Technology and profit maximization
Varian: Intermediate Microeconomics, 8e, chapters 18 and 19
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Theory of firm and market structure
Firms maximize profit.
The results of the interaction of
profit-maximizing firms depends on:
• market structure
• properties of the product
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Theory of firm and market structure
Firms maximize profit.
The results of the interaction of
profit-maximizing firms depends on:
• market structure
• properties of the product
The following four topics:
• theory of firm
• perfect competition
• monopoly and monopoly behavior
• oligopoly
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In this lecture you will learn
• what technology and a production function are
• what the difference between the short and long run is
• what it means that a firm maximizes profit and minimizes costs
• why fishermen in India need mobile phones
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Production and production plan
Production is the transforming inputs into outputs.
Inputs to production = factors of production (FP):
• labor
• land (resources)
• capital
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Production and production plan
Production is the transforming inputs into outputs.
Inputs to production = factors of production (FP):
• labor
• land (resources)
• capital
Production plan is a combination of inputs and outputs.
Technological constraints – only certain combinations of inputs
are feasible ways to produce a given amount of output.
There is usually a number of feasible production plans.
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Describing technological constraints
Production set (= technology) – set of all feasible production plans.
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Describing technological constraints
Production set (= technology) – set of all feasible production plans.
Production function – maximum possible output that you can get from a
given amount of input
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Describing technological constraints
Production set (= technology) – set of all feasible production plans.
Production function – maximum possible output that you can get from a
given amount of input
Izoquant – set of all possible combinations of inputs 1 and 2 that are just
sufficient to produce a given amount of output.
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Example – utility functions vs. production functions
Utility function – two consumers:
• consumer 1 – U1(x1, x2) = x1 + x2
• consumer 2 – U2(x1, x2) = (x1 + x2)2
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Example – utility functions vs. production functions
Utility function – two consumers:
• consumer 1 – U1(x1, x2) = x1 + x2
• consumer 2 – U2(x1, x2) = (x1 + x2)2
Both consumers have identical ICs and preferences =⇒ If they have the
same BL, they choose the same consumption bundle.
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Example – utility functions vs. production functions
Utility function – two consumers:
• consumer 1 – U1(x1, x2) = x1 + x2
• consumer 2 – U2(x1, x2) = (x1 + x2)2
Both consumers have identical ICs and preferences =⇒ If they have the
same BL, they choose the same consumption bundle.
Production function – two firms:
• firm 1 – f1(x1, x2) = x1 + x2
• firm 2 – f2(x1, x2) = (x1 + x2)2
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Example – utility functions vs. production functions
Utility function – two consumers:
• consumer 1 – U1(x1, x2) = x1 + x2
• consumer 2 – U2(x1, x2) = (x1 + x2)2
Both consumers have identical ICs and preferences =⇒ If they have the
same BL, they choose the same consumption bundle.
Production function – two firms:
• firm 1 – f1(x1, x2) = x1 + x2
• firm 2 – f2(x1, x2) = (x1 + x2)2
Both firms have identical isoquants but different technologies.
At the same inputs they produce different outputs (if x1 + x2 = 1).
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Examples of technology – perfect substitutes
Passport photographs – a photographer or a photo booth.
Production function – f (x1, x2) = x1 + x2
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Examples of technology – fixed proportions
Ice cream cart – one sales man needs one cart.
Production function – f (x1, x2) = min{x1, x2}
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Examples of technology – Cobb-Douglas
Cobb-Douglas production function – f (x1, x2) = Axc
1 xd
2 :
• A measures the scale of production
• c and d measure how the output responds to changes in input
Cobb-Douglas utility function could be monotonically transformed
– so we usually had A = 1 and c + d = 1.
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Properties of technology: monotonicity and convexity
Monotonicity – if you increase at least one of the inputs, you should
produce at least as much as before (nonincreasing isoquant).
Logics of monotonicity: free disposal = if the firm can costlessly dispose of
any inputs, having extra inputs around can’t hurt it.
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Properties of technology: monotonicity and convexity
Monotonicity – if you increase at least one of the inputs, you should
produce at least as much as before (nonincreasing isoquant).
