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COST

BY

SOURAV DAS

98367 93076

COST FUNCTION

Definition

A mathematical formula used to predict the cost associated with a certain action or a certain level of

output. Businesses use cost functions to forecast the expenses associated with production, in order to

determine what pricing strategies to use in order to achieve desired profit margins.

The opportunity cost of an asset (or, more generally, of a choice) is the highest valued opportunity that

must be passed up to allow current use. Opportunity cost ia also called economic opportunity loss. And is

the value of the next best alternative foregone as the result of making a decision

Example – I have a TV and VCR. I can either watch a movie or rent TV and VCR.Out of 2 choices, I am

selecting one. In this example, If I watch movie, then the opportunity cost of watching the movie would

be the amount of rent that I would have earned.

Explicit costs are expenses for which one must pay with cash or equivalent. Because a cash transaction is

involved, they are relatively easily accounted for in analysis. These costs are never hidden, one has to pay

separately.

Example- Electricity Bill, wages to workers etc.

Implicit costs do not involve a cash transaction, and so we use the opportunity cost concept to measure

them. Implicit costs are related to forgone benefits of any single transaction. These are intangible costs

that are not easily accounted for.

Example, the time and effort that an owner puts into the maintenance of the company rather than working

on expansion.

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Cost Functions

Cost Minimization and the Cost Function

Because any profit maximizing firm wants to minimize the cost of producing any given level of

output, we start with the problem of minimizing a firm’s total cost. In terms of our single product - dual

input example, the problem is to

Isocost Lines

Isocost Lines give combinations of labor and capital

representing equal cost. While similar to budget lines, there is

an obvious difference: firms have an infinite number of

Isocost Lines.

minimize pKK + p

LL

subject to the constraint that f(K,L) = qo

To follow historical convention, we denote the prices by pK = r and pL = w. Thus our problem becomes

min. rK + wL

s.t. f(K,L) = qo

Rather than solve the problem mathematically, however, we are going to do a graphical

derivation. In this simple case, we can graph isocost lines, that represent equal expenditures on labor and

capital, much like those plotted.

The isocost lines are obviously straight lines of the form rK + wL = co. The further the line from

the origin, the higher the cost. Ideally we would like to be at the least cost, but we must also achieve our

output objective. In the case illustrated, we must operate at the point K0, L0, where the isocost line is just

tangent to the isoquant.

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Cost-Minimizing Factor Inputs

Co is the minimum cost of producing Qo units of output. It

means using Ko and Lo to do so. Similarly, C1 is the minimum

cost of producing Q1 units of output. Obviously these values

depend on r and w.

By repeating this operation for each different level of output, we can come up with a general cost

function C(q,r,w).

Properties of the Cost Function with Respect to Output

Because the general cost function contains more than we want to use at the moment, let us hold r

and w constant and concentrate on the relation between cost and quantity. We simplify our cost function

to C = C(q).

Total Cost

The following illustrates three cost functions that embody specific assumptions about the relation

between cost and quantity. In all three cases, total cost is an increasing function of quantity produced.

(How could it be otherwise?) However costs increase at different rates. In the first case (A), total costs

increase at a faster rate than the quantity produced. Here, we save that we have decreasing returns to

scale. In the second case (B), the function is a straight line going through the origin. Here, costs increase

in proportion to output. This case is called constant returns to scale. (You should see that this is the case

with either the Cobb-Douglas or the Leontief production functions). In the third case, (C), costs increase

less than in proportion to quantity.

This is the case of increasing returns to scale.

Average Cost

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To further the picture, consider the corresponding average cost curves is presented. In Case A,

average cost is increasing as a function of output; in Case B, average cost is constant; in Case C, average

cost is decreasing as a function of output.

Three Cases of Average Cost Functions

Case A

Decreasing Returns to

Scale

Case B

Constant

Returns to Scale

Case C

Increasing Returns to

Scale

Here, average cost

rises with output

Here, average costs is

constant

Here, average cost falls

with output

Marginal Cost

Still a third property of interest is the marginal cost curve. Mathematically this is defined as the

derivative of the total cost function (or, if you will the cost of producing an incremental unit of output.

There is a relationship between the two:

If marginal cost is above average cost, then average cost is rising. If marginal cost is below

average cost, then it is falling.

This is a sufficiently important relationship that we will give two proofs, one not using calculus,

and another using calculus.

The non-calculus proof

Let C(qo) be the cost of producing qo units of output. The marginal cost of producing the qoth unit

of output is C(qo) - C(qo-1). The average cost of producing qo units of output is C(qo)/qo, while the

average cost of producing qo-1 units of output is C(qo-1)/(qo-1). Suppose marginal cost is greater than

average cost. Then

C(qo) -C(qo-1) > C(qo)/qo

In turn this means that

C(qo) > C(qo-1) + C(qo)/qo

implying that

C(qo)/qo > [C(qo-1)/(qo-1)][(qo-1)/qo] + C(qo)/qo2

Rearranging terms

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[C(qo)/qo][1-1/qo] > [C(qo-1)/(qo-1)][(qo-1)/qo]

The two right most terms on both sides of the inequality are equal. Thus, canceling them, we get

C(qo)/qo > C(qo-1)/(qo-1)

In short, if marginal cost is above average cost, then average cost is rising.

