Intermediate Microeconomic Theory

1
Intermediate Microeconomic Theory

• Cost Curves

2
Cost Functions

• We have solved the first part of the problem
  given factor prices, what is cheapest way to
  produce q units of output?
• Given by conditional factor demands for each
  input i, xi(w1,, wn, q)
• 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.

3
Cost functions

• Key to firms output decision is the firms cost
  function
• 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 n inputs for production, with
  conditional demand functions for each input given
  by
• x1(w1,, wn, q)
• xn(w1,, wn, q)
• What would be the generic form for its cost
  function?

4
Cost functions Example

• 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
• If wL 8 and wK 2, what will be optimal way to
  produce some amount q?
• What will be this firms cost function given
  these prices and technology?
• So, how much will it cost to produce q 10
  optimally?

5
Cost functions and Opportunity Costs

• It is important to remember that a cost function
  includes all costs of production, including
  opportunity costs.
• With Cobb-Douglas technology assumed, figuring
  out costs is easy because we have implicitly
  assumed only two inputs.
• Things can be more complicated though.

6
Cost functions with Opportunity Costs

• 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.
• 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).
• Making chairs also required time. Specifically, I
  can turn my time into chairs according to the
  production function q L0.5.
• If I currently have 1000 in savings at 10
  annual interest, and any time I spent making
  chairs would mean less time working at my current
  job which pays 20/hr, what would be my cost
  function for making chairs?

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Short run vs. Long(er) run

• It is often important to distinguish between the
  Short-Run (SR) and Long(er)-Run (LR) when
  considering costs.
• Short-run some factors of production are fixed
  (i.e. cant be adjusted).
• Long(er)-run previously fixed factors of
  production can be adjusted.

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Short Run Cost Curve

• 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).
• How does this relate to the chair making example?
• What might be some other production processes
  that have fixed inputs the short run?

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Short Run Cost Curve Analytically

• Short-run cost function where x2 fixed at x2f
  (and only two inputs)
• CSR(q) w1x1(qx2 x2f) w2x2f
• Short run cost function where there are n inputs,
  where inputs 1 to k are variable and k to n are
  fixed
• So, short-run cost function can be written
• CSR(q) cv(q) F

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Short Run vs. Long Run Costs Analytically

• Example Consider again a firm where
• q f(L, K) L0.5 K0.5, wL 8, wK 2.
• 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.
• What is short-run cost function?

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Short-Run Cost function

• 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

• In modeling optimal firm behavior, it will often
  be helpful to think of costs graphically via
  cost curves.
• The first thing we want to think about is how the
  cost of producing one more unit changes over
  the production cycle.
• 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?
• What if we graphed cost of one more?
• What might we call the cost of producing one
  more unit and its associated curve?

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Marginal Costs Graphically

• Marginal Cost Curve MC(q)
• Denotes the cost of producing a little bit
  more, given you have already produced q units
• So MC(q) C(q1) C(q)/1
• Actually rate of change, however, so
• MC(q) C(q?q) C(q)/?q
• And taking the limit as ?q goes to 0,
• Therefore, 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 functions and Returns-to-Scale

• We can also describe returns-to-scale via a cost
  curve/marginal cost curve
• If marginal costs are decreasing over a range of
  output levels, we say that technology exhibits
  increasing returns-to-scale (IRS) over that
  range.
• If marginal costs are constant over a range of
  output levels, we say that technology exhibits
  constant returns-to-scale (CRS) over that range.
• If marginal costs are increasing over a range of
  output levels, we say that technology exhibits
  decreasing returns-to-scale (DRS) over that
  range.
• Graphically?

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Cost functions and Returns-to-Scale

• Consider Cobb-Douglas production function f(L,K)
  L0.5K0.5, with wL 8 and wK 2.
• 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.
• Recall that the (Short-Run) cost function for
  this technology was
• Does this cost function exhibit DRS, CRS, IRS?

CSR(q) q2/3 48
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Cost functions and Returns-to-Scale

• 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

• In modeling optimal firm behavior, it will often
  be helpful to think of two other cost curves as
  well.
• Average Cost Curve AC(q)
• Denotes the average cost of producing each unit,
  given q units are produced.
• AC(q) C(q)/q
• 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

• Consider our example, C(q) q2/3 48
• (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)
• What is equation for AC(q)?
• What is equation for AVC(q)?
• What is equation for MC(q)?

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Cost Curves (cont.)

• How do these curves relate to each other?
• 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)

• Given AVC(q) MC(q) MC(q-1) MC(1)/q,
• AVC is essentially the average Marginal cost of
  producing each unit, given firm has produced q
  units.
• Therefore,
• If MC(q) lt 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)
• Alternatively, if MC(q) gt 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)

• Now, recall AC(q) C(q)/q cv(q) F/q
  
• AVC(q) F/q
• 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) lt AC(q) over some range of q,
  then AC(q) must be decreasing over that range.
• Alternatively, if MC(q) gt 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

• Recall our discussion of long-run vs. short-run.
• For example, consider a firm deciding how large
  of a plant to build.
• 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?