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?
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