The Calculus Proof

To see the relation between marginal and average cost, recall that average cost is given by C(q)/q,

and the derivative of average cost with respect to q is

C’(q)/q -C(q)/q2

or, with a little rewriting

(1/q)(C’(q)-C(q)/q).

The intuitive reasoning is simple: average cost can be rising only when the cost of incremental

units exceeds the average cost of the previous units.

Some Graphs

The following gives graphs of the marginal cost curve corresponding to the three cases of the

average cost curve. In case A, the average cost curve is upward sloping, indicating MC > AC. In Case B,

marginal cost is constant and equal to average cost. Finally, in case C, average cost is decreasing so that

MC < AC. Note that, even in this case marginal cost never becomes negative. Why?

Three Cases of Marginal Cost Functions

Case A

Decreasing Returns to

Scale

Case B

Constant

Returns to Scale

Case C

Increasing Returns to

Scale

Marginal cost is above

average cost

Marginal cost and

average cost are equal

Marginal cost is below

average cost

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As you will note, I have drawn marginal cost curves that have constant slope. This need not be

the case, as we will see in the case of the U-shaped average cost curve.

The U-Shaped Average Cost Curve

As a practical matter, most economists believe that the typical competitive firm is characterized

by a U-shaped average cost curve, where average cost initially falls as a function of the level of output,

reaches a minimum and then begin to rise. At the minimum level of AC, MC = AC. At the minimum AC,

the slope of the AC curve is zero, which is sufficient to prove that MC = AC. Finally note that the MC

cost curve must cut the AC curve from below. Thus we may draw the average cost curve and the

marginal cost curve.

The U-Shaped Average Cost Curve

Here average cost first falls, and then rises. As you can see

the marginal cost curve "cuts" the average cost curve at the

minimum average cost.

Note that I have drawn the MC curve so that it is initially falling. This need not be the case, and

we have no special interest in this issue.

Fixed and Variable Cost

It is sometimes useful to divide the firm’s costs into fixed and variable components. Fixed costs,

FC, are the costs the firm would incur if it had no output. That is, FC = C(0). Variable costs are those

that vary with output. They may either be defined as the difference between total and fixed costs,

VC(q) = C(q) - C(0).

It should be obvious that what is defined as “fixed” and “variable” depends on what time period

we are talking about. In the long run, there truly are no fixed costs, but in the short run, there may well

be.

Here is a relation between the average total cost and average variable cost for a U-shaped average

cost curve. As you will note

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The marginal cost curve cuts the ATC and the AVC costs at their minimum

The minimum of AVC occurs at a lower level of output that the minimum of the ATC cost

curve

Average Variable Cost is not equal to marginal cost

Average Variable Cost is less than average cost

As output grows infinitely large, AVC approaches ATC

Average Variable Cost and Average Total Cost for the U-

Shaped Average Cost Curve

Average fixed cost is continually declining, and the minimum

of AVC occurs at a lower level of output that the minimum of

ATC. MC cuts both the ATC and AVC curve at their

respective minimum.

Other Factors Affecting Cost

We originally wrote the cost function as C(q,r,w) to indicate that cost was affected by things other

than output. Here are some trick questions:

First Trick Question

Acme Widgets’ costs are now $100, of which $50 represents factor payments to workers and

$50 represents payments to capital. (To make life easier, assume that Acme Widgets leases

all plant and equipment.). Now suppose that wage rates are increase by 100%. Will total

costs go from $100 to $150

Clearly not. If Acme continued to use the same amounts of capital and labor, the answer would

be yes. But, given the new slope of the isocost lines (the situation is illustrated in Figure 8-8) Acme will

substitute capital for labor and reduce the cost increase to less than 50%.

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Isocost Lines with a Factor Price Shift

A change in factor prices causes isocost lines to rotate. As this

graph shows,, the minimum cost of producing Qo units is not

50% higher. Rather than stay at (Ko, Lo), the firm will move

to another point such as (K1, L1) on an isocost line below

C=$150.

Second Trick Question

Is it possible that the higher price of labor will lead Acme to lower total costs for the given

level of output?

This is a good trick question for examinations, and sometimes people do claim that higher factor

payments can lead to lower costs. But this is nonsense. Suppose, for instance, that Acme could find a

new solution involving $60 of capital payments and $30 of labor costs (at the new wage rate). At the old

wage rate, costs would only have been $80 = $60 + $20, which would violate our assumption that $100

represented the least cost. Any manager claiming that higher wager rates lead to lower costs is simply

admitting that he wasn’t doing his job in the first place.

Third Trick Question

Does an increase in total cost mean that average cost has gone up, and that marginal cost has

gone up?

Why a Rise in Total Cost Doesn't Mean a Rise in Marginal

Cost

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Here, a rise in total cost leads to a different production

technology with the lower marginal cost.

Another trick question. A possible illustration of why not is given. The average cost curve shifts

up and to the right from AC to AC*, and the marginal cost curve shifts from MC to MC*. So MC has

declined.

The intuition behind this is quite simple. A rise in labor cost, say, may lead a firm to purchase a

new piece of capital. Thus the average cost will rise (see the previous example to see why it cannot fall),

but, equipped with the new machine, marginal cost may well be less.

