UNIT II:FIRMS & MARKETS

UNIT II: FIRMS & MARKETS

• Theory of the Firm• Profit Maximization• Perfect Competition• Review• 7/14 MIDTERM

6/28

Theory of the Firm

Today we will build a model of the firm, based on the model of the consumer we developed in UNIT I. Where consumers attempt to maximize utility, firms attempt to maximize profit.

We saw how changes in prices affect consumers’ optimal decisions and derived a demand function: P = f(Qd).

Now we will see how changes in prices affect firms’ profit maximizing decisions and derive a supply curve: P = f(Qs).

Later, we will put supply and demand together, and begin our analysis of markets and market structures.

Theory of the Firm

The Good News!

In moving from the consumer to the firm, we replace the troublesome notion of utility with something nice and hard-edged: profit. Where utility is subjective and thus hard to measure, now we’ll be talking about simple, measurable quantities, physical units of inputs (tons of steel, hours of labor) and outputs, and an account for everything in dollars and cents (“the bottom line”).

Theory of the Firm

• The Technology of Production • Short-run v. Long-run• Isoquants• Returns to Scale• Cost Curves• Cost Minimization• Profit Maximization

Theory of the Firm

First, we need to write down our model:

Profit ( = Total Revenue(TR) – Total Cost(TC)

TR(Q) = PQ TC(Q) = rK + wL

P Price L LaborQ Quantity K Capital

w Wage Rate Q = f(K,L) r Rate on Capital

The firm wants to

maximize this

difference

Theory of the Firm



TR = PQ TC(Q) = rK + wL



Economic costs include opportunity costs

Economic < Normal (accounting)

If a firm earns positive profits, all factors are earning more than they could in alternative uses

Theory of the Firm






INPUTS

The production function describes a relationship between the quantity of physical inputs (K, L) and quantity of outputs (Q).

Theory of the Firm






OUTPUT

The production function describes a relationship between the quantity of physical inputs (K, L) and quantity of output (Q).

The Technology of Production

Technology: a list of all possible production plans, i.e., all the ways to transform inputs into outputs.

The production function: describes a relationship between inputs and the maximum quantity of output (all inputs are used technologically efficiently).

Technological constraints: Nature (& science) imposes certain physical constraints on a firm: only so much output can be created out of so much input(s). Thus the firm must limit its choice to technologically feasible production plans.

The Technology of Production

The Production Function

Q = F(K, L), For a given technologyQ OutputL LaborK Capital

Cobb-Douglas Production Function Q = AKL

where A is a constant (scale).

Production in the Short-Run

We define the short-run as the period in which at least one factor of production is fixed:

L

QTP: Q = f(L)

Total Product Function

The maximum quantity of output the can be

produced for a given quantity of input,

holding other factors constant.


We define the short-run as the period in which at least one factor of production is fixed:

L

Q

Law of Diminishing Marginal Returns

As firm adds more of a variable factor (holding others

constant), the increment to output

will eventually decrease.

Total Product FunctionTP: Q = f(L)


L

Q

Q

Average Product

AP = Q/L Output per unit labor

(input)

A ray from the origin to any point on the TP

curve is AP

AP

TP

L


L

Q

Q

Average Product

AP = Q/L Output per unit labor

(input)

AP is greatest at Point A

AP

TPA

L


L

Q

Q

Marginal Product

MPL=dQ/dL Increase in output from a

unit increase in labor (input)

The slope of TP curve is MP

MP is greatest at point B MP is negative at point C

TP

MP

A

B

C

L


L

Q

Q

Marginal Product

MPL=dQ/dL Increase in output from a

unit increase in labor (input)

The slope of TP curve is MP

MP = AP at AP max

TP

AP

MP

A

B

L

C


L

Q

Q

Average and Marginal Products

In general, average will equal marginal,

where average is greatest, e.g., the

height of people in the room.

TP

AP

MP

A

B

L

C

Production in the Long-Run

In the long run, all factors of production are variable.

K,L

Q Total Product FunctionTP: Q = f(K,L)

Again, we have 3

variables…



L


K



K,L


The production function determines

the maximum possible output for a given combination of

inputs (a boundary condition).

