Ch10-1

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Summary: Production functions

Production in the short run (one input fixed)Production in the long run (all inputs can be adjusted)The relationship between TP, AP, MP

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Production in the long run: Isoquants, the special cases of perfect substitutes and perfect complement input, the Cobb – Douglas production function.

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Properties of the typical production function (monotonic, convex).Marginal product: how to calculate it. Its relations to the MRTS.Returns to scale (long run term)Diminishing marginal product (short ran term) Production with more than one plant: The role of equating marginal products.

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CHAPTER 10: COST CHAPTER 10: COST

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INTRODUCTION

In the last chapter we described the production process employed by a firm. Here we continue the investigation of the firm behavior and explore the firm’s cost function and it geometrical description, the cost curve. Cost curves are important in studying the determination of optimal output choices.

Consider the cost function c( Q(l,k)) which gives the minimum cost of producing output level Q when factor prices are (w,r).

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Cost in the short run

First we consider how costs vary with output in the short run. Short run cost can be broken down into two components, variable (VC) and fixed (FC) cost, such that total cost is the sum of the two.

Fixed costs, FC, are the costs that must be paid regardless of what level of output the firm produces. Other costs change when output changes: these are the variable costs, VC.

The total costs of the firm can always be written as the sum of the variable costs, and the fixed costs:

TC = VC + FC.

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Recall our short run production function when capital was fixed at K0. If the cost of capital is r per unit of capital, fixed cost is

FC = rK0

For this short run production function, we can vary labor all we wanted. If the cost of labor is w per unit, variable cost is

VC = wL.Total cost is then just the sum of variable and fixed costs:

TC = wL + rK0.

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Since K0 doesn’t vary with the level of output, FC will not vary with the level of output.

VC however depends on how much labor we use. In the short run. L solely determines how much output we produce. Remember that:

Q = F(k0, l)= F0(L),

l = F0-1 (Q),

Substitute it into VC=wl to get

VC = wF0-1 (Q), cost as a function of the amount of output we

produce.

Similarly, total cost will also depend on the amount of output we produce: TC = wF0

-1 (Q) + rK0.

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Output as a Function of One Variable Input

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Cost in the short runImportant to note that fixed cost is just a horizontal line at rK0.

Variable cost intersects the origin because if we don’t use labor it doesn’t cost us anything.

Total cost is just the variable cost curve shifted up to intersect the vertical axis at FC = rK0.

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AVERAGE COST FUNCTION, AVERAGEVARIABLE COST FUNCTION, AVERAGE FIXED

COST FUNCTION

Oftentimes, we are interested in cost per unit of output: the concept of average.

The average cost function measures the total cost per unit of output. The average variable cost function measures the variable costs per unit of output, and the average fixed cost function measures the fixed cost per unit of output.

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AVERAGE COST FUNCTION, AVERAGEVARIABLE COST FUNCTION, AVERAGE

FIXED COST FUNCTION

Qrk

QFCQAFC 0)( ==

The average fixed cost function is the fixed cost divided by the quantity of output

The average variable cost function is the variable cost divided by the quantity of output

QQwf

Qwl

QQVCQAVC )()()(

10−

===

The average total cost function is the total cost divided by the quantity of output

QrkwlAVCAFCQATC 0)( +

=+=

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(1) AVERAGE FIXED COST CURVE

What do the average cost function, average variable cost function and average fixed cost function look like?

The easiest one is the average fixed cost function:

Q

$

AFC(Q)

when Q = 0, AFC(Q) is infinite

and as Q increases the average fixed cost decreases toward zero.

QFCQAFC =)(

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(2) AVERAGE VARIABLE COST CURVE

Consider the average variable cost function:

Start at Q = 0 and consider producing one unit. Then the average variable costs at Q = 1 is just the variable cost of producing this one unit.

Now increase the level of production to 2 units. We would expect that, at worst, variable costs would double, so that average variable costs would remain constant.

Better yet, if we can reorganize production in a more efficient way as the scale of output is increased, the average variable costs might even decrease initially.

But eventually we would expect the average variable costs to rise. Why? Well, if fixed factors are present, they will eventually constrain the production process.

QkQVC

kQ)(

)(AVC 00 =

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For example, suppose that the fixed costs are due to the rent on a building of fixed size.

Then as production increases, average variable costs - the per-unit production costs - may remain constant or decreasing for a while.

