Cost minimization and cost functions
Varian: Intermediate Microeconomics, 8e, 20 and 21
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In this lecture you will learn
• how to define a cost function
• what the conditional demand for factor is
• what follows from revealed cost minimization
• what different cost functions look like
• how to measure cost inefficiency
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Profit maximization and cost minimization
Profit maximization (last lecture) – what production plan maximizes the
firm’s profit (for a given technology and input and output prices).
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Profit maximization and cost minimization
Profit maximization (last lecture) – what production plan maximizes the
firm’s profit (for a given technology and input and output prices).
Cost minimization – 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).
In the rest of this lecture and the next 4 lectures we assume competitive
input markets. =⇒ Prices of inputs (w) are given.
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Cost minimization
The firm chooses a combination of inputs that minimizes its costs of
producing a given output (at given input prices and technology):
min
x1,x2
w1x1 + w2x2
pro f (x1, x2) = y
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Cost minimization
The firm chooses a combination of inputs that minimizes its costs of
producing a given output (at given input prices and technology):
min
x1,x2
w1x1 + w2x2
pro f (x1, x2) = y
Cost function c(y) gives the minimum costs necessary for producing a
given output y (at given prices and technology).
Isocost – all combinations of inputs x1 and x2 that correspond to a given
level of costs C:
w1x1 + w2x2 = C ⇐⇒ x2 =
C
w2
−
w1
w2
x1
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Cost minimization – graphical solution
If the isoquant is monotonic, smooth and convex and we have the inner
solution, then in the optimum holds:
slope of the isoquant = TRS(x∗
1 , x∗
2 ) = −
w1
w2
= slope of the isocost
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Firm optimum vs. consumer optimum
Consumer Firm
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Firm optimum vs. consumer optimum
Consumer – the point on the BL
with a maximum utility
Firm – the point on the isoquant
corresponding to minimum costs
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Conditional demand for inputs
Conditional demand for inputs – what quantity of input minimizes the
costs of producing a given output
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Conditional demand for inputs
Conditional demand for inputs – what quantity of input minimizes the
costs of producing a given output
Difference between the demand and conditional demand for input:
• demand – what x maximizes the profit for (p, w)
• conditional demand – what x minimizes the costs of (y, w)
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Deriving the cost function – convex isoquants
Production function: y =
√
x1 + 3
√
x2
Input prices: w1 = 1 a w2 = 1
What are the conditional demand functions and the cost function?
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Deriving the cost function – convex isoquants
Production function: y =
√
x1 + 3
√
x2
Input prices: w1 = 1 a w2 = 1
What are the conditional demand functions and the cost function?
Monotonic, smooth and convex isoquant =⇒ TRS = −w1/w2:
−
√
x2
3
√
x1
= −1
x2 = 9x1
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Deriving the cost function – convex isoquants
Production function: y =
√
x1 + 3
√
x2
Input prices: w1 = 1 a w2 = 1
What are the conditional demand functions and the cost function?
Monotonic, smooth and convex isoquant =⇒ TRS = −w1/w2:
−
√
x2
3
√
x1
= −1
x2 = 9x1
By substituting back into the pf, we get the conditional demands:
x1 = y2
/100 a x2 = 9y2
/100
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Deriving the cost function – convex isoquants
Production function: y =
√
x1 + 3
√
x2
Input prices: w1 = 1 a w2 = 1
What are the conditional demand functions and the cost function?
Monotonic, smooth and convex isoquant =⇒ TRS = −w1/w2:
−
√
x2
3
√
x1
= −1
x2 = 9x1
By substituting back into the pf, we get the conditional demands:
x1 = y2
/100 a x2 = 9y2
/100
Cost function:
c(y) = w1x1 + w2x2 = 1 × y2
/100 + 1 × 9y2
/100 = y2
/10
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Deriving the cost function – perfect complements
For a production of a 3D visualization (V) we need:
• 1 hour of labour (L)
• 2 hours of a computer (C)
Input prices: wL = 300 and wC = 100
What are the conditional demand functions and the cost function?