Logics of monotonicity: free disposal = if the firm can costlessly dispose of
any inputs, having extra inputs around can’t hurt it.
Convexity – if you have two ways to produce y units of output, (x1, x2)
and (z1, z2), then their weighted average will produce
at least y units of output.
Logics of convexity: If there are more ways to produce the same output
and the ways can be combined, we get a convex isoquant.
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Example: Deriving a convex isoquant – a 100 meter ditch
1 meter of ditch can be produced in two ways:
• using a small excavator: (hours of work, value of capital) = (a1, a2)
• using a pickaxe: (hours of work, value of capital) = (b1, b2)
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Marginal product
Marginal product of factor 1 (MP1) – what is the change in total product
if we increase x1 by one unit and x2 remains constant:
MP1(x1, x2) =
∂f (x1, x2)
∂x1
MP is similar to MU, but the value of MP means something.
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Technical rate of substitution (TRS)
TRS = slope of isoquant (similar to MRS = slope of IC)
By how many units we can reduce x2, if x1 increases by 1 unit
and we want to produce the same output y?
Calculation:
For changes in factors ∆x1 and ∆x2 on the same isoquant holds:
∆y = MP1(x1, x2)∆x1 + MP2(x1, x2)∆x2 = 0
By a simple manipulation of the equation, we get
TRS(x1, x2) =
∆x2
∆x1
= −
MP1(x1, x2)
MP2(x1, x2)
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Technical rate of substitution (TRS)
TRS = slope of isoquant (similar to MRS = slope of IC)
By how many units we can reduce x2, if x1 increases by 1 unit
and we want to produce the same output y?
Calculation:
For changes in factors ∆x1 and ∆x2 on the same isoquant holds:
∆y = MP1(x1, x2)∆x1 + MP2(x1, x2)∆x2 = 0
By a simple manipulation of the equation, we get
TRS(x1, x2) =
∆x2
∆x1
= −
MP1(x1, x2)
MP2(x1, x2)
Diminishing TRS = assumption of strict convexity – the absolute value
of TRS decreases as we move along the isoquant to the right.
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Diminishing marginal product (law of diminishing MP)
Why can’t we feed the world’s population using one flower pot?
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Diminishing marginal product (law of diminishing MP)
Why can’t we feed the world’s population using one flower pot?
Diminishing MP – if the amount of input i increases (beyond a certain
value) and other inputs remain constant, MPi falls.
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Diminishing marginal product (law of diminishing MP)
Why can’t we feed the world’s population using one flower pot?
Diminishing MP – if the amount of input i increases (beyond a certain
value) and other inputs remain constant, MPi falls.
Is diminishing MP and diminishing TRS the same?
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Diminishing marginal product (law of diminishing MP)
Why can’t we feed the world’s population using one flower pot?
Diminishing MP – if the amount of input i increases (beyond a certain
value) and other inputs remain constant, MPi falls.
Is diminishing MP and diminishing TRS the same?
No, but the logics are similar:
• diminishing MP – as x1 rises, MP1 for a given x2 decreases
• diminishing TRS – as x1 rises moving along the isoquant to the right,
the additional amount of input 1 necessary to compensate for the loss
of 1 unit of input 2 is increasing.
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The short run and the long run
The short run (SR) – there will be some factors of production
that are fixed at predetermined levels.
The long run (LR) – all the factors of production can be varied.
How long is the short run?
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The short run and the long run
The short run (SR) – there will be some factors of production
that are fixed at predetermined levels.
The long run (LR) – all the factors of production can be varied.
How long is the short run?
Cooper and Haltiwanger (RES, 2006) study the US manufacturing plants
(1972 to 1988): 1 of 10 plants does not make any capital adjustment in a
given year.
Plants make large changes and then wait. Depending on the size of the
change needed, SR lasts from several months to more than a year.
Other sectors may differ (e.g. services or transport).
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Quasifixed inputs
Quasifixed inputs are used only if the output is positive, but the quantity
is independent of the output.
Examples:
Electricity for lights in a production plant, some administrative positions
Quasifixed vs. fixed inputs:
Quasifixed inputs exist in the SR and LR, fixed inputs only in the SR.
