Transcript Intermediate Microeconomic Theory

Intermediate Microeconomic Theory
Cost Curves
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Cost Functions


We have solved the first part of the
problem: given factor prices, what is
cheapest way to produce q units of output?
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Given by conditional factor demands for
each input i, xi(w1,…, wn, q)
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However, this is only half the problem.
To model behavior of the firm, we also
have to derive how much output a firm will
find optimal to produce, given input and
output prices.
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Cost functions
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Key to firm’s output decision is the firm’s
cost function
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Cost of producing a given amount of output,
given some input prices and assuming the firm
acts optimally (i.e. cost minimizes).
Suppose a firm used n inputs for production,
with conditional demand functions for each
input given by:
 x1(w1,…, wn, q)
:
:
 xn(w1,…, wn, q)
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What would be the generic form for its cost
function?
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Cost functions: Example
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Consider a firm with Cobb-Douglas
technology of the form
q = f(x1, x2) = x10.25 x20.25
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Recall a cost function will be of the form:
C(q) = Σwi x1(w1, ,…, wn, q)
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What will be conditional factor demands for
this technology?
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So what will be this firm’s cost function if
w1 = 4 and w2 = 1?
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Short run vs. Long(er) run
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It is often important to distinguish between the
Short-Run (SR) and Long(er)-Run (LR) when
considering costs.
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Short-run: some factors of production are
fixed (i.e. can’t be adjusted).
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Long(er)-run: previously fixed factors of
production can be adjusted.
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Short Run Cost Curve
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The key aspect of a fixed factor of production is
that it will mean there will be some component
of cost that is the same regardless of how much
output (if any) is produced (in short run).
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What might be some production processes
that have fixed inputs the short run?
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Short Run Cost Curve
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Short-run cost function where x2 fixed at
x2f (and only two inputs)
CSR(q) = w1x1(w1,w2,q|x2 = x2f) + w2x2f
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Short run cost function where there are n
inputs, where inputs 1 to k are variable and
k to n are fixed:
k
C (q)   wi xi ( w1 , w2 ,...,wn , q | x ...x ) 
SR
f
k 1
i 1