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Deriving the cost function – perfect complements
For a production of a 3D visualization (V) we need:
• 1 hour of labour (L)
• 2 hours of a computer (C)
Input prices: wL = 300 and wC = 100
What are the conditional demand functions and the cost function?
Production function: V = min{L, C/2}
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Deriving the cost function – perfect complements
For a production of a 3D visualization (V) we need:
• 1 hour of labour (L)
• 2 hours of a computer (C)
Input prices: wL = 300 and wC = 100
What are the conditional demand functions and the cost function?
Production function: V = min{L, C/2}
Conditional demand functions:
V = L = C/2
L = V and C = 2V
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Deriving the cost function – perfect complements
For a production of a 3D visualization (V) we need:
• 1 hour of labour (L)
• 2 hours of a computer (C)
Input prices: wL = 300 and wC = 100
What are the conditional demand functions and the cost function?
Production function: V = min{L, C/2}
Conditional demand functions:
V = L = C/2
L = V and C = 2V
Cost function:
c(V ) = wL × L + wC × C = 300 × V + 100 × 2V = 500V
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Deriving the cost function – perfect substitutes
Book (B) can be produced using
• 1/5 of an hour using a hi-tech printer (H)
• 1/3 of an hour using a standard printer (S)
Input prices: wH = 10 and wS = 5
What are the conditional demand functions and the cost function?
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Deriving the cost function – perfect substitutes
Book (B) can be produced using
• 1/5 of an hour using a hi-tech printer (H)
• 1/3 of an hour using a standard printer (S)
Input prices: wH = 10 and wS = 5
What are the conditional demand functions and the cost function?
Production function: B = 5H + 3S
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Deriving the cost function – perfect substitutes
Book (B) can be produced using
• 1/5 of an hour using a hi-tech printer (H)
• 1/3 of an hour using a standard printer (S)
Input prices: wH = 10 and wS = 5
What are the conditional demand functions and the cost function?
Production function: B = 5H + 3S
I use the cheaper technology – the cost of one book printed on
• the hi-tech printer is wH/5 = 2
• the standard printer is wS /3 = 5/3
The conditional demand functions are H = 0 and S = B/3
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Deriving the cost function – perfect substitutes
Book (B) can be produced using
• 1/5 of an hour using a hi-tech printer (H)
• 1/3 of an hour using a standard printer (S)
Input prices: wH = 10 and wS = 5
What are the conditional demand functions and the cost function?
Production function: B = 5H + 3S
I use the cheaper technology – the cost of one book printed on
• the hi-tech printer is wH/5 = 2
• the standard printer is wS /3 = 5/3
The conditional demand functions are H = 0 and S = B/3
Cost function:
c(B) = wH × H + wS × S = 10 × 0 + 5 × B/3 = 5/3B
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Revealed cost minimization
A cost-minimizing firm chooses a combination of inputs in order to
produce a given output (at given input prices and technology) at costs
that are at least as low as the costs of alternative combinations of inputs.
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Revealed cost minimization – example
A firm produces output y using two different combinations of inputs:
• at input prices at time t (wt
1, wt
2) the firm chooses (xt
1, xt
2)
• at input prices at time s (ws
1 , ws
2 ) the firm chooses (xs
1, xs
2)
Weak axiom of cost minimization (WACM): If a firm produces
output y at minimum costs and technology hasn’t changed between
times t and s, then it holds that:
wt
1xt
1 + wt
2xt
2 ≤ wt
1xs
1 + wt
2xs
2 (1)
ws
1 xs
1 + ws
2 xs
2 ≤ ws
1 xt
1 + ws
2 xt
2 (2)
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Revealed cost minimization – example (cont’d)
If we copy the equation (1) and multiply the equation (2) by −1, we get
wt
1xt
1 + wt
2xt
2 ≤ wt
1xs
1 + wt
2xs
2
−ws
1 xt
1 − ws
2 xt
2 ≤ −ws
1 xs
1 − ws
2 xs
2
Since both equations have ≤, also the sum of the equations must have ≤:
(wt
1 − ws
1 )xt
1 + (wt
2 − ws
2 )xt
2 ≤ (wt
1 − ws
1 )xs
1 + (wt
2 − ws
2 )xs
2
Rearranging this equation and substituting ∆w1 for (wt
1 − ws
1 ), ∆x1 for
(xt
1 − xs
1), and so on, we find
∆w1∆x1 + ∆w2∆x2 ≤ 0.