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Production function in the SR
Production function in the SR f (x1, ¯x2), where input 1 is variable
and input 2 is fixed.
The function in the graph has a diminishing MP. For low x1 the MP1 may
be increasing (then the function would have an S-shape).
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Production function in the LR – returns to scale
What is the increase in output if all inputs increase t times (t > 1)?
If for all points on a production function f (x1, x2) holds that
• f (tx1, tx2) = tf (x1, x2), f (·) has constant returns to scale.
• f (tx1, tx2) > tf (x1, x2), f (·) has increasing returns to scale.
• f (tx1, tx2) < tf (x1, x2), f (·) has decreasing returns to scale.
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Production function in the LR – returns to scale
What is the increase in output if all inputs increase t times (t > 1)?
If for all points on a production function f (x1, x2) holds that
• f (tx1, tx2) = tf (x1, x2), f (·) has constant returns to scale.
• f (tx1, tx2) > tf (x1, x2), f (·) has increasing returns to scale.
• f (tx1, tx2) < tf (x1, x2), f (·) has decreasing returns to scale.
Examples:
• constant– replication the production technology, . . .
• increasing – pin factory, airlines, plane production, . . .
• decreasing – difficult to find real examples (
”
organization“ does not
change at the same rate)
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Examples – returns to scale of specific production functions
1) What are the returns to scale of f (x1, x2) = x
1/2
1 x
3/4
2 ?
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Examples – returns to scale of specific production functions
1) What are the returns to scale of f (x1, x2) = x
1/2
1 x
3/4
2 ?
Multiplying both inputs by t > 1, we get:
f (tx1, tx2) = (tx1)1/2
(tx2)3/4
= t5/4
x
1/2
1 x
3/4
2 = t5/4
f (x1, x2)
Increasing because f (tx1, tx2) > tf (x1, x2).
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Examples – returns to scale of specific production functions
1) What are the returns to scale of f (x1, x2) = x
1/2
1 x
3/4
2 ?
Multiplying both inputs by t > 1, we get:
f (tx1, tx2) = (tx1)1/2
(tx2)3/4
= t5/4
x
1/2
1 x
3/4
2 = t5/4
f (x1, x2)
Increasing because f (tx1, tx2) > tf (x1, x2).
2) What are the returns to scale of f (x1, x2) = min{x1, x2}?
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Examples – returns to scale of specific production functions
1) What are the returns to scale of f (x1, x2) = x
1/2
1 x
3/4
2 ?
Multiplying both inputs by t > 1, we get:
f (tx1, tx2) = (tx1)1/2
(tx2)3/4
= t5/4
x
1/2
1 x
3/4
2 = t5/4
f (x1, x2)
Increasing because f (tx1, tx2) > tf (x1, x2).
2) What are the returns to scale of f (x1, x2) = min{x1, x2}?
Multiplying both inputs by t > 1, we get:
f (tx1, tx2) = min{tx1, tx2} = t × min{x1, x2} = t × f (x1, x2).
Constant because f (tx1, tx2) = tf (x1, x2).
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EXAMPLE: Copy Exactly!
Intel operates dozens of plants produce and test computer chips.
Chip fabrication is such a delicate process that Intel found it difficult to
manage quality in a heterogeneous environment. That is why Intel moved
to its Copy Exactly! process
Each looks very much like another – constant returns to scale.
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EXAMPLE: Mobile technology in Indian fishing
Jensen (QJE, 2007) studied 15 markets in Kerala in southern India where
fishermen sell their daily catch (no freezers available).
What was the contribution of mobile phones?
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EXAMPLE: Mobile technology in Indian fishing
Jensen (QJE, 2007) studied 15 markets in Kerala in southern India where
fishermen sell their daily catch (no freezers available).
What was the contribution of mobile phones?
Coordination: before mobile phones there was often surplus of fish
in one market and shortage of fish in the neighbouring market.
With mobile phones the supply matches
the demand much better.
And the impact on prices?
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EXAMPLE: Mobile technology in Indian fishing (graph)
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Profit maximization and cost minimization
Profit maximization – what production plan maximizes the firm’s profit
(for a given technology and input and output prices).
In this lecture we assume competitive input and output markets. =⇒
Prices of inputs (w) and the price of output (p) are given.
