Outline - University of Colorado Boulder...1 =$50 million isocost line before the price of capital goes up 50 r 1 50 w 0 C 2 C 2 =$50 million isocost line after the price of capital

Outline

Introduction

Long-Run Total Cost Curve

Short-Run Total Cost Curve

2/44

Introduction

I In the last chapter - we solved the firm’s cost minimizationproblem. In these problems:

I We took quantity as givenI We varied wage w and rental rate r to get the firm’s labor and

capital demand curves

I In this chapter - we will look at how a firm’s total cost varieswith output Q

I What do you think will happen when a firm expands itsquantity?

I Will costs go up or down?

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What is the Long-Run Total Cost Curve? (3 steps)

Last Chapter: The firm’s cost minimization problem was

minL,K

TC = wL+ rK

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

We solved this problem for the firm’s labor and capital demandfunctions

Labor Demand: L⇤ = L(Q,w , r)Capital Demand: K ⇤ = K (Q,w , r)

We get the firm’s total cost curve by plugging these back into thetotal cost function

TC (Q,w , r) = wL(Q,w , r) + rK (Q,w , r)

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minL,K

TC = wL+ rK

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





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minL,K

TC = wL+ rK

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





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Cost Minimization

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Total Cost

For a given w and r , we can plot the curve TC = TC (Q).

TC (Q) = Long-Run Total cost curve. Shows how TC varies withoutput, holding constant input prices

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Total Cost

For a given w and r , we can plot the curve TC = TC (Q).

TC (Q) = Long-Run Total cost curve. Shows how TC varies withoutput, holding constant input prices

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Properties of Long-Run Total Cost Curve

1. Q = 0 =) TC = 0.I Why? In the long-run, there are no fixed costs. All inputs can

be varied.

2. TC (Q) is increasing in Q. (i.e., MC (Q) > 0)I As Q increases, the firm must use more inputs. The firm is on

a further out isocost line (higher TC )

Let’s figure out how we can find these curves mathematically (butfirst, let’s do a review problem)

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Review - Input Demand

Suppose production technology is characterized by Q = 50pLK .

a) State the firm’s cost minimization problem.

b) Find the firm’s labor demand and capital demand functions.(They will be functions of w , r , and Q.)

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Mathematically Finding Long-Run Total Cost

You should have gotten:

Labor Demand: L⇤ =Q

50

rr

w

Capital Demand: K ⇤ =Q

50

rw

r

Plug these functions in the total cost equation

TC ⇤ = wL⇤ + rK ⇤

= wQ

50

rr

w+ r

Q

50

rw

r

=Q

50

pwr +

Q

50

pwr

=Q

25

pwr

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50

rr

w


50

rw

r


TC ⇤ = wL⇤ + rK ⇤

= wQ

50

rr

w+ r

Q

50

rw

r

=Q

50

pwr +

Q

50

pwr

=Q

25

pwr

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50

rr

w


50

rw

r


TC ⇤ = wL⇤ + rK ⇤

= wQ

50

rr

w+ r

Q

50

rw

r

=Q

50

pwr +

Q

50

pwr

=Q

25

pwr

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50

rr

w


50

rw

r


TC ⇤ = wL⇤ + rK ⇤

= wQ

50

rr

w+ r

Q

50

rw

r

=Q

50

pwr +

Q

50

pwr

=Q

25

pwr

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50

rr

w


50

rw

r


TC ⇤ = wL⇤ + rK ⇤

= wQ

50

rr

w+ r

Q

50

rw

r

=Q

50

pwr +

Q

50

pwr

=Q

25

pwr

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How does TC (Q) shift when price of capital goes up?

L

K

1 million TVs

A

50

r0C1

C1 =$50 million isocost linebefore the price of capital goes up

50

r1

50

w0

C2

C2 =$50 million isocost lineafter the price of capital goes up

B

60

r1

60

w0

C3


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L

K

1 million TVs

A

50

r0C1


50

r1

50

w0

C2


B

60

r1

60

w0

C3


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L

K

1 million TVs

A

50

r0C1


50

r1

50

w0

C2


B

60

r1

60

w0

C3


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What happens when input prices change proportionally?