Technological efficiency.



K,L


One particularly convenient form is the

Cobb-Douglas production function

Q = AKL


Isoquant: all the possible combinations of inputs that are just sufficient to produce a given level of output.

L

K Well-behaved isoquants are monotonic and

convex

Q = 20

Q = 10



L

K Well-behaved isoquants are monotonic: More

of any input produces at least as much output.

MPL, MPK > 0.Q = 20

Q = 10



aL bL L

K

aK

bK

Well-behaved isoquants are convex: Combinations

of production plans produce at least as much

as either alone.

Q = 20

Q = 10

10 units of output can be produced

using aL units of L and aK units of K

or bL units of L and bK units of K

or (aL+bL)/2 of L and (aK+bK)/2 of K

or any linear combination of plans a and b



L

K

Q = 20

Q = 10

Marginal Rate of Technical Substitution (MRTS):

The rate at which one input can be exchanged for

another while keeping output constant.



L

K

Q = 20

Q = 10

MRTSKL = - MPL/MPK

Along an isoquant dQ = 0

dK*MPK +dL*MPL = 0 dK/dL = - MPL/MPK

= slope

MRTS: The rate at which one input can be exchanged for another while keeping output constant. This is the slope of the isoquant.

Returns to Scale

Constant returns: doubling all inputs doubles output. The firm can replicate production unit (e.g., build another plant).

L 2L L

K

2K

K

Q2 = 2Q1

Q = AKL

+ = 1Q2

Q1

Returns to Scale

Increasing returns: doubling all inputs more than doubles output. (aka “Economies of scale”).

L 2L L

K

2K

K

Q2 > 2Q1

Q = AKL

+ > 1Q2

Q1

Returns to Scale

Decreasing returns: doubling all inputs less than doubles output. Crowding (assume homogeneous factor quality).

L 2L L

K

2K

K

Q2 < 2Q1

Q = AKL

+ < 1

Q2

Q1

Returns to Scale

Decreasing returns: doubling all inputs less than doubles output. Crowding (assume homogeneous factor quality).

L 2L L

K

2K

K

Q2 < 2Q1

Q2

Q1

Firm should always be able to get at least

constant returns by dividing production

Returns to Scale

Do any of these cases violate with the law of diminishing marginal productivity?

Cost Curves

We can combine technological info about production possibilities with price info to characterize the firm’s cost structure.

In the short-run, economic efficiency = technological efficiency, b/c to produce a certain level of output, Q, given K, there is a unique level of L required. This is the cheapest way to produce Q.

In the long-run, all factors are variable, and the firm’s problem is to choose its optimal factor proportion.

L

Q

A firm will never choose to produce where MPL is negative.

Cost Curves


SR Total Product Function

TP: Q = f(L)

Lo L

Q

Qo

Cost Curves



TP: Q = f(L,K)

Qo Q

$

wLo

SR Cost Function

TVC = wL(Q)

Lo L

Q

Qo

Cost Curves



TP: Q = f(L,K)

Qo Q

$ SR Cost Functions TC = TFC + TVC

TVC = wL(Q)

TFC = rK

The shape of cost curves come from technology

Cost Curves

Q

$

$

Average Cost

AC = TC/Q Cost per unit output

A ray from the origin to any point on the

TC curve is AC

AC min at point AAC

A

Q

TC

Q

$

$

Marginal Cost

MC = dTC/dQ = dTVC/dQ

(b/c only VC can change)

= wdL/dQ= w/MPL

MC min at point B

TC

MC

B

Q

AC

Cost Curves

Cost Curves

Q

$

$

Marginal Cost

MC = dTC/dQ = dTVC/dQ

(b/c only VC can change)

= wdL/dQ= w/MPL

AC = MC at ACmin

TC

MC

B

Q

A

AC

Cost Curves

Q

$

$

Average Fixed Cost

AFC = TFC/Q

Average Fixed Cost

AVC = TVC/Q

TC

MC

B

Q

A

AC

TFC

AFC

TVC

AVC

Cost Curves

Deriving long-run cost curves:

Q1 Q2 Q3 Q

$

3 short-run total cost curves associated

w/different levels of K.

