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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Gives the total 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 3 inputs for production,
with conditional demand functions for each
input given by:
 x1(w1, w2, w3, q)
 x2(w1, w2, w3, q)
 x3(w1, w2, w3, q)

What would be the generic form for its cost
function?
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Cost functions: Example
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Consider a firm that builds a product with only two inputs (L,
K) with Cobb-Douglas technology of the form:
q = f(L,K) = L0.5 K0.5
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If wL = 8 and wK = 2, what will be optimal way to produce some
amount q?
What will be this firm’s cost function given these prices and
technology?
So, how much will it cost to produce q = 10 optimally? At q = 20?
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Cost functions and Opportunity Costs
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It is important to remember that a cost
function includes all costs of production,
including opportunity costs.
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With Cobb-Douglas technology assumed,
figuring out costs is easy because we have
implicitly assumed only two inputs.
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Things can be more complicated though.
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Cost functions with Opportunity Costs
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Suppose I am considering getting into the chair making business,
where I would deliver however many chairs I make to Ikea one
year from now.
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To make any chairs at all, I need to buy a saw which costs $400 (though I can re-sell it
for $200 at the end of the year)
Then, each chair I make requires 3 boards of wood (at $2/board, which I must pay for
up front)
Making chairs also required time. Specifically, I can turn my time into chairs according
to the production function q = L0.5.
If I can save at 10% annual interest, and any time I spent making chairs would mean
less time working at my current job which pays $20/hr (for which I get paid at the end
of the year), what would be my cost function for making chairs?
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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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How does this relate to the chair making
example?
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What might be some other production
processes that have fixed inputs the short
run?
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Short Run Cost Curve Analytically
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Short-run cost function where x2 fixed at x2f (and only two inputs)
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CSR(q) = w1x1(q|x2 = x2f) + w2x2f
Short run cost function where there are n inputs, where inputs 1 to k
are variable and k to n are fixed:
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So, short-run cost function can be written
k
C (q)   wi xi (q | x ...x ) 
SR
i 1
CSR(q) = cv(q)
f
k 1
f
n
n
w x
i  k 1
f
i i
+ F
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Short Run vs. Long Run Costs Analytically
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Example: Consider again a firm where:
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q = f(L, K) = L0.5 K0.5, wL = 8, wK= 2.
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From before, we know long-run (i.e. when both factors are variable)
cost function in this case will be
C(q) = 8q
Suppose in the short-run Capital (K) is fixed at 24 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 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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The first thing we want to think about is how the cost of producing “one
more unit” changes over the production cycle.
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Consider a “discrete” cost function where:
c(1) = $20
c(2) = $30
c(3) = $35
c(4) = $45
c(5) = $60
What would graph of this look like?
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What if we graphed cost of “one more”?
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What might we call the cost of producing “one more unit” and its
associated curve?
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Marginal Costs Graphically
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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+1) – C(q)]/1
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Actually rate of change, however,
so
MC(q) ≈ [C(q+Δq) – C(q)]/Δq
$
C(q)
q
And taking the limit as Δq goes to 0,
MC(q) 

C (q)
q
$
MC(q)
Therefore, we can get an idea of
what the MC(q) curve looks like
from the cost curve and vice versa.
q
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Cost functions and Returns-to-Scale
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We can also describe returns-to-scale via a cost
curve/marginal cost curve
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If marginal costs are decreasing over a range of output
levels, we say that technology exhibits increasing returnsto-scale (IRS) over that range.
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If marginal costs are constant over a range of output levels,
we say that technology exhibits constant returns-to-scale
(CRS) over that range.
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If marginal costs are increasing over a range of output
levels, we say that technology exhibits decreasing returnsto-scale (DRS) over that range.
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Graphically?
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Cost functions and Returns-to-Scale
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Consider Cobb-Douglas production function f(L,K) = L0.5K0.5, with
wL = 8 and wK = 2.
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Recall that the (Long-Run) cost function for this technology was
CLR(q) = 8q
Does this exhibit CRS, DRS, or IRS?
Now consider the same Cobb-Douglas production function f(L,K) =
L0.5K0.5, with wL = 8 and wK = 2, but where K is fixed at 24.
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Recall that the (Short-Run) cost function for this technology was
CSR(q) = q2/3 + 48
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Does this cost function exhibit DRS, CRS, IRS?
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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 exhibit DRS at some point.
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Cost Curves
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In modeling optimal firm behavior, it will often be helpful to think of two
other “cost curves” as well.
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Average Cost Curve – AC(q)
 Denotes the average cost of producing each unit, given q units are
produced.
AC(q) = C(q)/q
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Average Variable Cost Curve – AVC(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
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Cost Curves
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Consider our example, C(q) = q2/3 + 48
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(i.e. the cost function that arises from production function f(L,K)=
L0.5K0.5 with K fixed at 24 and wL = 8, wK= 2)
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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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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 MC curves
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Recall our discussion of long-run vs. short-run.
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For example, consider a firm deciding how large of a plant to build.
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Suppose there are three possible size plants.
Each plant size will be associated with its own cost curve and MC curve:
In the short-run, the firm is stuck with a given plant size, but over the longer-run
they can choose which plant size to use based on how much they plan to build.
How will cost curve and MC curve change from the short-run to the longer-run?
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