Transcript
Advanced Microeconomics
Ivan Etzo
University of Cagliari
ietzo@unica.itDottorato 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)
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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
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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.
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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
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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.
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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.
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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.
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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
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Profit Maximization: Short-run (graphically)
The slope of the isoprofit line equals the slope of the production function
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Profit Maximization: Short-run (graphically)
By employing x∗1 the firms reaches the highest isoprofit line
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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.
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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
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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
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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
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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.
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Short-Run Profit-Maximization: changes of w (graphically)
Figure : w ′ > w
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Short-Run Profit-Maximization: changes of p (graphically)
Figure : p′ < p
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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.
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The inverse factor demand curve
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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?
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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
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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).
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Comparative statics using algebraThe Weak Axiom of Profit 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
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Comparative statics using algebraThe Weak Axiom of Profit Maximization (WAPM)
∆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
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The Weak Axiom of Profit Maximization (WAPM)Revealed profitability (graphically)
It is possible to estimate the technology by observing a firm’s choices.
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The Weak Axiom of Profit Maximization (WAPM)Revealed profitability (graphically)
The point (y s′ , x s′) would yield a higher profit, but it is not chosen by thefirm. Why?
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The Weak Axiom of Profit Maximization (WAPM)Revealed profitability (graphically)
The point (y s′ , x s′) is not feasible.
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The Weak Axiom of Profit Maximization (WAPM)Revealed profitability (graphically)
We can observe a different choice in period s.
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The Weak Axiom of Profit Maximization (WAPM)Revealed profitability (graphically)
The related isoprofit line suggests another ”feasible” set of productionpossibilities.
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The Weak Axiom of Profit Maximization (WAPM)Revealed profitability (graphically)
We can combine the two observations.
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The Weak Axiom of Profit Maximization (WAPM)Revealed profitability (graphically)
The more observations we get the finer is the estimate of the existingtechnology.
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The Weak Axiom of Profit Maximization (WAPM)Revealed profitability (graphically)
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