Fourth Trick Question

Does technological change reduce costs?

We normally mean by technological change the ability to get more output from a given

combination of labor and capital. In our terms, technological change shifts the isoquants out. And, if all

of the isoquant (or at least the portion we are currently using) is shifted out, then technological change

does reduce costs. But some caveats are in order.

Don’t confuse changes in technology with changes in technique. In the example given above,

a 50% increase in wage rates leads to a change in technique. There are always several different ways of

achieving the same objective, using what we call techniques. We all know how to mow a lawn by hand,

or with a standard lawn mower, or with one of the larger mowers that commercial services use.

Switching from one to another is switching from one technique to another. The switch in technology

comes when a new type of mowing machine is invented.

For reasons we will get to later, it is important to distinguish between firm specific

technologies and industry technologies. A firm specific technology is one that only one firm can use,

protected either by something like trade secrets or patent rights, while industry technology represents

something that affects all firms in the industry.

An example: when Apple developed the Macintosh, it patented its software, and thus froze out

competitors. When IBM came out with the PC in the 1980’s it allowed for open architecture, including

the DOS system. Other firms quickly adopted DOS, and any advantage that IBM had quickly

disappeared.

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Technological change occurs gradually. Two examples will make the point. The original IBM

PC, not yet 15 years old, had 64 k of memory, a 64k floppy and an 8086 chip. The current system is more

likely to use 250 times as much memory, a floppy disk 20 times as large (not to speak of hard disks), and

a Pentium chip, with many times the power. Yet this change occurred gradually, and in fact represented a

continuation of the computer revolution of the last 40 years.

Fifth Trick Question

Last year, Acme Widgets produced widgets at $5 each. This year the plant manager reports

that he can cut production cost to $4 each. Does this mean he simply was not doing his job

last year?

While it is always possible that the manager was not doing his job last year, it may also mean

something more complicated. In World War II, the Pentagon contracted large production runs of planes,

tanks and the like to specific plants. They discovered over time that the cost of producing a plane or tank

at a specific plant declined, because, over time, the plant learned all sorts of production tricks.

Economists and production specialists labeled this phenomenon as the learning curve, reflecting the

commonsense notion of learning by doing. That is, the more we do of something, the better we get at it.

Thus a firm's cost function might be a function C(qt, Qt) the number of units it produces this period, qt, as

well as the cumulative number of units it has produced, Qt. Thus over time, the firm's average cost

function shifts down because of learning by doing.

Figure 8-10 shows how the production function could shift. Imagine a firm whose cost function

is

Ct = ctqt

That is, at any time, the marginal cost of producing the product is constant. However, over time,

marginal cost declines as a function of Qt, the cumulative number of units produced.

Learning by Doing: How the Cost Function Shifts

Imagine a firm whose cost function is Ct = ctqt, but where ct is

a declining function of the cumulative number of units

produced. ct might look like this.

Sixth Trick Question

Last year, Acme Widgets produced widgets for $5 each. This year Acme's Costs will be $4.

A new competitor, Wonder Widgets, figures that it can produce widgets for $4.50 each. Does

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this mean that Acme was inefficient? Does it mean that Wonder Widgets stole trade secrets

from Acme?

Again, Acme may have been inefficient, and Wonder Widgets may have stolen trade secrets from

Acme. One can learn a lot by simply observing competitor. If Wonder Widgets simply purchases a

widget from Acme, and turns it over to its production specialists for study, there will be a chorus of "So

that is what they do!" It is, if you will, just like looking at instructor's old exams. The knowledge will

pay off.

In fact, there is a general principle: much technological change can confer only a temporary

advantage on its developer. We rightly have patent laws to protect new technologies, for else there

would be little reason to develop new technology. But the “high tech” firms clearly recognize that the

result of new technology hitting the market is for it to be copied, sometimes illegally, but often legally.

Thus a high tech firm cannot rest on its laurels, but must continually innovate to keep a cost advantage.

Applications of Cost Functions to Some Problems of the Firm

We can show how some of this material applies by considering two problems that a firm faces in

making production decisions.

Allocating output between two different plants

The first involves allocating output between two different plants. Suppose a manufacturer has

two plants, "A" and "B". The average and marginal cost functions for the two plants are given in Figure

8-11. As you can see, plant "A" has a higher average cost and a higher marginal cost for each and every

level of output than plant "B". While this makes for a nice graph, it is not relevant for management

decision making.

Marginal and Average Cost for Two Different Plants

The example is constructed so that plant "A" has a higher

marginal and average cost than plant "B" for every level of

output. While this makes for a nice graph, it is irrelevant for

decision making.

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To see why, suppose management is considering producing q* units of output from the two

plants. Then it wants to find the allocation of output between the two plants that minimizes total cost of

production. Here, the only cost that is relevant is the marginal cost, and the least cost solution is achieved

when the marginal cost of each plant is equal. Even though "B" appears to be a more efficient plant, it

still pays to allocate some of its production to other the older plant.

The graphical solution is shown. Here, the marginal cost of plant "A" is measured from the left-

hand side, while the marginal cost of plant "B" is measured from the right hand side. As you can see the

two marginal costs are equated when qA is produced at plant "A" and qB = q*-qA is produced at plant "B".