f
n
n
w x
i  k 1
f
i i
So, short-run cost function can be written
CSR(q) =
cv(q) + F
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Short Run vs. Long Run Costs Analytically
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Consider again a firm where:
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q = f(L, K) = L0.25 K0.25, wL = 4, wK= 1.
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From before, we know long-run cost
function in this case will be
C(q) = 4q2
Suppose in the short-run Captial (K) is
fixed at 16 machine hrs.
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What is short-run cost function?
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Short-Run Cost function
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Given how cost functions are derived, will
the cost of producing any given level of
output be greater in the short-run or the
longer run?
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Cost functions with Opportunity Costs
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Suppose I want to start a ski instructor
school where I am the only teacher.
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To teach one lesson I need a pair of skis.
Each lesson takes 2 hrs. of my time (i.e. q = f(L)
= L/2, where q is number of ski lessons and L is
hours of my time)
What is my cost function if skis cost $400 (but I
can re-sell them for $200 at the end of the year),
my current job pays $20/hr., and I can save at a
rate of 10%?
What if I can rent skis for $5/hr. How will this
change my (long-run) cost function?
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Returns-to-Scale
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Returns-to-Scale is a concept describing
how costs change as scale of operation (i.e.
quantity produced) changes.
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What do people usually mean when they use
this term? (for example in NYT article on
Whirlpool and Maytag)
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Cost functions and Returns-to-Scale
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To describe Returns-to-scale technically,
consider first the “unit cost function” or the
cost minimizing way of producing one unit
of output (given input prices), as given by
C(1).
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If C(q) = q C(1), then technology exhibits
Constant-Returns-to-Scale (CRS)
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If C(q) > q C(1), then technology exhibits
Decreasing-Returns-to-Scale (DRS)
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If C(q) < q C(1), then technology exhibits
Increasing-Returns-to-Scale (IRS)
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Cost functions and Returns-to-Scale
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Consider Cobb-Douglas production
function f(x1, x2) = x10.25x20.25, with w1 = 4
and w2 = 1
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Recall that the (Long-Run) cost function for
this technology was CLR(q) = 4q2
Does this exhibit CRS, DRS, or IRS?
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What about when CLR(q) = 4q?
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What about when CLR(q) = 4q0.5?
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Cost functions and Returns-to-Scale
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What about CSR(q) = q4/4+ 16
(i.e. the short-run cost function for f(x1, x2) = x10.25x20.25, with w1 = 4 and w2 = 1,
and x2 fixed at 16)
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Cost functions and Returns-to-Scale
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From now on, we will generally be considering relatively
Short-run (i.e. at least one factor fixed), so cost functions
will generally exhibit both IRS and DRS.
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Cost Curves
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In modeling optimal firm behavior, it will often be helpful to think of costs
graphically via “cost curves”.
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Average Cost Curve – AC(q)
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Average Variable Cost Curve – AVC(q)
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Denotes the average cost of producing each unit, given q units are produced.
AC(q) = C(q)/q
As discussed above, we can often think of our SR cost function as:
C(q) = cv(q) + F
So AVC(q) = cv(q)/q
Marginal Cost Curve – MC(q)
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Denotes the cost of producing a “little bit” more, given you have already produced q units
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So MC(q) ≈ [C(q+Δq) – C(q)]/Δq = [cv(q+Δq) – cv(q)]/Δq
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Actually rate of change, however, so consider when Δq goes to zero,
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So
MC(q) 
C (q)
q
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Cost Curves
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Consider our example, C(q) = q4/4 + 16
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What is equation for AC(q)? What is Avg.
Cost per unit for producing 2 units? How
about 4 units?
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What is equation for AVC(q)? What is Avg.
Variable Cost per unit for producing 2 units?
How about 4 units?
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What is equation for MC(q)? What is
Marginal Cost for producing 2nd unit? How
about 4th?
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Cost Curves
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How about, C(q) = q3/3 – 5q2 + 60q + 20
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What is equation for AC(q)?
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What is equation for AVC(q)?
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What is equation for MC(q)?
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Marginal Costs Graphically
More generally, we can
get an idea of what
the MC(q) curve
looks like from the
cost curve and vice
versa.
$
C(q)
q
$
MC(q)
q
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Cost Curves (cont.)
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How do these curves relate to each other?
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First note: MC(q) ≈ [C(q) – C(q-1)]/1
Next, recall:
AVC(q) = cv(q)/q
= [C(q)-F]/q
(noting that cv(q) = C(q)- F)
= [C(q)-C(0)]/q (noting that C(0) = F)
= [(C(q)-C(q-1) + (C(q-1)-C(q-2))+…+(C(1)-C(0))]/q
So AVC(q) ≈ [MC(q) + MC(q-1) + …+ MC(1)]/q
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Cost Curves (cont)
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Given AVC(q) ≈ [MC(q) + MC(q-1) + …+ MC(1)]/q,
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AVC is essentially the average Marginal cost of producing each unit,
given firm has produced q units.
Therefore,
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If MC(q) < AVC(q) over some range of q, then AVC(q) must be
decreasing over that range (if you continually add something below
the average, average will go down)
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Alternatively, if MC(q) > AVC(q) over some range of q, then
AVC(q) must be increasing over that range (if you continually add
something above the average, average will go up)
So MC(q) must intersect AVC(q) at the q with the minimum Average
Variable cost (call it q*)
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MC(q) and AVC(q)
MC(q)
$
AVC(q)
q*
q
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Cost Curves (cont)
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Now, recall
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So AC(q) - AVC(q) = F/q
 (difference between AC(q) and AVC(q) decreases as q
increases)
Also, AC(q) ≈ [MC(q) + MC(q-1) + …+ MC(1)]/q + F/q
 Therefore, if MC(q) < AC(q) over some range of q, then AC(q)
must be decreasing over that range.
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AC(q) = C(q)/q = [cv(q) + F]/q
= AVC(q) + F/q
Alternatively, if MC(q) > AC(q) over some range of q, then
AC(q) must be increasing over that range.
So MC(q) also intersects AC(q) at the q with the minimum
Average Cost (call it q**).
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MC(q) and AVC(q)
MC(q)
$
AC(q)
AVC(q)
F
q*
q**
q
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Long-run vs. Short run AC & MC curves
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Recall our discussion of long-run vs. short-run.
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In SR, at least one factor is fixed.
For example, consider a firm deciding how
large of a plant to build.
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Suppose there are three possible size plants.
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Each plant size will be associated with its own
cost curve and therefore:
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Each plant size will have its own AC curve.
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Each plant size will have its own MC curve.
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