But, as the capacity of the building is reached, average variable costs will rise sharply.

Thus, a typical average variable cost curve looks like the one depicted in the graph here. Q

$

QkQVC

kQ)(

)(AVC 00 =

),(AVC 0klQ

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(3) AVERAGE COST CURVE

The average cost curve is the sum of the average variable cost curve and the average fixed cost curve.

The initial decline in average costs is due to the decline in average fixed costs as well as the decline in the average variable costs.

The eventual increase in average costs is due to the increase in average variable costs.

)()|(AVC)( 0 QAFCkQQAC +=

Thus, it will have the U-shape indicated in the figure here.

y

$

),(AC rwQ

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MARGINAL COST FUNCTION

The marginal cost curve measures the change in costs for a given change in output: the concept of additional.

Since the only component of the cost function that varies with output is the variable costs (fixed costs do not vary with output), we could write the definition of marginal costs in terms of the variable cost function:

)()(

)()(

QQCQQC

dQQdCQMC

∆−∆+

=

=

)()(

)()(

QQVCQQVC

dQQdVCQMC

∆−∆+

≅

= The two definitions are equivalent because:

TC(Q) = VC(Q) + FC

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How can we depict a marginal cost function graphically?

First, note that the variable costs are zero when zero units of output are produced and, hence, the marginal cost for the first small unit of output equals the average variable cost for that single unit of output.

]0)0( [ 1

)1(

1])0([])1([)1(

==

+−+=

VCVC

CFVCCFVCMC

Q

Fact #1: At the first small unit of the output, themarginal cost equals the average variable cost.

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Now suppose that we are producing in a range of output where average variable costs are decreasing.

Then it must be that the marginal costs are less than the average variable costs in this range.

For the way that you push an average down is to add in numbers that are less than the average. (If your cumulative GPA is dropping, then your current semester GPA must be lower than your previous cumulative GPA.)

Finally, suppose that we are producing in a range of output where average variable costs are increasing.

Then it must be that the marginal costs are greater than the average variable costs in this range.

It is the higher marginal costs that are pushing the average costs up.

Fact #2: The marginal cost curve must lie below the average variable cost curve to the left of the minimum point of the average cost curve, and above it to the right.

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Putting Fact #1 and Fact #2 together, we know that:

The marginal cost curve and the average variable cost curve must be at the same point when the output is zero.

The marginal cost curve must lie below the average variable cost curve at low output levels, intersect the average variable cost curve at its minimum point, and then lie above the average variable cost curve.

Q

$

)(AVC Q

)(QMC

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We have discussed the relationship between the average variable cost curve and the marginal cost curve. Exactly the same kind of argument applies for the average cost curve.

Q

$

)(AC Q

)(QMC

If average costs are falling, then marginal costs must be less than the average costs.

If average costs are rising, then the marginal costs must be largerthan the average costs.

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Q

$

)(QMC

)(AC Q

)(AVC Q The marginal cost and average variable cost are the same at the first unit of output.

The marginal cost curve passes through the minimum point of both the average variable cost and the average cost curves.

The average variable costcurve may initially slope down but need not. However, it will eventually rise, as long as there are fixed factors that constrain production.

The average cost curve will initially fall due to declining fixed costs but then rise due to the increasing average variable costs.

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AREA under THE MARGINAL COST CURVE

Here is another important relationship between the marginal costs and the variable costs:

The area under the marginal cost curve up to Q* gives us the variable cost of producing Q* units of output.

Why is that? Well, the marginal cost curve measures the cost of producing each additional unit of output. Thus, if we add up the cost of producing each additional unit of output we will get the total cost of production -- except for fixed costs.

Q

$

)(QMC

5

variable costs of producing 5 units

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EXAMPLE OF COST FUNCTIONS

Consider a specific cost function depending only on output level (but not on factor prices):

TC(Q) = Q2 + 1

Then, we have the following derived cost curves:

variable costs: VC(Q) = Q2

fixed costs: FC = 1

average variable costs: AVC(Q) = Q2/Q = Qaverage fixed costs: AFC(Q) = 1/Qaverage costs: AC(Q) = (Q2 + 1)/Q = Q + 1/Q

marginal costs: MC(Q) = QdQ

QCd 2)(=

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Cost minimization in the short run: Example

