advanced microeconomics - unica.it

Advanced Microeconomics

Ivan Etzo

University of Cagliari

[email protected] in Scienze Economiche e Aziendali, XXXIV ciclo

Ivan Etzo (UNICA) Lecture 2: Profit Maximization 1 / 35

Overview

1 The model of profit maximization

2 Short-run Profit-Maximization

3 Long-run Profit-Maximization

4 Profit maximization and Returns to Scale

5 The Weak Axiom of Profit Maximization (WAPM)


The model of profit maximization

Aim: to describe how the firm chooses the amount of output toproduce and the production plan to employ.

We need to make some assumptions regarding:1 The firm’s objective2 The output market3 The inputs market


Economic profit

A firm uses inputs x = 1, 2, · · · ,m to make products i = 1, 2, · · · , n.

Output levels are y1, · · · yn.

Input levels are x1, · · · xm.

Product prices are p1, · · · pn.

Input prices are w1, · · · wm.

We will study the profit-maximization problem of a firm that facescompetitive markets for both factors of production and output.

The competitive firm takes all output prices p1, · · · pn and all inputprices w1, · · · wm as given constants.


Economic profit

The economic profit generated by the production plan(x1, · · · xm, y1, · · · yn) is:

π =n∑

i=1

piyi −m∑i=1

wixi

Likewise inputs and output profit is typically a flow; e.g. Euros ofprofit earned per hour/week/month/year.

economic profit requires that all the costs must be valued at theiropportunity costs (⇒ benefits forgone by particular use ofresources).

remember the difference btw accounting profit and economic profit


Short run vs Long run profit maximization

In the short run there is at least one fixed factor, that is a factor ofproduction for which the quantity used is fixed.

In the long run all factors of production are variable. The least profitvalue is zero.

In the short run the firm can make negative profits because isobligated to employ the fixed factors.


Profit Maximization: Short-run

Let’s consider a firm which produces output y using the productionfunction f (x1, x2), where the input 2 is fixed at some level x̄2

The profit-maximization problem facing the firm can be written as

maxx1

pf (x1, x̄2)− w1x1 − w2x̄2

Applying the first-order condition we have

p∂f (x1, x̄2)

∂x1− w1 = 0

or

pMP1(x∗1 , x̄2) = w1

That is, the value of the marginal product of a factor should equal itsprice.


Profit Maximization: Short-run

pMP1(x∗1 , x̄2) = w1

Remember that if the firm employs more(or less) of factor 1 then theoutput change is

δy = MP1δx1

this additional output is worth

pMP1δx1

but it costsw1δx1

thus, as long as pMP1δx1 6= w1δx1 profits are not maximized.


Profit Maximization: Short-run (graphically)

Solving for y from the (short-run) profit function

π = py − w1x1 − w2x̄2

we obtain the equation that describes the isoprofit lines

y =π

p+

w2

px̄2 +

w1

px1

Represents all the combinations of output (y) and input (x1) thatgive a constant level of profit



The slope of the isoprofit line equals the slope of the production function



By employing x∗1 the firms reaches the highest isoprofit line


Short-Run Profit-Maximization: A Cobb-Douglas Example

Suppose the short-run production function is

y = x1/31 x̄2

1/3

The (first order) profit-maximizing condition is

pMP1 = w1

⇒ p1

3x−2/31 x̄

1/32 = w1

we can solve for x∗1 to obtain the factor demand function

x∗1 =

(p

3w1

)3/2

x̄1/22

which is the firm’ short-run demand for input 1 when the level ofinput 2 is fixed at x̄2 units.


Short-Run Profit-Maximization: A Cobb-Douglas Example

By substituting x∗1 in the production function we can derive theoutput supply function

y∗ =

(p

3w1

)1/2

x̄1/22


Short-Run Profit-Maximization: changes of w

What happens to the optimal choice of x1 when w1 varies?

We can analyze the comparative statics of firm behavior.

Let’s consider y = f (x), where x = f (w , p)The problem facing the firm is

maxx1

pf (x)− wx

The first and second order condition for maximization are:

pf ′(x(w , p))− w ≡ 0

pf ′′(x(w , p)) ≤ 0


Short-Run Profit-Maximization: changes of w

FOC:pf ′(x(w , p))− w ≡ 0

SOC:pf ′′(x(w , p)) ≤ 0

the FOC is an identiy, it must be satisfied for all values of w and p.Thus, we can differentiate the FOC with respect to w (or p)

pf ′′(x(w , p))dx(w , p)

dw− 1 ≡ 0

dx(w , p)

dw≡ 1

pf ′′(x(p,w))

From the SOC we know that pf ′′(x(w , p)) ≤ 0, because p > 0,therefore the factor demand curve slopes downward

dx(w , p)

dw< 0


Short-Run Profit-Maximization: changes of p

Similarly, differentiating the FOC with respect to p

f ′(x(w , p)) + pf ′′(x(w , p))dx(w , p)

dp≡ 0

dx(w , p)

dp= − f ′(x(w , p))

pf ′′(x(w , p))

where f ′(x(w , p)) is MPx , thus it must be positive and pf ′′(x(w , p))must be negative (see the SOC)Accordingly

dx(w , p)

dp> 0

Thus, the supply function must slope upwards (remember that x̄2 isfixed so when x1 decreases y must decrease as well.