The cheapest way to produce Q1 is using

SRTC1 …

SRTC(K3)

SRTC(K2)

SRTC(K1)

Cost Curves


Q1 Q2 Q3 Q

$



If K is continuously variable, the lower

envelope of all SRTCs

shows the lowest cost for each quantity

of output.

This is the long-run total cost curve

SRTC(K3)

SRTC(K2)

SRTC(K1)

Cost Curves


Q1 Q2 Q3 Q

$




envelope of all SRTCs

shows the lowest cost for each quantity

of output.

This is the long-run total cost curve

LRTC

Cost Curves


Q1 Q2 Q3 Q

$

We can’t say much about the shape of the

LRTC curve. (Shown is a cubic

function, giving us s-shaped curves).

Cobb-Douglas w/constant returns

give us a linear (upward-sloping)

curve.

LRTC

Cost Curves


Q1 Q2 Q3 Q

$

3 short-run average cost curves associated



envelop of all SRACs is the

long-run average cost curve.

LRACSRAC1

SRAC2

SRAC3

Cost Curves


Q1 Q2 Q3 Q

$

SRMC1

SRMC2 SRMC3 LRMC

3 short-run marginalcost curves associated


The long-run marginal cost curve is the locus of points

along srmcs associated

w/difference levels of output.

Cost Minimization

Consider a firm that produces output according to the following production function.

Q = 4K½L½

Assume that w = $18 and r = $36, and the firm currently has 16 units of capital.

How much will it cost this firm to produce 10 units of output in the short-run?

Cost Minimization

Consider a firm that produces output according to the following production function:

Q = 4K½L½


With K = 16, the firm’s short-run production function is:

Q = 16L½

For Q = 10, 10 = 16L½ => L = 100/256 = .39

Cost Minimization


Q = 4K½L½


To produce 10 units: 10 = 16L½ => L = 100/256 = .39

With K = 16, the firm’s total cost of production is:

TC = rK + wL = 36(16) + 18(.39) = $583.03

Cost Minimization


Q = 4K½L½


More generally: Q = 16L½ => L = (Q/16)2

With K = 16, the firm’s short-run total cost function is:

TC = rK + wL = 36(16) + 18(Q/16)2 = $576 + 18(Q/16)2

Cost Minimization in the Short-Run

How much will it cost this firm to produce Q units of output in the short-run?

Q

$ Total Cost FunctionTCsr = 576 + 18(Q/16)2

Fixed Costs = rK = 36(16) = $576

Cost Minimization in the Short-Run

How much will it cost this firm to produce 10 units of output in the short-run?

L = 0.39 L

K

K = 16

In the short-run, if the firm wants to produce more (or less) than 10

units, it would move along it’s short-run output-expansion

path, at K =16.

Q = 10

Cost Minimization in the Long-Run

How much will it cost this firm to produce 10 units of output in the long-run?

L

K

In the long-run, all factors are variable. Firms combine factors of production in a manner analogous to the way consumers choose a consumption bundle.

Q = 10



L

K

An isoquant is all the technologically efficient combinations of K ,L to produce a certain output, Q.

Q = 10

Isoquant

Slope = - MRTS



L

K

If we think of the all the combinations of

K&L that cost a certain amount (TC), we have an isocost

line:

K = TC/r – (w/r)L

Recall: TC = wL + rKQ = 10

Isocost lines Slope = - w/r



L* L

K

K*

Tangency between the isoquant and an isocost curve shows

the economically efficient combination

K*, L*. Hence, the condition

for optimal factor proportion is:

MRTS = w/rQ = 10


Isoquant

Slope = - MRTS



L* L

K

K*

The condition for optimal factor proportion is:

MRTS = w/r.

This is LR condition! Why? Because some

factors (K) are fixed in the SR.

Q = 10


Isoquant

Slope = - MRTS



L* L

K

K*

Another way to think about this: TC is a

projection of the firm’s long-run

output expansion path: the locus of

optimal factor bundles (K,L) for

different levels of Q.

