outline - university of colorado boulder...1 =$50 million isocost line before the price of capital...

Outline

Introduction

Long-Run Total Cost Curve

Short-Run Total Cost Curve

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?

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)

minL,K

TC = wL+ rK

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

minL,K

TC = wL+ rK

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

Cost Minimization

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

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

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)

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.)

Mathematically Finding Long-Run Total Cost

You should have gotten:

Labor Demand: L⇤ =Q

Capital Demand: K ⇤ =Q

Plug these functions in the total cost equation

TC ⇤ = wL⇤ + rK ⇤

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

1 million TVs

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

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

1 million TVs

What happens when input prices change proportionally?

1 million TVs

C1 = TCA isocost linebefore the input prices go up

C2 = TCA isocost lineafter the input prices go up

1 million TVs

C2 = TCA isocost lineafter the input prices go up

1 million TVs

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

Same inputs, Higher Cost

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%.

TC0 = w0L+ r0K

TC1 = w1L+ r1K

= (�w0)L+ (�r0)K

= �(w0L+ r0K )

TC1 = �TC0

TC0 = w0L+ r0K

TC1 = w1L+ r1K

= (�w0)L+ (�r0)K

= �(w0L+ r0K )

TC1 = �TC0

TC0 = w0L+ r0K

TC1 = w1L+ r1K

= (�w0)L+ (�r0)K

= �(w0L+ r0K )

TC1 = �TC0

TC0 = w0L+ r0K

TC1 = w1L+ r1K

= (�w0)L+ (�r0)K

= �(w0L+ r0K )

TC1 = �TC0

TC0 = w0L+ r0K

TC1 = w1L+ r1K

= (�w0)L+ (�r0)K

= �(w0L+ r0K )

TC1 = �TC0

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

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

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

Graphical Approach: AC and MC

TCTC (Q)

$1,500

TCTC (Q)

$1,500

AC = Slope of ray 0A = 30

TCTC (Q)

$1,500

MC = Slope of line BAC = 10

AC ,MC

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

AC ,MC

When MC < AC , AC is falling

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

AC ,MC

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

At D, MC = AC , AC is at minimum

AC ,MC

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

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)

MC (Q) =dTC (Q)

TC (Q,w , r) =Q

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

TC (Q) = 2Q

AC (Q) =TC (Q)

MC (Q) =dTC (Q)

TC (Q,w , r) =Q

TC (Q) = 2Q

AC (Q) =

TC (Q)

MC (Q) =

dTC (Q)

TC (Q,w , r) =Q

TC (Q) = 2Q

AC (Q) =TC (Q)

MC (Q) =

dTC (Q)

TC (Q,w , r) =Q

TC (Q) = 2Q

AC (Q) =TC (Q)

MC (Q) =

dTC (Q)

TC (Q,w , r) =Q

TC (Q) = 2Q

AC (Q) =TC (Q)

MC (Q) =dTC (Q)

TC (Q,w , r) =Q

TC (Q) = 2Q

AC (Q) =TC (Q)

MC (Q) =dTC (Q)

Cost Curves from Cobb-Douglas

$TC (Q)

$2AC (Q),MC (Q)

Cost Curves from Cobb-Douglas

$TC (Q)

$2AC (Q),MC (Q)

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.

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

Example of AC (Q) curve

AC (Q)

Economies of ScaleIndivisible Inputs

Diseconomies of ScaleIncreasing Managerial Costs

Minimum E�cient Scale

AC (Q)

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

Measuring the Extent of Economies of Scale

Output elasticity of Total Cost

✏TC ,Q =%�TC

dQ| {z }MC

TC|{z}1AC

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

✏TC ,Q =%�TC

%�Q=

dQ| {z }MC

TC|{z}1AC

✏TC ,Q =%�TC

%�Q=

dQ| {z }MC

TC|{z}1AC

✏TC ,Q =%�TC

%�Q=

dQ| {z }MC

TC|{z}1AC

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

The Model

I L = variable cost

The Model

I L = variable cost

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

STC Graphically

TFCrK̄

TVC (Q)

STC (Q)

STC Graphically

TFCrK̄

TVC (Q)

STC (Q)

STC Graphically

TFCrK̄

TVC (Q)

STC (Q)

STC Graphically

TFCrK̄

TVC (Q)

STC (Q)

STC Graphically

TFCrK̄

TVC (Q)

STC (Q)

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

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

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

2500K̄

STC (Q) = wL+ rK

2500K̄

| {z }TVC(Q)

+ r K̄|{z}TFC

2500K̄

STC (Q) = wL+ rK

2500K̄

| {z }TVC(Q)

+ r K̄|{z}TFC

2500K̄

STC (Q) = wL+ rK

2500K̄

| {z }TVC(Q)

+ r K̄|{z}TFC

2500K̄

STC (Q) = wL+ rK

2500K̄

| {z }TVC(Q)

+ r K̄|{z}TFC

2500K̄

STC (Q) = wL+ rK

2500K̄

| {z }TVC(Q)

+ r K̄|{z}TFC

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

What happens in short-run at higher quantity?

The long-run?

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

The long-run?

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

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

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

Short-Run Average Cost Curve

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

SAC (Q) =STC (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) =

Short-run Cost Curves

$AFC (Q) always falls and approaches zero

AFC (Q)

SMC (Q)

AVC (Q)

SAC (Q)

AFC (Q)

SMC (Q) U-shaped

SMC (Q)

AVC (Q)

SAC (Q)

AFC (Q)

SMC (Q)

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

and is also U-shaped

AVC (Q)

SAC (Q)

AFC (Q)

SMC (Q)

AVC (Q)

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

SAC (Q)

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

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

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

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

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)

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.

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)

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)

outline - university of colorado boulder...1 =$50 million isocost line before the price of capital...

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