aug1

Outline for Friday, August 1

� Remember

� Quiz Monday

� Review Thursday

� Cost curves� Cost curves

� Profit maximization

Cost minimization

� Fixed proportions: q0=aL=bK

� Perfect substitutes: q0 = aL OR q0 = bK, whichever is cheaper

� Cobb-Douglas� Cobb-Douglas

� Produce exactly q0: q = LAKB

� Two options…

� Matching slopes: MRTS = -MPL/MPK = -w/r

� Cost shares: SLC = wL or SKC = rK

�Shortcut: BLw = AKr

How does cost change with q?

� To find the cost function

we must solve for L* and K* as functions of q

**),,( rKwLrwqC +=

we must solve for L* and K* as functions of q

� So when we do the algebra,

� Leave q as a variable

� It’s not a known number

� Imagine the COO or the bar owner is planning for all contingencies

What is the cost function?

� C(q, w, r) is

� The cheapest way of making q when

� the wage of labor w

� the rental rate of capital is r

� Abbreviated C(q) when w and r are fixed


� C(q, w, r) is

� The cheapest way of making q when

� the wage of labor w

� the rental rate of capital is r

� Abbreviated C(q) when w and r are fixed

� The job of the COO, who chooses inputs

� A black box to the CEO, who chooses q


� Remember

� Quiz Monday

� Review Thursday




� How do we graph cost minimization?

� Graph the optimal bundle of inputs as q increases, the output expansion path

Figure 7.7 Expansion Path and Long-Run Cost Curve

(a) Expansion Path

$3,000isocost

$4,000 isocost

© 2007 Pearson Addison-Wesley. All rights reserved.

7–9

x

y

z

10075500 L, Workers per hour

150

200

100

Expansion path

$2,000isocost

100 isoquant

150 isoquant

200 isoquant


� How do we graph cost minimization?

� Graph the optimal bundle of inputs as q increases, the output expansion path

� Graph the cost of the optimal bundle as q increases, ignoring the inputs, the cost curve

Figure 7.7 Expansion Path and Long-Run Cost Curve (cont’d)

Y

Z4,000

3,000

Long-run cost curve

(b) Long-Run Cost Curve


7–11

X

Y

0 q, Units per hour

2,000

200100 150


� C(q) is different in the short run and the long run

Short-run cost


� In the long run, we can choose capital

� In the short run, capital is fixed so C(q) is much simpler

rKwLrwqC +=),,( 0

qKLfL

rKwLrwqC

=

+=

),( satisfies where

),,(

0

0

Short-run cost exercises

� What is the lowest cost at which a firm can make q output if its technology is q = 2L+3K0, where capital is fixed at K0, the wage is 2 and the rental rate is 10?

� Solve for L(q)

� Plug it into C(q) = wL(q) + rK0


� What is the lowest cost at which a firm can make q output if its technology is q = min{2L, 3K0}, where capital is fixed at K0, the wage is 2 and the rental rate is 10?

� Solve for L(q)



� What is the lowest cost at which a firm can make q output if its technology is q = L2K0

3, where capital is fixed at K0, the wage is 2 and the rental rate is 10?

� Solve for L(q)


Short-run cost




rKwLrwqC +=),,( 0

� This is our short-run total cost curve (TC), how cost varies with quantity

qKLfL

rKwLrwqC

=

+=

),( satisfies where

),,(

0

0

Short-run cost




rKwLrwqC +=),,( 0

� This is our short-run total cost curve (TC), how cost varies with quantity

� TC/L is the average cost per unit of labor (AC)

� The slope of TC is short-run marginal cost (MC)

� AC and MC cross at AC’s lowest value

qKLfL

rKwLrwqC

=

+=

),( satisfies where

),,(

0

0

Short-run cost

� How can we find MC?

LL MP

w

MPw

q

L

L

C

q

CMC =⋅=

∆

∆⋅

∆

∆=

∆

∆=

1

Short-run cost

� How can we find MC?

� AC?LL MP

w

MPw

q

L

L

C

q

CMC =⋅=

∆

∆⋅

∆

∆=

∆

∆=

1

� AC?