Any other point would involve higher cost.

Allocating Production Between Two Plants

The least cost solution requires that the marginal cost of

operation be the same. If the firm is going to produce q* units

of output, then optimization occurs at qA.

Allocating output between a new type of plant and old plants

To see a similar problem, suppose a firm has several plants of type "A" such as illustrated in

Figure 8-13. It now becomes possible to build a new type of plant, type "B". As you can see, type B

represents a technological innovation, representing lower cost. What should the firm do? (The first thing

to recognize is that long run marginal cost is now given by the minimum average cost of plant "B". The

firm can build as many of those as it wishes, and operate them at the minimum of the LRAC.

Marginal and Average Cost for Two Different Plants

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A firm is operating a number of plants of type "A". and it now

becomes possible to build new, more efficient plants of type

"B". This does not mean the firm should shut down its

existing plants, merely change the scale of operation. The

existing plants should not begin producing qA units of output.

That does not mean that it should junk its existing plants. It should continue to operate them

where the marginal cost equals the new long run marginal cost.

Choosing Plant Size

Now consider a different problem. A firm is considering building one of three different plants,

whose average costs are as depicted in Figure 8-14. Its long run average cost curve is then a peculiar

scallop-shaped figure, representing the minimum of each of these average cost curves.

Long Run Average Cost when Different Sized plants are

possible

A firm can choose between three different plants. Its long run

average cost curve is now the scallop-shaped figure, the

minimum of the average cost for each of the three plants.

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It is also possible to extend this analysis to the case of many different sized plants, so that the firm

can select from a continuum of plant sizes. While, this discussion is much beloved by professors of

economics, we will not go into it here.1

Cost functions

In talking about cost functions, we have presupposed a simple cost function C = C(q). While the

precise definition of the cost function will depend on the exact nature of the firm's production function,

we often illustrate the properties of a cost function with one of two simple standard functions:

The Linear Cost Function

C = a + b q

The Quadratic Cost Function

C = a + b q + c q2

For example, the cost function

C = 3+ 5q

is a linear cost function, while

C = 4 + 5q + q2

is a quadratic cost function. The difference between the two functions is straightforward. The first cost

function simply contains q, while the second term contains q2.

Let us illustrate some of our basic notions using these cost functions.

Fixed Cost

Fixed cost is, quite simply the costs when q = 0. It is a simple matter to show that, in the case of

both our linear and quadratic cost functions, the first term, a, represents fixed cost.

Variable Cost

Variable cost is the cost varying with output, or, if you will, total cost less fixed cost. In the case

of the linear cost function, variable cost will be the term bq; in the case of the quadratic cost function,

variable cost will be bq + cq2.

Average Cost

Average cost is total cost divided by quantity, or C(q)/q. In the case of the linear cost function,

average cost will be

a/q + b.

In the case of the quadratic cost function, average cost will be

1 Jacob Viner, who originally did the work, was then at the University of Chicago and later at Princeton University. Viner

stumbled on this discussion by accident. He did it wrong, and only discovered his error when one of his Ph.D. students,

assigned the task of drafting the graphs reported that he could not draw them the way Viner wanted them. For this, Viner

got an important article published and the Ph.D. student got a footnote. It should serve as a reminder to readers of these

notes to check the logic carefully behind each graph.

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a/q + b + cq.

We could also compute average fixed cost and average variable cost.

Marginal Cost

Marginal cost is simply the incremental cost of producing an additional unit of the product. The

idea is the incremental cost C, associated with producing an incremental unit of output, Q. When we

are dealing with simple production functions such as linear or quadratic functions, we customarily

compute the marginal cost by taking the derivative of the cost function. That is

In words, marginal cost is the derivative of the cost function with respect to output. For those rusty on

elementary differential calculus, remember that, in the case of our linear cost function

C'(q) = b

and in the case of the quadratic cost function

C'(q)= b + 2cq

Two Problems

Here are two simple exercises. One is worked, and one is left for you to do on your own.

Suppose a firm has a cost function

C = 10 +20q + 4q2.

Compute total cost, total variable cost, total fixed cost, average cost, average variable cost,

average fixed cost, and marginal cost when q=10.

Find the value of q at which MC = AC.

What level of output minimizes AC?

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At what level of output is marginal cost equal 60?

Answers

Total cost is 10 + 20 (10) + 4(10)2 = 610. Total variable cost is the last two terms, or 600.

Average cost is 10. To compute average cost, average variable cost, and average foxed cost,

just divide by 10 to get 61,60, and 1 respectively. Marginal cost is given by 20 + 8q = 100

when q = 10.

Average cost is 10/q + 20 + 4q. We want to find the value at which

10/q + 20 + 4q = 20 + 8q.

Simplifying, we get that

10/q = 4q

and we can solve that equation to get q2 = 2.5, or q = 1.58

We know that MC = AC at the minimum of the AC function. Thus, from the answer above,

we know that the minimum occurs at 1.58.

Recall that MC = 20 + 8q. That equals 60 when q = 5.

A Second Problem

Suppose the cost function is

C = 5+ 10q2.