L

K

1 million TVs

A

TCA

r0

TCA

w0

C1

C1 = TCA isocost linebefore the input prices go up

TCA

r1

TCA

w1

C2

C2 = TCA isocost lineafter the input prices go up

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L

K

1 million TVs

A

TCA

r0

TCA

w0

C1


TCA

r1

TCA

w1

C2

C2 = TCA isocost lineafter the input prices go up

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L

K

1 million TVs

A,B

TCB

r1

TCA

w0=

TCB

w1

C1,C3


TCA

r1

TCA

w1

C2

C2 = TCA isocost lineafter the input prices go upC3 = TCB isocost lineafter the input prices go up

Same inputs, Higher Cost

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We can think about this problem mathematically. Before the pricerise, we have:

TC0 = w0L+ r0K

After the price rise:

TC1 = w1L+ r1K

= (�w0)L+ (�r0)K

= �(w0L+ r0K )

TC1 = �TC0

If w and r increase by some x%

I L⇤ and K ⇤ remain unchanged

I TC ⇤(Q) also increases by x%.

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TC0 = w0L+ r0K


TC1 = w1L+ r1K

= (�w0)L+ (�r0)K

= �(w0L+ r0K )

TC1 = �TC0




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TC0 = w0L+ r0K


TC1 = w1L+ r1K

= (�w0)L+ (�r0)K

= �(w0L+ r0K )

TC1 = �TC0




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TC0 = w0L+ r0K


TC1 = w1L+ r1K

= (�w0)L+ (�r0)K

= �(w0L+ r0K )

TC1 = �TC0




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TC0 = w0L+ r0K


TC1 = w1L+ r1K

= (�w0)L+ (�r0)K

= �(w0L+ r0K )

TC1 = �TC0




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TC0 = w0L+ r0K


TC1 = w1L+ r1K

= (�w0)L+ (�r0)K

= �(w0L+ r0K )

TC1 = �TC0




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What does the firm care about?

Does a firm just care about total costs? What else might they careabout when they are making decisions?

I Per unit cost: Long-Run Average Cost

I Cost to make the next unit: Long-Run Marginal Cost

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Long-Run Average and Marginal Cost

LR Average Cost: AC (Q) =TC (Q)

Q= slope of ray from origin to point

on TC curve

LR Marginal Cost: MC (Q) =dTC (Q)

dQ= slope of TC (Q) at a certain

point

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Long-Run Average and Marginal Cost

LR Average Cost: AC (Q) =TC (Q)

Q= slope of ray from origin to point

on TC curve

LR Marginal Cost: MC (Q) =dTC (Q)

dQ= slope of TC (Q) at a certain

point

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Graphical Approach: AC and MC

Q

TCTC (Q)

A

50

$1,500

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Q

TCTC (Q)

A

50

$1,500

0

AC = Slope of ray 0A = 30

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Q

TCTC (Q)

A

50

$1,500

B

C

MC = Slope of line BAC = 10

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Q

AC ,MC

A0

A00

50

$30

$10

MC

AC

When MC < AC , AC is fallingWhen MC > AC , AC is risingAt D, MC = AC , AC is at minimum

D

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Q

AC ,MC

A0

A00

50

$30

$10

MC

AC


D

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Q

AC ,MC

A0

A00

50

$30

$10

MC

AC


D

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Q

AC ,MC

A0

A00

50

$30

$10

MC

AC

When MC < AC , AC is falling

When MC > AC , AC is risingAt D, MC = AC , AC is at minimum

D

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Q

AC ,MC

A0

A00

50

$30

$10

MC

AC

When MC < AC , AC is fallingWhen MC > AC , AC is rising

At D, MC = AC , AC is at minimum

D

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Q

AC ,MC

A0

A00

50

$30

$10

MC

AC


D

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Mathematically Finding AC and MC

Let’s use our example from earlier. Recall, that when theproduction function was Q = 50

pKL we solved for total cost as:

TC (Q,w , r) =Q

25

pwr

Suppose w = 25 and r = 100.

We then have:

TC (Q) = 2Q

How do we find AC (Q) and MC (Q)?