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Revealed cost minimization – example (cont’d)
What follows from the result ∆w1∆x1 + ∆w2∆x2 ≤ 0?
E.g. if the price of factor 1 w1 changes and the price of factor 2 w2
remains constant, then
∆w1∆x1 ≤ 0.
It never holds that ∆w1 > 0 and ∆x1 > 0 or ∆w1 < 0 and ∆x1 < 0. =⇒
The conditional factor demand of a competitive firm can’t be increasing.
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APPLICATION: Costs and inefficiency
We can estimate cost functions from the input prices and output data.
Different cost functions in one industry – possible explanations:
• firms have different technologies
• firms do not minimize costs
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APPLICATION: Costs and inefficiency
We can estimate cost functions from the input prices and output data.
Different cost functions in one industry – possible explanations:
• firms have different technologies
• firms do not minimize costs
Piacenza (J Prod Anal, 2006) – the costs of Italian public transport firms is
on average 11% above the minimum costs of producing the same output.
The inefficiency is influenced by the type of transport subsidy:
• Cost plus: the size of subsidy is a function of transport costs.
• Fixed price: transport firms have a subsidized, but fixed, price.
What type of subsidies generates a higher inefficiency?
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APPLICATION: Costs and inefficiency (graph)
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APPLICATION: Cost minimization in the US health sector
Before 1983: Medicare would reimburse a share of hospitals’ capital and
labor costs equal to Medicare patient-days/total patient-days.
After 1983: Capital costs paid as before, labor costs covered by a flat rate
based on the patient’s diagnosis (any additional labor cost covered fully by
the hospital) = the isocost’s slope changes.
What was the reaction of hospitals?
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APPLICATION: Cost minimization in the US health sector
Before 1983: Medicare would reimburse a share of hospitals’ capital and
labor costs equal to Medicare patient-days/total patient-days.
After 1983: Capital costs paid as before, labor costs covered by a flat rate
based on the patient’s diagnosis (any additional labor cost covered fully by
the hospital) = the isocost’s slope changes.
What was the reaction of hospitals?
Acemoglu a Finkelstein (JPE, 2008): 10% increase in the K/L ratio.
A bigger increase in hospitals with a higher share of Medicare patients.
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Costs
Total costs: c(y) = cv (y) + F (+QF)
• variable costs cv (y) – costs of variable inputs (SR and LR)
• fixed costs F = costs of fixed inputs (only SR): a constant for y ≥ 0
• quasifixed costs QF = costs of quasifixed inputs (SR and LR)
QF =
a constant if y > 0
0 if y = 0
Average costs:
AC(y) =
c(y)
y
=
cv (y)
y
+
F
y
= AVC(y) + AFC(y)
Marginal costs:
MC(y) =
dc(y)
dy
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Average costs
AFC(y) = F
y – decreasing; the same fixed costs spread over a higher y
AVC(y) = cv (y)
y – increasing from a given y; limited by the fixed input
AC(y) = AFC(y) + AVC(y) – typically U-shaped
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Average and marginal costs
For discrete output MC(y) and AVC(y) are equal for y = 1:
MC(1) =
cv (1) + F − cv (0) − F
1
=
cv (1)
1
= AVC(1)
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Average and marginal costs
For discrete output MC(y) and AVC(y) are equal for y = 1:
MC(1) =
cv (1) + F − cv (0) − F
1
=
cv (1)
1
= AVC(1)
MC(y) crosses the AC(y) and AVC(y) curves in their minimum:
AC (y∗
) =
c(y∗)
y∗
=
c (y∗)y∗ − c(y∗)
y∗2
= 0 ⇐⇒ c (y∗
) =
c(y∗)
y∗
AVC (ˆy) =
cv (ˆy)
ˆy
=
cv (ˆy)ˆy − cv (ˆy)
ˆy2
= 0 ⇐⇒ cv (ˆy) =
cv (ˆy)
ˆy
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Average and marginal costs (graph)
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Marginal costs and total variable costs
Variable costs necessary for a production of y units of output =
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Marginal costs and total variable costs
Variable costs necessary for a production of y units of output =
the area below the MC curve for the output between 0 and y.
