Microeconomics Producer Theory

Ornella Tarola

Producer’s final objective:

Maximizing profits

1. Technology

However, producer faces 2 constraints:

1. Technical constraint

2. Economic constraint

vs.

Technically

possible

Economically

viable

1. Technology

1. Technology

Inputs: •capital

•labour

•land

•administrative

organisation

•…

Technology Output(s):

•goods

•services

• A technology is a process by which inputs are

converted to an output.

Example: labor, computer, projector, electricity, software

are combined to produce this lecture.

• Usually several technologies will produce the same

product - a blackboard and chalk can be used instead

of a computer and a projector.

• Which technology is “best”?

• How do we compare technologies?

1. Technology

The technology’s production function states the

maximum amount of output possible from an input

bundle:

Y=f(x1,x2,…xn)

1. Technology

outp

ut

Pro

duction f

unction

→“t

echnolo

gy”

inputs

x’ input level

output level

y’

y”

technology set

Technically inef-

ficient plans

technically

efficient plan

y” = f(x’) is an output level that is feasible from x’ input units. y’ = f(x’) is the maximal output level obtainable from x’ input units.

Production function: y=f(x)

The technology’s production function

states the maximum amount of output

possible from an input bundle.

1. Technology

1. Technology

Marginal product

The marginal product of an input is the additional quantity of output that

is produced by using one more unit of that input.

The marginal product is the rate-of-change of the output level as the

level of input changes, holding all other input levels fixed

Example: marginal product of labour:

LyLyMPL 0L if

Diminishing Returns to an Input

There are diminishing returns to an input when an increase in the

quantity of that input, holding the levels of all other inputs fixed, leads

to a decline in the marginal product of that input.

1. Technology

1. Technology In economics terms:

The source of the diminishing

returns of programmers lies in

the nature of the production

function for a programming

project: Each programmer must

coordinate his or her work with

that of all the other programmers

on the project, leading to each

person spending more and more

time communicating with others

as the number of programmers

increases.

1. Technology

Marginal products vs. Returns-to-scale

→Marginal products describe the change in output level as a

single input level changes.

→Returns-to-scale describes how the output level changes

as all input levels change in direct proportion (e.g. all input

levels doubled, or halved).

1. Technology Illustration:

Labour product

+ =

Case 1: Increasing labour product

1. Technology

+ =

Case 2: Decreasing labour product

Illustration:

Labour product

1. Technology

Illustration:

Labour product

+ =

Case 3: Constant labour product

1. Technology

Illustration:

Returns-to-scale

+ =

Case 1: Increasing returns-to-scale

1. Technology

Illustration:

Returns-to-scale

+ =

Case 2: Decreasing returns-to-scale

1. Technology

Illustration:

Returns-to-scale

+ =

Case 3: Constant returns-to-scale

1. Technology

If there is only 1 factor of production, marginal product and returns to

scale concepts are equivalent.

=

→ constant returns to scale

→ constant marginal product

Back to case of more than 1 input

At what rate can a firm substitute one input for another without

changing its output level?

1. Technology

Definition:

The slope of the isoquant is the rate at which input 2 must be

given up as input 1’s level is increased so as not to change the

output level.

The slope of an isoquant is its technical rate-of-substitution.

→ technical rate-of-substitution

1. Technology x2

x1

Y is constant

along the

isoquant

1. Technology

2

2

1

1

dxx

ydx

x

ydy

Tota

l outp

ut

variation

Tota

l in

tput

1

variation

Tota

l in

tput

2

variation

Outp

ut

va

ria

tion f

ollo

win

g

unit input

1 v

ariation

Outp

ut

va

ria

tion f

ollo

win

g

unit input

2 v

ariation

Analytic expression of the technical rate of substitution

2

2

1

1

0 dxx

ydx

x

y

2

1

1

2

/

/

xy

xy

dx

dx

Slope of the isoquant = technical rate of substitution

1. Technology

Moving along the isoquant: dy=0

Utilità marginale e curve di indifferenza

• Muovendosi lungo una curva di indifferenza … l'utilità addizionale che si riceve dal consumo di un'unità in più di C deve compensare la perdita di utilità derivante dalla diminuzione nel consumo di V:

• 0=U’c(Delta C)+U’v(Delta V)

Muovendosi lungo la curva l'utilità non cambia. La compensazione tra i beni lascia i consumatori indifferenti

Utilità marginale e scelta del consumatore

• Riordinando i termini:

VC UUCV '/'/

VC/U USMS '' Ovvero …

Ma abbiamo visto anche che:

VC/P PSMS Questo implica che:

Rapp ut mar=rapp. prezzi

2. Firm’s profit

So far:

technical constraint: given present technology, what

can the firm produce?

Now:

economic constraint: in the set of technologically

feasible production possibilities, which one is

economically efficient?