AC (Q) =TC (Q)

Q=

2Q

Q= 2

MC (Q) =dTC (Q)

dQ= 2

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TC (Q,w , r) =Q

25

pwr

Suppose w = 25 and r = 100. We then have:

TC (Q) = 2Q


AC (Q) =TC (Q)

Q=

2Q

Q= 2

MC (Q) =dTC (Q)

dQ= 2

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TC (Q,w , r) =Q

25

pwr


TC (Q) = 2Q


AC (Q) =

TC (Q)

Q=

2Q

Q= 2

MC (Q) =

dTC (Q)

dQ= 2

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TC (Q,w , r) =Q

25

pwr


TC (Q) = 2Q


AC (Q) =TC (Q)

Q

=2Q

Q= 2

MC (Q) =

dTC (Q)

dQ= 2

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TC (Q,w , r) =Q

25

pwr


TC (Q) = 2Q


AC (Q) =TC (Q)

Q=

2Q

Q= 2

MC (Q) =

dTC (Q)

dQ= 2

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TC (Q,w , r) =Q

25

pwr


TC (Q) = 2Q


AC (Q) =TC (Q)

Q=

2Q

Q= 2

MC (Q) =dTC (Q)

dQ

= 2

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TC (Q,w , r) =Q

25

pwr


TC (Q) = 2Q


AC (Q) =TC (Q)

Q=

2Q

Q= 2

MC (Q) =dTC (Q)

dQ= 2

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Cost Curves from Cobb-Douglas

Q

$TC (Q)

$2AC (Q),MC (Q)

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Cost Curves from Cobb-Douglas

Q

$TC (Q)

$2AC (Q),MC (Q)

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Relationship between LR AC and MC

I If average cost is decreasing as quantity is increasing, thenAC (Q) > MC (Q).

I If average cost for 100 units is $90/unit and the next unitcosts $80 to make, then average cost will go down.

I If average cost is increasing as quantity is increasing, thenAC (Q) < MC (Q).

I If average cost for 100 units is $90/unit and the next unitcosts $100 to make, then average cost will go up.

I If average cost is unchanged as quantity is increasing, thenAC (Q) < MC (Q).

I If average cost for 100 units is $90/unit and the next unitcosts $90 to make, then average cost will stay the same.

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23/44

Economies and Diseconomies of Scale

We look at the average cost curve for this

I Economies of Scale: a characteristic of production in whichaverage cost decreases as output goes up

I Diseconomies of Scale: a characteristic of production inwhich average cost increases as output goes up

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Example of AC (Q) curve

Q

AC

AC (Q)

Q 0

Economies of ScaleIndivisible Inputs

Q 00

Diseconomies of ScaleIncreasing Managerial Costs

Minimum E�cient Scale

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Q

AC

AC (Q)

Q 0


Q 00



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Q

AC

AC (Q)

Q 0


Q 00



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Q

AC

AC (Q)

Q 0


Q 00



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Economies of Scale and Returns to Scale

I If average cost decreases as output increases, we haveeconomies of scale and increasing returns to scale

I If average cost increases as output increases, we havediseconomies of scale and decreasing returns to scale

I If average cost stays the same as output increases, we haveneither economies nor diseconomies of scale and constantreturns to scale

26/44

Measuring the Extent of Economies of Scale

Output elasticity of Total Cost

✏TC ,Q =%�TC

%�Q

=dTC

dQ| {z }MC

Q

TC|{z}1AC

=MC

AC

Putting it all together (Table 8.3 in textbook)Economies/

Value of ✏TC ,Q MC vs. AC AC (Q) Diseconomies of Scale✏TC ,Q < 1 MC < AC Decreases Economies of Scale✏TC ,Q > 1 MC > AC Increases Diseconomies of Scale✏TC ,Q = 1 MC = AC Constant Neither

27/44



✏TC ,Q =%�TC

%�Q=

dTC

dQ| {z }MC

Q

TC|{z}1AC

=MC

AC



27/44



✏TC ,Q =%�TC

%�Q=

dTC

dQ| {z }MC

Q

TC|{z}1AC

=MC

AC



27/44



✏TC ,Q =%�TC

%�Q=

dTC

dQ| {z }MC

Q

TC|{z}1AC

=MC

AC



27/44


I Recall: What is the di↵erence between the long-run andshort-run?

I At least one input is fixed =) Some costs are fixed!