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Example – marginal cost curves for two plants
A firm has two plants with cost functions c1(y1) a c2(y2).
How to divide the production of y units between the plants?
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Example – marginal cost curves for two plants
A firm has two plants with cost functions c1(y1) a c2(y2).
How to divide the production of y units between the plants?
Optimal outputs y∗
1 and y∗
2 are such that MC1(y∗
1 ) = MC2(y∗
2 ) = c.
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Numerical example – cost functions
Total costs:
• c(y) = y2 + 1
• variable – cv (y) = y2
• fixed – F = 1
Average and marginal costs:
• AFC(y) = 1/y
• AVC(y) = y2/y = y
• AC(y) = y + 1/y
• MC(y) = 2y
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Long-run average costs (LAC)
If the quasifixed costs are 0 and the production function exhibits
• constant returns to scale, LAC(y) is constant,
• increasing returns to scale, LAC(y) is decreasing,
• decreasing returns to scale, LAC(y) is increasing.
Why?
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Long-run average costs (LAC)
If the quasifixed costs are 0 and the production function exhibits
• constant returns to scale, LAC(y) is constant,
• increasing returns to scale, LAC(y) is decreasing,
• decreasing returns to scale, LAC(y) is increasing.
Why? If t > 1 and the production function has
• constant returns to scale, then
LAC(ty) =
c(ty)
ty
=
t · c(y)
ty
= LAC(y).
• increasing returns to scale, then
LAC(ty) =
c(ty)
ty
<
t · c(y)
ty
= LAC(y).
• decreasing returns to scale, then
LAC(ty) =
c(ty)
ty
>
t · c(y)
ty
= LAC(y).
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Short-run and long-run average costs
SR: for a fixed plant size k∗, the optimal output is y∗
LR: the firm chooses the optimal plant size for each output
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Short-run and long-run average costs
SR: for a fixed plant size k∗, the optimal output is y∗
LR: the firm chooses the optimal plant size for each output
For the output y∗ holds: SAC = LAC
For all other outputs y = y∗ holds: SAC > LAC
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Discrete levels of plant size
The LAC curve (dark blue) if a firm chooses from 4 plant sizes:
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Discrete levels of plant size
The LAC curve (dark blue) if a firm chooses from 4 plant sizes:
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Long-run marginal costs (LMC)
The long-run marginal cost LMC curve:
• left – a firm chooses among 3 plant sizes
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Long-run marginal costs (LMC)
The long-run marginal cost LMC curve:
• left – a firm chooses among 3 plant sizes (black curve)
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Long-run marginal costs (LMC)
The long-run marginal cost LMC curve:
• left – a firm chooses among 3 plant sizes (black curve)
• right – a firm can choose any continuous plant size
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Long-run marginal costs (LMC)
The long-run marginal cost LMC curve:
• left – a firm chooses among 3 plant sizes (black curve)
• right – a firm can choose any continuous plant size
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What should you know?
• Cost minimization – what combination of
inputs minimizes costs of a given output
(for a given technology and input prices).
• Cost function – the minimum costs necessary
for producing a given output.
• Conditional demand for input – how much
input minimizes the cost of production of
a given output.
X Demand for input – profit-maximizing firm
buys such quantity of input that w = pMP.
• If a firm minimizes costs, its conditional
demand function cannot be increasing.
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What should you know? (cont’d)
• The average cost function AC is usually U-shaped,
because AFC is decreasing and AVC increasing
(beyond certain quantity).
• In the minimum, AC and AVC equal to MC.
• The area below MC from 0 to y equals VC(y).
• The LAC curve is the lower envelope of the
short-run average cost curves.
• Fixed cost: SR, can be positive for y = 0
Quasificed cost: SR and LR, zero for y = 0
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