Output for a simple production process is given by Q = 2KL, where K denotes capital, and L denotes labor. The price of capital is $25 per unit and capital is fixed at 8 units in the short run. The price of labor is $5 per unit. What is the total cost of producing 80 units of output? Substituting the fixed K into the production function yields: Q=16L. Q=80 L=80/16 = 5.The cost is thus: TC = rK + wL = 25x8 + 5x5 = 225

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The Relationship Between MP, MC and AP, AC

When the labor is the only variable:

Qwl

QQVC

QQTCMC

∆∆

=∆

∆=

∆∆

=)()()(

However, w does not change so:

LMPw

lQw

QlwMC 1

)(

1=

∆∆

=∆∆

=

Thus, MC increases where MPL decreases

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Similarly, when the labor is the only variable:

Qwl

QQVC

QQVCAVC ===

)()(

However, this can be re written as:

LAPw

lQw

QwlAVC 1

)(

1===

Thus, AVC increases where APL decreases and vise versa

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Graphically, the relationship between MP, MC,AP and AVC are the following:

Fig. 10.9

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MARGINAL COST CURVES FOR TWO PLANTS

Suppose that you have two plants that have two different cost functions: C1(y1) and C2(y2). [note: we hold the factor price, w, constant and hence omit them in the cost function expressions.]

You want to produce y units of output in the cheapest way.

In general, you will want to produce some amount of output in each plant. The question is, how much should you produce in each plant?

Set up the minimization problem:

min C1(y1) + C2(y2){y1, y2)

such that y1 + y2 = y

Note: Since C1(y1) and C2(y2) are cost functions, the problem of coming up with the minimum-cost input mix for each plant has already been solved. Rather, the issue here is to come up with the output mix that minimizes the total cost of the two plants.

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From the constraint y1+y2=y, we isolate y1=y-y2. Than we substitute it into the cost minimization:

)()(min 22212

yCyyCy

+−

•To find the optimal y2 we differentiate with respect to y2 and equate to zero. After some algebra, we get:

212

2

1

1 MCMCdydC

dydC

=⇔=

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Now how do we solve it?

It turns out that at the optimal division of output between the two plants we must have the marginal cost of producing output at plant 1 equal to the marginal cost of producing output at plant 2, and total production is givenTo prove this, suppose the marginal costs were not equal.

Then it would pay to shift a small amount of output from the plant with higher marginal costs to the plant with lower marginal costs.

Thus, if the output division is optimal, switching output from one plant to the other cannot lower costs because the marginal costs of producing output at the two plants are equal.

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Let TC(y) be the overall cost function that gives the cheapest way to produce y units of output -- that is, the cost of producing y units of output given that you have divided output in the best way between the two plants.

3$)()()(*

*

*2

*22

*1

*11 =≡=

dyydC

dyydC

dyydC

The marginal cost of producing an extra unit of output must be the same no matter which plant you produce it in:

mc1

mc2 mc

y2y1

y = y1 + y2

Marginalcost

Marginalcost

Marginalcost

y1* y2

* y* = y1* + y2

*

The marginal cost curve for the two plants taken together is just the horizontal sum of the two marginal cost curves.

$3

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Minimizing cost with two plants: Example

A firm has two plants. The cost function at plant A is given by: TCA = 6QA

2 + 16.The cost at plant B is given by: TCB = 2QB

2 + 240.The firm wants to produce 32 units of output.

What is the level of production at each plant?What is the total cost of production of 32 units of output?

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Minimizing cost with two plants: Example

The two conditions for cost minimization are: MCA = MCB, and QA + QB = 32.First we find MCA and MCB.

MCA= dTCA/dQA = 12QA

MCB= dTCB/dQB = 4QB.

MCA = MCB12QA = 4QB

QB = 12/4 QA or QB = 3QA

Now we remember that QA+QB = 32. Substituting the red relations:

QA + 3QA = 32. 4QA = 32 QA = 8, QB = 24.

Total cost of production is: TCA(QA) + TCB(QB)6x82 + 16 + 2x242 + 240 = 400 + 1392 = 1792

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•Where the optimal y1 and y2 obey:

212

2

1

1 MCMCdy

dTCdy

dTC=⇔=

In terms of the two plants example we did,

If MC1 = w/MPL1 and MC2 = w/MPL2 MPL

1 = MPL2. Which is the same

as maximizing production in two plants by equating marginal productivities

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Ch10-1