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Profit maximization and cost minimization
Profit maximization – what production plan maximizes the firm’s profit
(for a given technology and input and output prices).
In this lecture we assume competitive input and output markets. =⇒
Prices of inputs (w) and the price of output (p) are given.
Cost minimization (next lecture) – what combination of inputs minimizes
the cost of producing a given output (for a given technology and input
prices) – derivation of the cost function.
In the second step the firm chooses the profit-maximizing output
(for a given cost function and demand).
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The economic profit
We distinguish two types of costs:
• Explicit costs – accounting costs
• Implicit costs – opportunity costs of the firm-owned inputs
Example:
If the owner works in her firm and does not pay any wage to herself, she
has no accounting costs, but the firm has an implicit cost.
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The economic profit
We distinguish two types of costs:
• Explicit costs – accounting costs
• Implicit costs – opportunity costs of the firm-owned inputs
Example:
If the owner works in her firm and does not pay any wage to herself, she
has no accounting costs, but the firm has an implicit cost.
Two types of profit:
• Accounting profit = revenues – explicit costs
• Economic profit = revenues – explicit costs –
implicit costs
We will always use the economic profit.
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Profit maximization
A firm chooses quantities of inputs 1 and 2 to maximize its profit:
max
x1,x2
π = pf (x1, x2) − w1x1 − w2x2
It follows from the first order condition that :
pMP1(x∗
1 , x∗
2 ) = w1
pMP2(x∗
1 , x∗
2 ) = w2
The firm maximizes profit if of all inputs the value of their marginal
product equals to their prices.
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Demand for input
Demand for input – profit-maximizing quantities of input
The inverse demand for input 1 derived from the first order condition:
w1 = pMP1(x1, x∗
2 )
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Example of profit maximization in the SR
Production function: f (x1, x2) = x
1/2
1 x
1/2
2 , where ¯x2 = 16
Prices: (p, w1, w2) = (40, 10, 20)
What is the optimum quantity of the variable input and profit?
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Example of profit maximization in the SR
Production function: f (x1, x2) = x
1/2
1 x
1/2
2 , where ¯x2 = 16
Prices: (p, w1, w2) = (40, 10, 20)
What is the optimum quantity of the variable input and profit?
Short-run production function: f (x1, 16) = 4x
1/2
1
Profit function: π = p4x
1/2
1 − w1x1 − w2 ¯x2
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Example of profit maximization in the SR
Production function: f (x1, x2) = x
1/2
1 x
1/2
2 , where ¯x2 = 16
Prices: (p, w1, w2) = (40, 10, 20)
What is the optimum quantity of the variable input and profit?
Short-run production function: f (x1, 16) = 4x
1/2
1
Profit function: π = p4x
1/2
1 − w1x1 − w2 ¯x2
Deriving with respect to x1 we get pMP1 = w1 (FOC):
p
2
√
x1
= w1
x1 =
2p
w1
2
x1 = 64
Profit of the firm: π = p4x
1/2
1 − w1x1 − w2 ¯x2 = 320
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Profit maximization in the SR (graph)
Isoprofit lines – combinations of x and y earning a constant profit π:
π = py − w1x1 − w2 ¯x2 ⇐⇒ y =
π
p
+
w2 ¯x2
p
+
w1
p
x1
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Profit maximization in the SR (graph)
Isoprofit lines – combinations of x and y earning a constant profit π:
π = py − w1x1 − w2 ¯x2 ⇐⇒ y =
π
p
+
w2 ¯x2
p
+
w1
p
x1
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Comparative statics
If the price of input w1 increases, the optimal x1 decreases (Fig. A).
If the price of output p increases, the optimal x1 increases (Fig. B).
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Revealed profitability
A profit-maximizing firm shows that the chosen combination of inputs and
outputs represents a feasible production plan, which is at least as
profitable as other feasible production plans.
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Revealed profitability – example
Two different choices the firm makes at different sets of prices:
• at prices at time t (pt, wt
1, wt
2) the firm chooses (yt, xt
1, xt
2),
• at prices at time s (ps, ws
1 , ws
2 ) the firm chooses (ys, xs
1, xs
2).