� TC?

q

rKwL

q

CAC 0+

==

0rKwLCTC +==

Short-run cost

� We can break down short-run TC into

� Fixed costs – Costs that do not vary with q

Sunk costs – Fixed, a cost that we cannot change

� Variable costs – Costs that vary with q� Variable costs – Costs that vary with q

Costs that we can change

Short-run cost


� Sunk costs – Pay no matter what

Fixed but not sunk – Pay if q > 0

e.g., franchise fee, license fee, protection moneye.g., franchise fee, license fee, protection money

� Variable costs – Pay depending on q

SUNK

FIXED, NOT

SUNK

VARIABLE

Short-run cost


� Sunk costs – Pay no matter what

Fixed but not sunk – Pay if q > 0

e.g., franchise fee, license fee, protection moneye.g., franchise fee, license fee, protection money

� Variable costs – Pay depending on q

� Ignore sunk costs

� They cannot be recovered

� The firm may make a loss

Short-run cost



� Variable costs – Costs that vary with q

Costs that we can changeCosts that we can change

0

0 F VC

rKF

wLVC

rKwLCTC

=

=

+=+==

Short-run cost

� It will be useful later to compare per-unit costs

� MC w/MPL

� AFC – average fixed cost rK0/q� AFC – average fixed cost rK0/q

� AVC – average variable cost w/APL

� AC – average cost AFC+AVCLAP

w

Lq

w

q

wL==

/

Short-run cost

� It will be useful later to compare per-unit costs

� MC w/MPL

� AFC – average fixed cost rK0/q

� AVC – average variable cost w/APL� AVC – average variable cost w/APL

� AC – average cost AFC+AVC

� MPL crosses APL at its peak � MC crosses AVC at its minimum value

� MC crosses AC at its minimum value

Short-run cost



� Variable costs – Costs that vary with q

Costs that we can changeCosts that we can change

� In the long run, capital is variable so we can produce q at lower cost per unit – lower AC

Figure 7.9 Long-Run Average Cost as the Envelope of Short-Run Average Cost Curves

SRAC1 SRAC2SRAC3

SRAC3

LRAC


7–28

a

bd

e

SRAC SRAC

c

q2

q1

q, Output per day

10

0

12

Short run and long run cost


� Are our SR or LR AC curves U-shaped?




++

BwArBAABAB )/()/(

+=

=

+

=

+

−

b

r

a

wqC

b

r

a

wqC

rAr

Bww

Bw

ArqC

CompsPerfect

SubsPerfect

BA

DouglasCobb

)/(1

,min

Figure 7.7 Expansion Path and Long-Run Cost Curve (cont’d)

Y

Z4,000

3,000

Long-run cost curve

(b) Long-Run Cost Curve


7–31

X

Y

0 q, Units per hour

2,000

200100 150

For perfect complements and perfect substitutes




� No, but real cost curves are� No, but real cost curves are

� The shape of the AC curve tells us how cost changes with scale

� Choose t > 1, compare C(tq) to tC(q)

� C(tq) < tC(q) Economies of scale

� C(tq) > tC(q) Diseconomies of scale

� Refers to costs, not technology




� C(tq) < tC(q) Economies of scale� C(tq) < tC(q) Economies of scale

AC rises falls with q


AC rises with q




� C(tq) < tC(q) Economies of scale� C(tq) < tC(q) Economies of scale

AC rises falls with q


AC rises with q

� So, U-shaped AC curves have diseconomies of scale eventually

� And L-shaped AC curves have no economies or diseconomies of scale eventually

Application (Page 205) Average Cost of Cement Firms: L-shaped cost curve

7

6


7–35

3.0 3.32.01.0 2.51.50.5

AC

q, Cement, million tons per year

5

4

0



� Think about this in terms of the expansion path, which tells us how L* and K* vary with qwhich tells us how L* and K* vary with q


� Output Expansion Path …

37

K

… in the long-run

5/27/2008M. L. Williams, Department of Economics, PSU

L

c

a

b


� Output Expansion Path

� We cannot choose the optimal level of capital in the short run

38

K

… in the long-run

5/27/2008M. L. Williams, Department of Economics, PSU

L

acb … in the short-run



� Think about this in terms of the expansion path, which tells us how L* and K* vary with qwhich tells us how L* and K* vary with q

� We are constrained to one cross section of technology in the short run

� To meet our production targets, we must use more or less capital than we would like to

Cost curves recap

� In the short run, some costs are fixed� Sunk costs, which cannot be recovered, must be ignored

� We cannot choose capital levels

� In the long run, average costs are lower� We can reach any of the SR AC curves we want

� Costs that are fixed, but not sunk, may still matter

� LR AC curves� Slope down where there are economies of scale

� Slope up where there are diseconomies of scale

� Are flat where there are no economies of scale

� Are often U-shaped or L-shaped


� Remember

� Quiz Monday

� Review Thursday



Profit maximization

� The firm wants to maximize profits

π = R(q, p) – C(q, r, w)

Profit maximization


π = R(q, p) – C(q, r, w)

� We can use cost minimization to find C(q, r, w)

� We know that revenue is money collected, R = pq� We know that revenue is money collected, R = pq

Profit maximization


π = R(q, p) – C(q, r, w)



� Why do we assume that the firm is a price-taker?