Compute the same answers asked for above.

1. A person managing dry cleaning store for $30,000 per year decides to open a dry cleaning

store. The revenues of the store during the first year of operation are $100,000 and the expenses

are $35,000 for salaries, $10,000 for supplies, $8,000 for rent, $2,000 for utilities, and $5,000 for

interest on a bank loan. Calculate: a. the explicit cost

Explicit costs = salaries + supplies + rent + utilities + interest on the bank loan = 35000 + 10000 + 8000 +

2000 + 5000 = $60000.

b. the implicit costs

Implicit costs = entrepreneur forgone salary = $30000.

c. the accounting profit

Accounting profit = TR - explicit costs = 100000 - 60000 = $40000

d. the economic profit

Economic profit = TR - explicit costs - implicit costs = 100000 - 60000 - 30000 = $10000

e. Indicate whether the person should open the dry cleaning store

Since the person would earn an economic profit of $10000 per year, he or she should open the dry

cleaning store: positive economic profits.

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2. Answer the following questions:

a. What is the slope of the line that goes through (2, 5), (8, 2)? m = (2 - 5) / (8 - 2) = - 3 / 6 = - 1 / 2

slope = -0.5

b. What is the slope of the line for which an equation is:

6x + 3y - 12 = 0

3y = 12 - 6x

y = 4 - 2x

slope = -2

c. Is the following equation an equation for a demand or for a supply curve?

2q + 6p - 8 = 0

6p = 8 - 2q

p = 4/3 -(1 / 3)q

This is a demand curve because the slope is negative.

3. You are given the Total Cost function TC = (1/3)Q^3 - 8.5Q^2 +60Q+27

a. Derive the AC and MC functions AC = TCQ = (1/3)Q^2 - 8.5Q + 60 + 27/Q

MC = dTR/dQ = Q^2 - 17Q +60

b. Explain the (graphical, i.e. in terms of slopes) relationship between TC and MC MC is the slope of TC.

c. Determine the level of output at which the total cost function is minimized and the level of total

cost (Hint: You will need to check the second derivative). We need to solve: Q^2- 17Q + 60 = 0,

(Q - 12)*(Q - 5) = 0, so the optimum are Q= 5 and 12. Checking the second derivative to see which one

of these is a minimum,

dMC / dQ = 2Q - 17

2(5) - 17 < 0, thus Q = 5 corresponds to a maximum.

2(12) - 17 > 0, thus, Q = 12 corresponds to a minimum.

At Q = 12, TC = (1 /3)*123 - 8.5*122 + 60*12 + 27 = 99.

4. A market demand is such that if price is $10 per unit, the quantity demanded is 100. For each $1

increase in price, the quantity demanded decreases by 20. The quantity supplied is zero until the

price reaches $5. Above that price, the quantity supplied increases by 10 for each $1 increase in

price.

a. Determine the equation of the supply and demand curve. First, one needs two data points. An example is: P = 10, Q = 100; P = 11, Q = 80.

Then, we can calculate the slope: Slope = (10 - 11) / (100 - 80) = -0.05 = a.

Using (Q = 100, P = 10), we have P = b - 0.05Q, where b is the vertical intercept.

Solving for b yields b = 10 + 0.05*100 = 15.

The equation of the demand curve is therefore P = 15 - 0.05Q.

Now we can do the same thing for the supply:

The 2 data points can be for example (Q = 0, P = 5) and (Q = 10, P = 6).

Slope = (6- 5) / (10 - 0) = 0.1.

We know that the vertical intercept is 5 (see above), thus, the equation of the supply curve is

P = 5 + 0.1Q.

b. What are the demand and supply functions?

For the demand:

0.05Q = 15 - P, so

Q = 300 - 20P.

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Similarly for the supply function, inverting the equation of the supply curve yields:

Q = -50 + 10P.

c. What is the equilibrium price and quantity?

Setting demand = supply yields:

-50 + 10P = 300 - 20P, 350 = 30P,

P = 350 / 30 = $11.67

Plugging in the demand or supply function to get quantities yields: Q = Q = -50 + 10(11.67) = 66.67.

5. Using the demand function

Q = 300 - 10P (1)

a. Determine the equation of the total revenue (TR), average revenue (AR) and marginal revenue

(MR). TR = PQ. In order to calculate it, one needs to invert the demand function so that we can find the

demand curve equation:

10P = 300 - Q, P = 30 - 0.1Q.

Now, if we multiply both side of the demand curve equation by Q, we will have an expression for total

revenue: TR = PQ = (30 - 0.1Q)*Q = 30Q - 0.1Q2

Average revenue is defined as TR / Q: AR = (30Q - 0.1Q2 ) / Q = 30 - 0.1Q

Marginal revenue is defined as the first derivative of total revenue:

MR = dTR / dQ = 30 - 0.2Q

b.c. see class notes/ not included d. Determine mathematically at what quantity is total revenue maximized. Check that you have

maximized not minimized total revenue. In order to maximize total revenue, one needs to set the first derivative of the total revenue function (the

marginal revenue function) equal to 0:

30 - 0.2Q = 0, 30 = 0.2Q, Q = 30 / 0.2 = 150. At quantities of 150, total revenue is optimized. In order

to know whether we have a maximum or a minimum, one needs to check the second derivative:

d2TR / dQ2 = dMR / dQ = -0.2.