(Constant return depicted)

Q = 10


To find total cost in terms of Q, we use the cost minimization condition and the production function to find substitute expressions in terms of Q for K and L. Cost minimization requires that the firm produce using a combination of inputs for which the ratios of the marginal products, or the marginal rate of technical substitution, equals the ratio of the input prices:

MRTS = w/r



Q = 4K1/2L1/2

w = 18; r = 36

MRTS = MPL/MPK

MPL = 2K1/2L-1/2

MPK = 2K-1/2L1/2 MRTS = K/L.

= w/r = 18/36 = L = 2K.


The firm’s optimal factor proportion

(given technology and factor prices).


Q = 4K1/2L1/2

L = 2K => Q = 4K1/2(2K)1/2

= 4(2)1/2K

Q = 4(2)1/2K => K = Q/[4(2)1/2]Q = 5.66K => K = Q/5.66For Q =10 => K = 1.77; L = 3.54

TC = wL + rKTC(Q=10) = 36(1.77)+18(3.54) = $127.28


To produce 10 units of output, we solve for K and L in terms of Q and substitute in the

total cost function.Producing 10 units

costs $127.28. At this point, the firm is using 1.77 units of

capital and 3.54 units of labor.


Q = 4K1/2L1/2

L = 2K => Q = 4K1/2(2K)1/2

= 4(2)1/2K

Q = 4(2)1/2K => K = Q/[4(2)1/2]Q = 5.66K => K = Q/5.66For Q =10 => K = 1.77; L = 3.54

TC = wL + rKTC(Q=10) = 36(1.77)+18(3.54) = $127.28


Comparing this to the short-run, the cost of production is lower in the long-run, because the firm is now able to adjust K as well as L to minimize the cost of the production of a given amount of output, which is more efficient.

How much will it cost this firm to produce Q units of output in the long-run?

TC = 18L + 36K = 9Q/(2)1/2 + 9Q/(2)1/2 = 18/(2)1/2(Q)= 12.73Q

MC = 12.73 = AC


We can also solve for the firm’s long run

total cost function for any level of output.

.


Graphically: Q = 4K1/2L1/2

w = 18; r = 36

Q

$ TCsr = 576 + 18(Q/16)2

Fixed Costs = rK = $576

TClr = 12.73Q

MC = 12.73

Profit Maximization


$ To maximize profits, look for Q where distance between TC and TR

is greatest.

This will be where they have the same

slope.. slope.

TC

TR = PQ

max

Q1 Q2 Q3 Q* Q

Profit Maximization


$ Marginal Analysis:

If TC is rising faster than TR, reduce Q.

If TR is rising faster than TC, increase Q.

TC

TR = PQ

max

Q1 Q2 Q3 Q* Q

Profit Maximization


$ Marginal Analysis:

Recall: slope TR = MRslope TC = MC

Hence, to maximize profits:

MR = MC

TC

TR = PQ

max

Q1 Q2 Q3 Q* Q

Profit Maximization

Demand for the firm’s output is given by Q = 100 – 2P. Find the firm’s profit maximizing level of output.

Q = 4K1/2L1/2

w = 18; r = 36Q = 100 – 2P => P = 50 – 1/2Q

TR = PQ = (50 – 1/2Q)Q= 50Q – 1/2 Q2

MR = 50 – Q = MC = 12.73

=> Q* = 37.27; P* = 31.37

Profit Maximization

We solved the firm’s optimization problem focusing on the profit output level, Q*, but it is important to emphasize that the optimization principle also tell us about input choices.

When the firm chooses an output level Q* that maximizes for given factor prices (w, r), the firm has simultaneously solved for L* and K*.

To produce Q* = 37.23 (given the production function, Q = 4K1/2L1/2, and optimal factor proportion, L = 2K), we find: L* = 6.58; K* = 3.29.

Finally, = TR–TC = PQ–12.73Q = (31.37–12.73)37.27 = $695.

Next Time

6/30 Profit Maximization

Pindyck & Rubenfeld, Ch. 7.

or Besanko, Ch. 8

Varian Ch. (parts of) Ch 19, 22, 23.

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UNIT II:FIRMS & MARKETS

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