Perfect competition:

→competitive firm takes all output prices and

all input prices as given. Thus profit can be

written as:

2. Firm’s profit

_ =

Economic Profit

π = p × y (w × L+r × K) _

2. Firm’s profit

π = p × y - (w × L+r × K)

given, exogenous

(i.e. we are in a perfect competition setting)

→so firm will have to decide upon: y, L, K

→ furthermore, we know: y=f(L,K) [production function]

→ thus if firm chooses L & K, then y is unequivocally determined

i.e. if firm chooses inputs, output is determined simultaneously

2. Firm’s profit

What if there is more than 1 input in production?

When will the firm’s profit be maximized?

→ find the optimal combination of inputs that

maximize profits

→ for every isoquant (i.e. input combination

maintaining output level constant) find the

profit maximizing input combination

2. Firm’s profit

Given output y is constant along an isoquant, revenue

(i.e. p×y) will be constant as well!

Thus, maximizing profits for a given isoquant amounts

to minimize costs!

π = p × y - (w × L+r × K)

constant along

isoquant given by output market;

constant for individual firm

→ so maximizing π = minimizing costs (along isoquant)

total costs

2. Firm’s profit

• A firm is a cost-minimizer if it produces any given

output level y 0 at smallest possible total cost.

• c(y) denotes the firm’s smallest possible total cost for

producing y units of output.

• c(y) is the firm’s total cost function.

2. Firm’s profit

Firm’s profit maximization can be divided into 2 steps:

1. Minimize total cost of firm, for every possible production level

2. Maximize profits, given cost function c(y)

MinL,K (w × L+r × K) such that y=f(K,L) c(y)

total costs production

function

cost

function

Maxy π = p × y – C such that C=c(y)

profit function cost

function

2. Firm’s profit

C’ w×L + r×K

C” w×L + r×K

C’ < C”

L

K

Slopes = -w/r.

Graphically:

Iso-cost Lines

cost decrease

2. Firm’s profit

L

K

production function

f(K,L) y’

total costs

w×L + r×K C

All input bundles

yielding y’ units

of output. Which is

the cheapest?

L*

K*

→ the cost-minimizing input bundle is (K*,L*)

→ slope of isocost = slope of isoquant

K

LK,y

LLK,y

capital ofproduct marginal

labour ofproduct marginal

r

w

2. Firm’s profit

Firm’s cost function: c(y)

L

K

K*

L* L** L***

K**

K***

C’=c(y’)

C’’=c(y’’)

C’’’=c(y’’’)

2. Firm’s profit Returns-to-scale and Costs

If a firm’s technology exhibits decreasing returns-to-scale then doubling its output level requires

more than doubling all input levels.

• →total production cost more than doubles.

If a firm’s technology exhibits increasing returns-to-scale then doubling its output level requires

less than doubling all input levels.

• →total production cost less than doubles.

If a firm’s technology exhibits constant returns-to-scale then doubling its output level requires just

the doubling all input levels.

• →total production cost just doubles.

total costs

y

constant returns-to-scale

increasing returns-to-scale

decreasing returns-to-scale

2. Firm’s profit

Now that we have determined the firm’s cost function c(y), the profit

maximization problem becomes:

Maxy π = p×y – C such that C=c(y)

Maxy π = p×y – c(y)

total costs

y

p×y c(y)

2. Firm’s profit

…analytically:

MCcost

marginalprice

output

0

..

y

ycp

y

ycypMax

ycCtsCypMax

y

y

In words: as long as the revenue of 1 supplementary output (p) unit

is larger than its cost (MC), the firm will produce!

3. Supply of the firm

From profits to firm supply!

Firm will produce the amount of output y which maximizes its

profits!

The profit-maximizing output y* is reached when:

marginal cost (MC) = output price (p)

Thus, as p is given in competitive market, MC is crucial in

producer theory.

MC is the supply curve, when firm is in perfect

competition!!!

y = f(L)

y

3. Supply of the firm

L MC

L

MC

p

firm supply

Firm’s supply curve

40

Microeconomics Producer Theory

Exercises

41

APPLICATION 1

42

Short-Run & Long-Run Total Costs

Exercise:

Suppose a producer using 2 inputs, is constraint by his factory size, i.e. the

factory cannot be extended or reduced in the short run.

Will the short run costs of the producer be larger or smaller then the long run

costs?

Solution:

Suppose the production function is y=f(K,L), with K is fixed in the short run

This means that isoquants reduce to a dot, i.e. there is only one possible

combination of production factors for each given production level.

So, if the producer cannot choose anymore the least expensive input

combination for each production level in the short run, it is intuitively clear that

short run cost are at best the level of long run costs, but usually larger.

Graphically:

43

Short-Run & Long-Run Total Costs

K

L

Short-run output

expansion path

Long-run costs are:

c(y’)=w×L’ + r×K’

c(y’’)=w×L’’ + r×K’’

c(y’’’)=w×L’’’ + r×K’’’

Short-run costs are:

cs(y’)=w×L1 + r×K’’

cs(y’’)=w×L’’ + r×K’’

cs(y’’’)=w×L2 + r×K’’

y’’’

y’’

y’

K’ K’’ K’’’

Fixed level of capital in the short run!

L’

L’’’

L’’

Long-run output

expansion path

If y=y’, what will be the optimal input

combination in the short-run?