases. Beyond that, different production functions could leadto almost any shape.]

y* output

$

C(y)

STC(y;z2(y*

))


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The key features are that

any short-run total cost curve must lie on or above the long-run cost

curve, the curves touch at output levels (y* in the figure) for which the short-run

fixed factor matches the (long-run) conditional factor demand,

if the cost curves dont have kinks, the short-run curve is tangent to thelong-run curve where they touch.

Again a similar result holds for the average cost curves:

the long-run average cost curve is the lower envelope of the short-run

average total cost curves, the two are tangent when they touch.


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(3) Since the slope of a cost curve is the corresponding marginal cost, thefact that the short-run curve is tangent to the long-run curve where theytouch means that

SMC(y;z2(y)) = MC(y) for every y

i.e., when the short-run fixed factor is at the optimal level for producingoutput y, the short-run and long-run marginal costs are equal at y.


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(4) Not only must the curves be tangent where they touch, but also

the short-run curve must be more sharply curved upward at that pointsince it can never lie below the long-run curve.

Since the second derivative with respect to output determines thecurvature, and the second derivative with respect to output is thederivative of the first derivative, i.e., the derivative of the corresponding

marginal cost with respect to output, this means

i.e., short-run marginal cost is increasing at least as fast as long-run

marginal cost at the point of tangency of the cost curves. For theaverage cost curves, the short-run curve must be more sharply curvedupward than the long-run curve at points at which they are tangent.

2( ; ( ))MC y z y dMC

y dy


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Combining these observations with other things we know (e.g., marginal isbelow average when average is decreasing, marginal equals average atthe minimum of average, and marginal is above average when averageis increasing), we can demonstrate the relationship between short-runand long-run marginal and average cost curves.

Starting with a U-shaped long-run average cost curve, AC(y), and thecorresponding long-run marginal cost curve, MC(y), consider an output

level, y*.


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When z2 = z2(y*), the corresponding short-run average total cost is tangentto the long-run average cost at y = y*. Since long-run average cost isdeclining at y*, short-run average total cost must be declining there aswell. That means the minimum of short-run average total cost must be atan output larger than y*.

$SATC(y;z2(y

*))

SATC(y;z2(y**))

y*

AC(y)

SMC(y;z2(y*))

MC(y)

SMC(y;z2(y**))

y** output


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The other point is the minimum point of the short-run average total costcurve, since the marginal and average are always equal where averageis minimized.

$SATC(y;z2(y

*))

SATC(y;z2(y**))

y*

AC(y)

SMC(y;z2(y*))

MC(y)

SMC(y;z2(y**))

y** output


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A similar analysis is carried out at y** in the figure. Since long-run averagecost is increasing at y**, short-run average total cost (for z2 = z2(y**))must also be increasing at y**, and the minimum point for this short-runaverage total cost must occur at an output less than y**.

$SATC(y;z2(y

*))

SATC(y;z2(y**))

y*

AC(y)

SMC(y;z2(y*))

MC(y)

SMC(y;z2(y**))

y** output


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Sample problems

1. Consider a short-run production function with one variable input, x. Whenthe output is 100, for the corresponding level of the variable input, theaverage product of the variable input is 20 and the marginal product ofthe variable input is 10. The per-unit price for the variable input is $500.

a. What is the level of the variable input when output is 100?

b. Approximately how much extra variable input would be needed to raiseoutput from 100 to 101?

c. What is the short-run marginal cost at output 100?

d. Is the short-run marginal cost greater than, equal to, or less than short-run average variable cost at output 100? Explain.


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2. Consider a firm with production function q=x(y0.5) where x and y are theamounts of the two inputs and q is the amount of output.

a. What is the marginal rate of technical substitution at input bundle (x, y) =(5, 4)?

b. What is the marginal rate of technical substitution at a general input bundle(x,y)?

c. If input prices are p = 6 and p =9, what is the optimal input bundle to produceoutput 3?

d. If input prices are p = 6 and p = 9, what is the firms lowest long-run cost ofproducing output 3?

e. If input prices are p = 6 and p= 9, for a general output, q, what is the relationshipbetween the optimal amounts of inputs x and y used to produce q?

f. Suppose in the short run the amount of the second input is fixed at y =4. Find

the total product, marginal product, and average product as functions of x.

g. Suppose in the short run the amount of the second input is fixed at y =4 and theinput prices are p = $16 and p = $5. Find all seven short-run cost functions.

11. Extended Cost Function Example 2

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