� Recall that a monopolist faces the whole demand curve

� When there are many firms, however…

Profit maximization

P � Charge too much � no one buys

� Charge too little � must satisfy all buyers

Market demand curve

Firm's demand curve

Q

Price-taking


π = R(q, p) – C(q, r, w)





� When there are many firms, a single firm will take the market price as given

Price-taking


π = R(q, p) – C(q, r, w)





� When there are many firms, a single firm will take the market price as given

Price-taking

� We assume that the firm is a price-taker

� Buyers and sellers see and trade at the same price

� Transaction costs are low

� Search costs are low

� Firms cannot differentiate themselves

� There is no product differentiation: branding, quality, location

� Firms freely enter and exit the market

� No fixed costs of entry or exit

� There are too many competitors, so the firm’s demand curve is horizontal

Price-taking




� Firms freely enter and exit the market� Firms freely enter and exit the market

� These assumptions make up the model of perfect competition

� A firm only chooses how much to produce

Price-taking




� Firms freely enter and exit the market� Firms freely enter and exit the market

� These assumptions make up the model of perfect competition

� A firm only chooses how much to produce

� We have not assumed that firms are identical

Profit maximization in the long run


π = pq – C(q)

� How much will a firm produce in the long run?



π = pq – C(q)


� It will produce more as long as p > MC� It will produce more as long as p > MC

� It will produce less if p < MC

� It will shut down if p < AC



π = pq – C(q)





� As long as P > AC, the firm will follow the rule P=MC



π = pq – C(q)






� This is the rule any firm will follow in the long run

� Price is Marginal Revenue, so this rule is also MC = MR



� In other words, MC is the firm’s supply curve when P>AC

(a) Firm

P>AC

150

LRAC

LRMC

q, Hundred metric tons of oil per year

10

S1

0© 2007 Pearson Addison-Wesley. All rights reserved.

8–55



� In other words, MC is the firm’s supply curve when P>ACP>AC

� Market supply

� If there are other firms that are just as productive, the

firm’s output can be reproduced – zero profit

Figure 8.10 Long-Run Firm and Market Supply with Identical Vegetable Oil Firms

(b) Market

LRAC

(a) Firm

S1


8–57

Q, Hundred metric tons of oil per year

Long-run market supply10

0150

LRMC

q, Hundred metric tons of oil per year

10

0




� Market supply



� If other firms are less productive, they will only enter at higher prices

Application (Page 246) Upward-Sloping Long-Run Supply Curve for Cotton

Iran

United States1.56

1.71 S


8–59

0.71

0 1 2 3

United States

Nicaragua, Turkey

Brazil

AustraliaArgentina

Pakistan

4 5 6 6.8

Cotton, billion kg per year

1.081.15

1.27

1.43

1.56

Profit maximization in the long run: summary



� Market supply



� If other firms are less productive, they will only enter at higher prices

� You will not have to calculate supply curves

Profit maximization in the short run

� The firm cannot change its sunk costs, so…

� Instead of producing whenever P>AC…

� The firm will produce whenever P>AVC


� The firm cannot change its sunk costs, so…

� Instead of producing whenever P>AC…

� The firm will produce whenever P>AVC

� Again, the firm will produce at P=MC� Again, the firm will produce at P=MC

Figure 8.4 How the Profit-Maximizing Quantity Varies with Price – in the short run

e3

e4

7

8

AC

S


8–63

q3 = 215 q

4 = 285q1 = 50 q

2 = 140

e1

e2

0

q, Thousand metric tons of lime per year

6

7

5

AVC

MC


� As long as P > AVC, the firm will follow the rule P=MC

� The market supply curve is the horizontal sum of curves, as seen beforecurves, as seen before

Profit maximization graphs

� You should be able to tell from a graph

� Will a firm operate or shut down/exit at this price?

� What area represents revenue?

� How much is cost?How much is cost?

� How much is profit?

� What does a per-unit tax do?

� Raises MC and thus AC

� Reduces the quantity or shuts the firm down

� Reduces profit


� Remember

� Quiz Monday

� Review Thursday



� Next time: Welfare

�Read Chapter 9 for Monday

aug1

Economy & Finance

functions of q

cost function

lakbtwo optionsmatching

bar owner

known numberimagine

wrcost shares

bkperfect substitutes

forall contingencies