The Model

Assume: 2 inputs and capital (K , L)Capital is fixed at K̄w and r are taken as given by the firm

I L = variable cost

I K = short-run fixed costs

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The Model


I L = variable cost


28/44




The Model


I L = variable cost


28/44


Short-Run Total Cost Curve STC (Q) = minimized total cost toproduce Q units of output when at least one input is fixed at aparticular level

Total-Variable Cost Curve TVC (Q) = A curve that shows thesum of expenditures on variable inputs, such as labor, at theshort-run cost minimizing input combination

Total-Fixed Cost Curve TFC = A curve that shows the cost offixed inputs. It does not vary with output (because its fixed!)

STC (Q) = TVC (Q) + TFC

= TVC (Q) + r K̄

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= TVC (Q) + r K̄

29/44






= TVC (Q) + r K̄

29/44






= TVC (Q) + r K̄

29/44

STC Graphically

Q

$

TFCrK̄

TVC (Q)

r K̄

r K̄

STC (Q)

30/44

STC Graphically

Q

$

TFCrK̄

TVC (Q)

r K̄

r K̄

STC (Q)

30/44

STC Graphically

Q

$

TFCrK̄

TVC (Q)

r K̄

r K̄

STC (Q)

30/44

STC Graphically

Q

$

TFCrK̄

TVC (Q)

r K̄

r K̄

STC (Q)

30/44

STC Graphically

Q

$

TFCrK̄

TVC (Q)

r K̄

r K̄

STC (Q)

30/44

ExampleAgain, let’s suppose that our production function is Q = 50

pLK

and that capital is fixed at K̄ . Find the short-run total cost curve.

Step 1) Find the labor demand curve.

Demand Function: L =Q2

2500K̄

Step 2) Plug values of L and K into the total cost equation

STC (Q) = wL+ rK

= w

✓Q2

2500K̄

◆

| {z }TVC(Q)

+ r K̄|{z}TFC

Notice something about TVC (Q). As we increase K̄ the totalvariable costs actually fall. Why is this?

I Having more fixed capital means you use less labor

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pLK




2500K̄


STC (Q) = wL+ rK

= w

✓Q2

2500K̄

◆

| {z }TVC(Q)

+ r K̄|{z}TFC



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pLK




2500K̄


STC (Q) = wL+ rK

= w

✓Q2

2500K̄

◆

| {z }TVC(Q)

+ r K̄|{z}TFC



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pLK




2500K̄


STC (Q) = wL+ rK

= w

✓Q2

2500K̄

◆

| {z }TVC(Q)

+ r K̄|{z}TFC



31/44


pLK




2500K̄


STC (Q) = wL+ rK

= w

✓Q2

2500K̄

◆

| {z }TVC(Q)

+ r K̄|{z}TFC



31/44


pLK




2500K̄


STC (Q) = wL+ rK

= w

✓Q2

2500K̄

◆

| {z }TVC(Q)

+ r K̄|{z}TFC



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How are Long-Run and Short-Run Total Costs Related?

L

K

K̄

Q1

A

What happens in short-run at higher quantity?

Q2

B

The long-run?

C

Generally, costs are higher in the SRAt A, costs are the same in LR and SR

32/44


L

K

K̄

Q1

A


Q2

B

The long-run?

C


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L

K

K̄

Q1

A


Q2

B

The long-run?

C


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L

K

K̄

Q1

A


Q2

B

The long-run?

C


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L

K

K̄

Q1

A


Q2

B

The long-run?

C


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L

K

K̄

Q1

A


Q2

B

The long-run?

C


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L

K

K̄

Q1

A


Q2

B

The long-run?

C


32/44


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STC (Q) always lies above TC (Q) except at the point where K̄ isthe cost-minimizing amount of capital used in the long-run

34/44

Short-Run Average and Marginal Cost Curves

SR Average Cost: SAC (Q) =STC (Q)

Q= firm’s total cost per unit of

output when it has at leastone fixed input

SR Marginal Cost: SMC (Q) =dSTC (Q)

dQ= slope of STC (Q) at a cer-

tain point. The change in theshort-run total cost if outputis increased by one unit

35/44

Short-Run Average and Marginal Cost Curves

SR Average Cost: SAC (Q) =STC (Q)

Q= firm’s total cost per unit of

output when it has at leastone fixed input

SR Marginal Cost: SMC (Q) =dSTC (Q)

dQ= slope of STC (Q) at a cer-

tain point. The change in theshort-run total cost if outputis increased by one unit

35/44

Short-Run Average Cost Curve

We can also break these down into variable and fixed costs.