Since -0.2 < 0, we conclude that we have maximized not minimized total revenue at Q = 150.

SUNK COST

Sunk costs are retrospective costs that cannot be recovered, and are therefore irrelevant to future

investment decisions in the project which incurs them.

CHECKPOINTS

o Only prospective costs should impact an investment decision. Therefore, sunk costs are

not to be considered when deciding whether to undertake a project.

o A sunk cost is distinct from an economic loss. A loss may be caused by a sunk cost,

however.

o Sunk costs are irrecoverable.

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Sunk Costs Examples

Let’s say a company spent $5 million building an airplane. Before the plane is complete, the managers

learn that it is obsolete and no airline will buy it. The market has evolved and now the airlines want a

different type of plane.

The company can finish the obsolete plane for another $1 million, or it can start over and build the new

type of plane for $3 million. What should the managers decide? Should they spend that last $1 million to

finish up the plane that’s almost done, or should they spend the $3 million to build the new plane?

At first glance, you may think the company should just finish the old plane. It’s only another million

bucks and they already spent $5 million. But in reality, the five million is irrelevant. It is a sunk cost. The

only relevant cost is the $3 million dollars. The managers should consider whether or not to spend $3

million on the new plane, and nothing regarding the old plane should affect the decision.

DIFFERENCE BETWEEN SUNK COST AND ECONOMIC LOSS

The sunk cost is distinct from economic loss. For example, when a car is purchased, it can subsequently

be resold; however, it will probably not be resold for the original purchase price. The economic loss is the

difference between these values (including transaction costs).

The sum originally paid should not affect any rational future decision-making about the car, regardless of

the resale value. If the owner can derive more value from selling the car than not selling it, then it should

be sold, regardless of the price paid.

In this sense, the sunk cost is not a precise quantity, but an economic term for a sum paid in the past,

which is no longer relevant to decisions about the future. The sunk cost may be used to refer to the

original cost or the expected economic loss. It may also be used as shorthand for an error in analysis due

to the sunk cost fallacy, irrational decision-making or, most simply, as irrelevant data.

Sunk Costs Fallacy

The sunk cost fallacy is when someone considers a sunk cost in a decision and subsequently makes a poor

decision.

An example of the sunk cost fallacy is paying for a movie ticket, finding out the movie is terrible, and

staying to watch anyway just to get your money’s worth. When you find out the movie is terrible, you

should make a decision whether to sit through the bad movie or to do something more meaningful with

your time – the price you paid for the ticket should not affect your decision. The ticket price is a sunk

cost.

Another example of the sunk cost fallacy is paying for an all-you-can-eat buffet, eating until you’re full,

and then going back for more just to get your money’s worth. When you are full, you should decide

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whether you want to eat more or to stop eating – the fact that you paid for unlimited food should not

affect your decision. The price of the buffet is a sunk cost.

COST CURVES

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EXPANSION PATH

The change in the optimal input bundle when output changes depends on the nature of the production

function. The path of all optimal input bundles as output increases is called the output expansion path.

The blue line in the following figure is an example of an output expansion path.

In this example, as output increases initially more of both inputs are used. Then, between the second and

third isocost lines the amount of input 2 used starts to decline.

If the optimal amount of an input increases with output then the input is called normal, while if the

amount used decreases then the input is called inferior.

Example: a production function with fixed proportions

Consider the production function F (z1, z2) = min{z1, z2}. Given the shape of its isoquants, the output

expansion path of this production function is a ray from the origin, as in the following figure. For any

input prices, the firm uses y units of each input to produce y units of output (see its conditional input

demands), so that its output expansion path is the line z2 = z1.

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Example: a production function with fixed proportions

Consider the production function F (z1, z2) = min{z1/2, z2}. Given the shape of its isoquants, the output

expansion path of this production function is a ray from the origin, as in the following figure. For any

input prices, the firm uses 2y units of input 1 and y units of input 2 to produce y units of output (see its

conditional input demands), so that its output expansion path is the line z2 = z1/2.

Example: a general production function with fixed proportions

A general production function with fixed proportions takes the form F (z1, z2) = min{az1, bz2} for some

constants a and b. Given the shape of its isoquants, the output expansion path of this production function

is, as in the previous case, a ray from the origin. For any input prices, the firm uses y/a units of input 1

and y/b units of input 2 to produce y units of output (see its conditional input demands), so that its output

expansion path is the line z2 = (a/b)z1.

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Example: a production function in which the inputs are perfect substitutes

Consider the production function F (z1, z2) = z1 + z2, in which the inputs are perfect substitutes. Given the

shape of its isoquants, its output expansion path is

the z2 axis if w1 > w2 (if the price of input 1 exceeds that of input 2 then the firm uses none of

input 1)

the z1 axis if w1 < w2 (if the price of input 2 exceeds that of input 1 then the firm uses none of

input 2)

the set of all pairs (z1, z2) if w1 = w2 (if the prices of the inputs are the same then every

combination of inputs is optimal). [That is, in this case there is not a single output expansion

path.]