Weak axiom of profit maximization (WAPM): If the firm maximizes
profit and the production function of the firm hasn’t changed between
times s and t, then we must have:
pt
yt
− wt
1xt
1 − wt
2xt
2 ≥ pt
ys
− wt
1xs
1 − wt
2xs
2 (1)
ps
ys
− ws
1 xs
1 − ws
2 xs
2 ≥ ps
yt
− ws
1 xt
1 − ws
2 xt
2 (2)
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Revealed profitability – example
Two different choices the firm makes at different sets of prices:
• at prices at time t (pt, wt
1, wt
2) the firm chooses (yt, xt
1, xt
2),
• at prices at time s (ps, ws
1 , ws
2 ) the firm chooses (ys, xs
1, xs
2).
Weak axiom of profit maximization (WAPM): If the firm maximizes
profit and the production function of the firm hasn’t changed between
times s and t, then we must have:
pt
yt
− wt
1xt
1 − wt
2xt
2 ≥ pt
ys
− wt
1xs
1 − wt
2xs
2 (1)
ps
ys
− ws
1 xs
1 − ws
2 xs
2 ≥ ps
yt
− ws
1 xt
1 − ws
2 xt
2 (2)
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Revealed profitability – example
Two different choices the firm makes at different sets of prices:
• at prices at time t (pt, wt
1, wt
2) the firm chooses (yt, xt
1, xt
2),
• at prices at time s (ps, ws
1 , ws
2 ) the firm chooses (ys, xs
1, xs
2).
Weak axiom of profit maximization (WAPM): If the firm maximizes
profit and the production function of the firm hasn’t changed between
times s and t, then we must have:
pt
yt
− wt
1xt
1 − wt
2xt
2 ≥ pt
ys
− wt
1xs
1 − wt
2xs
2 (1)
ps
ys
− ws
1 xs
1 − ws
2 xs
2 ≥ ps
yt
− ws
1 xt
1 − ws
2 xt
2 (2)
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Revealed profitability – example
Two different choices the firm makes at different sets of prices:
• at prices at time t (pt, wt
1, wt
2) the firm chooses (yt, xt
1, xt
2),
• at prices at times (ps, ws
1 , ws
2 ) the firm chooses (ys, xs
1, xs
2).
Weak axiom of profit maximization (WAPM): If the firm maximizes
profit and the production function of the firm hasn’t changed between
times s and t, then we must have:
pt
yt
− wt
1xt
1 − wt
2xt
2 ≥ pt
ys
− wt
1xs
1 − wt
2xs
2 (1)
ps
ys
− ws
1 xs
1 − ws
2 xs
2 ≥ ps
yt
− ws
1 xt
1 − ws
2 xt
2 (2)
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Revealed profitability – example
Two different choices the firm makes at different sets of prices:
• at prices at time t (pt, wt
1, wt
2) the firm chooses (yt, xt
1, xt
2),
• at prices at time s (ps, ws
1 , ws
2 ) the firm chooses (ys, xs
1, xs
2).
Weak axiom of profit maximization (WAPM): If the firm maximizes
profit and the production function of the firm hasn’t changed between
times s and t, then we must have:
pt
yt
− wt
1xt
1 − wt
2xt
2 ≥ pt
ys
− wt
1xs
1 − wt
2xs
2 (1)
ps
ys
− ws
1 xs
1 − ws
2 xs
2 ≥ ps
yt
− ws
1 xt
1 − ws
2 xt
2 (2)
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Revealed profitability – example (cont’d)
We copy equation (1) and transpose the two sides of equation (2) to get
pt
yt
− wt
1xt
1 − wt
2xt
2 ≥ pt
ys
− wt
1xs
1 − wt
2xs
2
−ps
yt
+ ws
1 xt
1 + ws
2 xt
2 ≥ −ps
ys
+ ws
1 xs
1 + ws
2 xs
2.
Since both equations have ≥, also the sum of the equations must have ≥:
(pt
− ps
)yt
− (wt
1 − ws
1 )xt
1 − (wt
2 − ws
2 )xt
2
≥ (pt
− ps
)ys
− (wt
1 − ws
1 )xs
1 − (wt
2 − ws
2 )xs
2.