SAC (Q) =STC (Q)

Q=

TVC (Q) + TFC

Q= AVC (Q) + AFC (Q)

I Average short-run total cost is comprised of average variablecosts and average fixed costs.

AVC (Q) =TVC (Q)

Qand AFC (Q) =

TFC

Q

36/44

Short-run Cost Curves

Q

$AFC (Q) always falls and approaches zero

AFC (Q)

SMC (Q)

B

AVC (Q)

SAC (Q)

A

37/44


Q

$

AFC (Q)

SMC (Q) U-shaped

SMC (Q)

B

AVC (Q)

SAC (Q)

A

37/44


Q

$

AFC (Q)

SMC (Q)

AVC (Q) minimum when it crosses SMC (Q)

and is also U-shaped

B

AVC (Q)

SAC (Q)

A

37/44


Q

$

AFC (Q)

SMC (Q)

B

AVC (Q)

SAC (Q) starts above AVC (Q) and AFC (Q)and approaches AVC (Q)

SAC (Q)

A

37/44

Relationship between LR and SR Average & Marginal CostCurves

What will be higher? Firms long-run average cost or short-runaverage cost curves

The short-run

I In the short-run we can’t vary K̄ , this means our total costsare higher. So, our short-run average costs must be higher.

I Except for the case where K̄ is the long-run cost-minimizingamount of capital

38/44


What will be higher? Firms long-run average cost or short-runaverage cost curvesThe short-run



38/44


What will be higher? Firms long-run average cost or short-runaverage cost curvesThe short-run



38/44


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I Long-run average cost curve forms a boundary (envelope)around the set of short-run cost curves corresponding todi↵erence levels of output at di↵erent amount of the fixedinput

I Short-run average cost curve lies above long-run curve, exceptfor the level of output where the fixed capital is optimal

40/44


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I SMC (Q) = MC (Q) when Long-Run and Short-Run Total(Average) Costs are equal

How to draw SMC :

I Must cut through SAC at its minimum

I Must be equal to MC (Q) at Q⇤ when STC = LTC (orSAC = AC )

From the previous slide:

At A: SAC = AC because the firm has the optimal plant sizeto produce 1 million units

G : SMC (Q) = MC (Q) because the firm has the optimalplant size to produce 1 million units

42/44

Economies of Scope

So far, we have been assuming that the firm only produces onegood. How could we think about relaxing this assumption?

I Firms may make 2 or more goods, Q1,Q2, . . .

I Total costs = TC (Q1,Q2, . . .)I E�ciencies may occur when they produce two goods

I For example - The same manufacturing process is used soinstead of leaving the capital unused for part of the day, theycan put it to use.

I Or maybe labor is working to its fullest potential. If the firmproduced more than one product, they would have more workto fill their day (e.g., graphic design, managers)

43/44

Economies of Scope



I Total costs = TC (Q1,Q2, . . .)

I E�ciencies may occur when they produce two goodsI For example - The same manufacturing process is used so

instead of leaving the capital unused for part of the day, theycan put it to use.


43/44

Economies of Scope






43/44

Economies of Scope






43/44

Economies of Scope

Economies of Scope: A production function characteristic inwhich the total cost of producing given quantities of two goods inthe same firm is less than the total cost of producing thosequantities in two single-product firms.

TC (Q1,Q2) < TC (Q1, 0) + TC (0,Q2)

44/44

Economies of Scope

Economies of Scope: A production function characteristic inwhich the total cost of producing given quantities of two goods inthe same firm is less than the total cost of producing thosequantities in two single-product firms.

TC (Q1,Q2) < TC (Q1, 0) + TC (0,Q2)

44/44

Documents

Outline - University of Colorado Boulder...1 =$50 million isocost line before the price of capital goes up 50 r 1 50 w 0 C 2 C 2 =$50 million isocost line after the price of capital