Short-Run Profit-Maximization: changes of w (graphically)

Figure : w ′ > w


Short-Run Profit-Maximization: changes of p (graphically)

Figure : p′ < p


Profit Maximization: Long-run

Let’s consider a firm which produces output y using the productionfunction f (x1, x2), where now the input 2 is also variableThe profit-maximization problem facing the firm can be written as

maxx1,x2

pf (x1, x2)− w1x1 − w2x2

The two FOC are:

p∂f (x1, x2)

∂x1− w1 = 0

p∂f (x1, x2)

∂x2− w2 = 0

or

pMP1(x∗1 , x∗2 ) = w1

pMP2(x∗1 , x∗2 ) = w2

That is, in the long-run the value of the marginal product of eachfactor should equal its price.Ivan Etzo (UNICA) Lecture 2: Profit Maximization 19 / 35

The inverse factor demand curve

We have derived the factor demand curve of factor 1 by solving for x∗1the first order profit-maximizing condition

Similarly we can derived the inverse factor demand curve of factor 1by solving for w∗1 the first order profit-maximizing condition.

The inverse factor demand curve measures what the factor pricesmust be for some given quantity of inputs to be demanded.


The inverse factor demand curve


Profit maximization and Returns to Scale

Suppose a firm’s production function exhibits constant RTS

the long-run profit maximizing output is y∗ = f (x∗1 , x∗2 )

and the profits are π∗ = py∗ − w1x∗1 − w2x∗2Let’s say that the firm doubles both inputs levels, what happen to theoutput? And to the profits?

Output should double (CRTS) and profits should double as well(Competitive markets assumption: all prices are constant) →Contradiction!!

Explanation: the only reasonable level of profits in the long-run iszero.

Is this conclusion reasonable?


Profit maximization and Returns to Scale

Let’s assume that the firm expands indefinitely.1 The firm could get too large → coordination problems → decreasing

returns to scale2 The firm increases its market share noticeably → the assumption of

perfect competition no longer holds3 Other firms can also increase output → industry supply increases →

output price decreases and profits decrease


Comparative statics using algebraThe Weak Axiom of Profit Maximization (WAPM)

Consider the choices of a profit maximizing firm in two differentperiods, t and s

(y t , x t1, x

t2) with (pt ,w t

1 ,wt2)

(y s , x s1 , x

s2) with (ps ,w s

1 ,ws2 )

If the technology hasn’t changed between t and s, then we must havethat

a) pty t − w t1x t

1 − w t2x t

2 ≥ pty s − w t1x s

1 − w t2x s

2

andb) psy s − w s

1 x s1 − w s

2 x s2 ≥ psy t − w s

1 x t1 − w s

2 x t2

The satisfaction of these inequalities is called The Weak Axiom ofProfit Maximization (WAPM).



Using simple algebra, let’s transpose the two sides of equation b)

−psy t + w s1 x t

1 + w s2 x t

2 ≥ −psy s + w s1 x s

1 + w s2 x s

2

then add equation a)

a) pty t − w t1x t

1 − w t2x t

2 ≥ pty s − w t1x s

1 − w t2x s

2

to gety t(pt − ps)− x t

1(w t1 − w s

1 )− x t2(w t

2 − w s2 )

≥ y s(pt − ps)− x s1(w t

1 − w s1 )− x s

2(w t2 − w s

2 )

rearrange to yield

(y t − y s)(pt − ps)− (x t1 − x s

1)(w t1 − w s

1 )− (x t2 − x s

2(w t2 − w s

2 ) ≥ 0

∆y∆p − ∆x1∆w1 − ∆x2∆w2 ≥ 0



∆y∆p−∆x1∆w1 −∆x2∆w2 ≥ 0

This equation contains all the results about profit-maximizing choicesE.g. let’s consider ∆p = 0 and ∆w2 = 0it remains −∆x1∆w1 ≥ 0 or

∆x1∆w1 ≤ 0

Which states that if w1 ↑ then x1 ↓ in order for the profit to bemaximized, or in other words that the factor demand curve slopesdownward.Similarly, if ∆w1 = 0 and ∆w2 = 0, then

∆y∆p ≥ 0

thus, the profit-maximizing supply curve of a competitive firm musthave a positive (or zero) slope.Ivan Etzo (UNICA) Lecture 2: Profit Maximization 26 / 35

The Weak Axiom of Profit Maximization (WAPM)Revealed profitability (graphically)

It is possible to estimate the technology by observing a firm’s choices.

Usually also ps , πs , and w s are known, therefore



It is possible to estimate the technology by observing a firm’s choices.



The point (y s′ , x s′) would yield a higher profit, but it is not chosen by thefirm. Why?



The point (y s′ , x s′) is not feasible.



We can observe a different choice in period s.



The related isoprofit line suggests another ”feasible” set of productionpossibilities.



We can combine the two observations.



The more observations we get the finer is the estimate of the existingtechnology.


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