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Example: a general production function in which the inputs are perfect substitutes

The general form of a production function in which the inputs are perfect substitutes is F (z1, z2) = az1 +

bz2, for some constants a and b. Given the shape of its isoquants, its output expansion path is

the z2 axis (the firm uses none of input 1) if w1 > (a/b)w2

the z1 axis (the firm uses none of input 2) if w1 < (a/b)w2

the set of all pairs (z1, z2) (every combination of inputs is optimal) if w1 = (a/b)w2. [That is, in this

case there is not a single output expansion path.]

Example: a Cobb-Douglas production function

Consider the Cobb-Douglas production function F (z1, z2) = z11/2z2

1/2. The conditional input demand

functions for this production function are

z1 = y(w2/w1)1/2 and z2 = y(w1/w2)1/2.

Thus the output expansion path satisfies z2/z1 = w1/w2, or

z2 = (w1/w2)z1.

As in all the previous examples, the output expansion path is thus a ray from the origin.

Long run and short run cost functions

In the long run, the firm can vary all its inputs. In the short run, some of these inputs are fixed. Since the

firm is constrained in the short run, and not constrained in the long run, the long run cost TC(y) of

producing any given output y is no greater than the short run cost STC(y) of producing that output:

TC(y) STC(y) for all y.

Now consider the case in which in the short run exactly one of the firm's inputs is fixed. For concreteness,

suppose that the firm uses two inputs, and the amount of input 2 is fixed at k. For many (but not all)

production functions, there is some level of output, say y0, such that the firm would choose to use k units

of input 2 to produce y0, even if it were free to choose any amount it wanted. In such a case, for this level

of output the short run total cost when the firm is constrained to use k units of input 2 is equal to the long

run total cost: STCk(y0) = TC(y0). We generally assume that for any level at which input 2 is fixed, there is

some level of output for which that amount of input 2 is appropriate, so that for any value of k,

TC(y) = STCk(y) for some y.

(There are production functions for which this relation is not true, however: see the example of a

production function in which the inputs are perfect substitutes.)

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For a total cost function with the typical shape, the following figure shows the relations between STC and

TC.

Examples of long run and short run cost functions

Long run and short run average cost functions

Given the relation between the short and long run total costs, the short and long run average and marginal

cost functions have the forms shown in the following figure.

Note:

The SMC goes through the minimum of the SAC and the LMC goes through the minimum of the

LAC.

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When SAC = LAC we must have SMC = LMC (since slopes of total cost functions are the same

there).

In the case that the production function has CRTS, the LAC is horizontal, as in the following figure.

Derivation of Long-Run Marginal Cost Curve

The Derivation of Long-Run Marginal Cost Curve!

Since marginal cost curve is important both from the viewpoint of the short run and the long run, it will

be useful to know how the long-run marginal cost curve is derived.

The long-run marginal cost curve can be directly derived from the long-run total cost curve, since the

long-run marginal cost at a level of output is given by the slope of the total cost curve at the point

corresponding to that level of output.

Besides, the long-ran marginal cost curve can be derived from the long-run average cost curve, because

the long-ran marginal cost curve is related to the long-run average cost curve in the same way as the

short-ran marginal cost curve is related to short-run average cost curve. In Fig. 19.12, it is depicted that

how the long-run marginal cost curve LMC is derived from a long-run average cost curve LAC

enveloping a family of short-run average and marginal cost curves.

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If the output OA is too produced in the long ran, then it must be produced on the long-run average cost

curve LAC at point H which is a tangency point with the short-run average cost curve SAC1. Thus, when

output OA is to be produced in the long run, it will be produced with the plant corresponding to the short-

run average cost curve SAC1 and the short-run marginal cost curve SMC1.

Corresponding to the tangency point H between the short-run average cost curve SAC1 and the long-run

average cost curve LAC, there is a point N on the short run marginal cost curve SMC. This means that the

production of output OA in the long run involves the marginal cost AN. Therefore point N must lie on the

long-run marginal cost curve corresponding to output OA. If output OB is to be produced in the long run,

it will be produced at point Q which is the tangency point between LAC and SAC2.

Q is also the point on the short-run marginal cost curve SMC2, corresponding to output OB. (Q is the

common point between SAC2 and SMC2 because Q is the minimum point of SAC2, at which the SMC2,

cuts it from below). Thus Q must also lie on the long-run marginal cost curve corresponding to output

OB. Similarly, if output OC is to be produced in the long run, it will be produced at point M which is the

tangency point between LAC and SAC3 Corresponding to point M, the relevant point on the SMC3 is K

which means that the long-run marginal cost of producing OC is CK.

Thus point K must lie on the long-run marginal cost curve corresponding to output OC. By connecting

points N, Q and K we obtain the long-run marginal cost curve LMC. It will be seen from Fig. 19.12 that

the long-run marginal cost curves is flatter than the short-run marginal cost curves.

It should also be remembered that the relationship between the long-run marginal cost curve LMC and the

long-run average cost curve LAC is the same as that between the short-run marginal cost curve and the

short -run average cost curve.

Thus, when the long-run marginal cost curve LMC lies below the long-run average cost curve, the latter

will be falling, and when the long-run marginal cost curve lies above the long-run average cost, the latter

will be rising. When the long- run marginal cost is equal to the long- run average cost, the latter will be

neither rising nor falling.