Rearranging this equation and substituting ∆p for (pt − ps), ∆y for
(yt − ys), and so on, we find
∆p∆y − ∆w1∆x1 − ∆w2∆x2 ≥ 0.
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Revealed profitability – example (cont’d)
What follows from the result ∆p∆y − ∆w1∆x1 − ∆w2∆x2 ≥ 0?
• If the output price p changes and w1 and w2 remain constant, then
∆p∆y ≥ 0.
It never holds that ∆p > 0 and ∆y < 0, or ∆p < 0 and ∆y > 0.
=⇒ The supply curve of a competitive firm can’t be decreasing.
• If the price of input 1 w1 changes and p a w2 remain constant, then
∆w1∆x1 ≤ 0.
It never holds that ∆w1 > 0 and ∆x1 > 0, or ∆w1 < 0 and ∆x1 < 0.
=⇒ The factor demand of a competitive firm can’t be increasing.
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Estimating technology using WAPM
Can WAPM be used for estimating technology?
Yes, if the firm maximizes profit and its technology hasn’t changed, we can
use its choices to estimate its technology .
Suppose we have one input x1 and one output y. In period
• t, the firm chooses a production plan (xt
1, yt) and the isoprofit
function is πt = pty − wt
1x1 ⇐⇒ y = πt/pt + (wt
1/pt)x1.
• s, the firm chooses a production plan (xs
1, ys) and the isoprofit
function is πs = psy − ws
1 x1 ⇐⇒ y = πs/ps + (ws
1 /ps)x1.
It follows from WAPM that production plans above the isoprofit lines are
not feasible. Otherwise the firm could increase its profit by choosing them.
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Estimating technology using WAPM (cont’d)
White area = production plans above the isoprofit line. They are not
feasible, so they can be part of the technology.
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Estimating technology using WAPM (cont’d)
White area = production plans above the isoprofit line. They are not
feasible, so they can be part of the technology.
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Estimating technology using WAPM (cont’d)
The more production plans we observe at different prices, the tighter is the
estimate of the technology (and the production function).
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APPLICATION: ATM withdrawal fees
Ishii (2005) “Compatibility, Competition, and Investment in Network
Industries: ATM Networks in the Banking Industry”
People often pay higher withdrawal fees at ATMs of other banks (a
surcharge). Does it reduce competition? Should this practice be banned by
the government?
In order to answer these questions, we need to
estimate a structural model and among other find
out, what are the costs of ATM.
We do not observe the costs.
We need to estimate them.
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APPLICATION: ATM withdrawal fees (cont’d)
We estimate the bank revenue as a function of the number of ATMs d
It follows from the revealed profit maximization that the bank chooses d
such that
MR of an ATM d ≥ MC ≥ MR of an ATM d + 1
Results:
• The size of fee surcharge influences the bank’s demand.
=⇒ Banks build an inefficiently large network of ATMs.
• MC are higher than fee revenue. An additional ATM reduces the
bank’s revenue by more than by how much they increase his surplus.
• If the government banned the surcharge, competition would increase,
profits decrease and consumer surplus increase = good for consumers.
• But banks would be motivated to reduce the network of ATMs,
which would probably increase profits of banks and reduce CS.
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What should you know?
• Technology – all technologically feasible
combinations of inputs and outputs.
• Production functions – the maximum
output attainable with given inputs.
• We assume that isoquants are convex
and monotonic.
• Technical rate of substitution measures
the slope of the isoquant.
• SR: at least one input is fixed
LR: all inputs are variable
• Returns to scale – how output changes if
inputs are changed in the same proportion.
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What should you know? (cont’d)
• Profit maximization – what production plan
maximizes firm’s profit (for a given technology
and input and output prices).
X Cost minimization – what combination of
inputs minimizes costs of a given output
(for a given technology and input prices).
• Profit is the difference between revenues and
costs. Impicit costs are included.
• If the firm is maximizing profits, then the value
of the MP of each factor that it is free to vary
must equal its factor price.
• The logic of profit maximization implies that
the supply function of a competitive firm can’t
be decreasing and that each factor demand
function can’t be increasing.
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