Relationship of LAC and LMC with SAC and SMC:

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It is important to note that LAC and SAC curves are related in an important way with SMC and LMC

curves. This relationship shows, as will be seen from Fig. 19.13, that at the level of output at which a

particular SAC curve is tangent to the LAC curve, the corresponding SMC curve intersects the LMC

curve.

This relationship can be proved using a short-run total cost curve and the long run total cost curve and this

has been done in Figure 19.13 where a short-run total cost curve STC and the long-run total cost curve

LTC are drawn.

It will be seen from Figure 19.13 that long-run total cost curve LTC lies below the short-run total cost

curve STC at all levels of output except at output OA at which the two curves are tangent. This means

that L4C is less than SAC at all levels of output other than OA.

As will be seen from the bottom of panel figure 19.13 that long-run average cost LAC will be equal to the

short-run average cost SAC at output OA at which LTC curve is tangent to the STC curve. But the long-

run marginal cost LMC must also be equal to short-run marginal cost SMC at output OA i.e. the tangency

point P.

This is because marginal cost is given by the slope of the total cost curve at any point, and the LTC curve

and STC curve have the same slope at the tangency point P. It will be seen from the bottom panel of Fig.

19.13 that at output level OA, where the SAC is equal to the LAC, SMC is also equal to the LMC.

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Economies of Scope

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Economies of scope are cost advantages that result when firms provide a variety of products rather than

specializing in the production or delivery of a single product or service. Economies of scope also exist if a

firm can produce a given level of output of each product line more cheaply than a combination of separate

firms, each producing a single product at the given output level. Economies of scope can arise from the

sharing or joint utilization of inputs and lead to reductions in unit costs. Scope economies are frequently

documented in the business literature and have been found to exist in countries, electronic-based B2B

(business-to-business) providers, home healthcare, banking, publishing, distribution, and

telecommunications.

METHODS OF ACHIEVING ECONOMIES OF SCOPE

Flexible Manufacturing

The use of flexible processes and flexible manufacturing systems has resulted in economies of scope

because these systems allow quick, low-cost switching of one product line to another. If a producer can

manufacture multiple products with the same equipment and if the equipment allows the flexibility to

change as market demands change, the manufacturer can add a variety of new products to their current

line. The scope of products increases, offering a barrier to entry for new firms and a competitive synergy

for the firm itself.

Related Diversification

Economies of scope often result from a related diversification strategy and may even be termed

"economies of diversification." This strategy is made operational when a firm builds upon or extends

existing capabilities, resources, or areas of expertise for greater competitiveness. According to Hill,

Ireland, and Hoskisson in their best selling strategic management textbook, Strategic Management:

Competitiveness and Globalization, firms select related diversification as their corporate-level strategy in

an attempt to exploit economies of scope between their various business units. Cost-savings result when a

business transfers expertise in one business to a new business. The businesses can share operational skills

and know-how in manufacturing or even share plant facilities, equipment, or other existing assets. They

may also share intangible assets like expertise or a corporate core competence. Such sharing of activities

is common and is a way to maximize limited constraints.

As an example, Kleenex Corporation manufactures a number of paper products for a variety of end users,

including products targeted specifically for hospitals and health care providers, infants, children, families,

and women. Their brands include Kleenex, Viva, Scott, and Cottonelle napkins, paper towels, and facial

tissues; Depends and Poise incontinence products; Huggies diapers and wipes; Pull-Ups, Goodnites, and

Little Swimmers infant products; Kotex, New Freedom, Litedays, and Security feminine hygiene

products; and a number of products for surgical use, infection control, and patient care. All of these

product lines utilize similar raw material inputs and/or manufacturing processes as well as distribution

and logistics channels.

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Mergers

The merger wave that swept the United States in the late 1990s and early 2000s is, in part, an attempt to

create scope economies. Mergers may be undertaken for any number of reasons. "As a rule of thumb,"

explained Rob Preston in an article about the trouble with mergers, "'scope' acquisitions—moves that

enhance or extend a vendor's product portfolio—succeed more often than those undertaken to increase

size and consolidate costs." Pharmaceutical companies, for example, frequently combine forces to share

research and development expenses and bring new products to market. Research has shown that firms

involved in drug discovery realize economies of scope by sustaining diverse portfolios of research

projects that capture both internal and external knowledge spillovers.

Linked Supply Chains

Today's linked supply chains among raw material suppliers, other vendors, manufacturers, wholesalers,

distributors, retailers, and consumers often bring about economies of scope. Integrating a vertical supply

chain results in productivity gains, waste reduction, and cost improvements. These improvements, which

arise from the ability to eliminate costs by operating two or more businesses under the same corporate

umbrella, exist whenever it is less costly for two or more businesses to operate under centralized

management than to function independently.

The opportunity to gain cost savings can arise from interrelationships anywhere along a value chain. As

firms become linked in supply chains, particularly as part of the new information economy, there is a

growing potential for economies of scope. Scope economies can increase a firm's value and lead to

increases in performance and higher returns to shareholders. Building economies of scope can also help a

firm to reduce the risks inherent in producing a single product or providing a service to a single industry.