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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?

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.

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Cost functions

Key to firm’s output decision is the firm’s 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?

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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 firm’s cost function given these prices and technology?

So, how much will it cost to produce q = 10 optimally?

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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.

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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. can’t 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(q|x2 = x2f) + w2x2

f

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

n

ki

fii

fn

fk

k

iii

SR xwxxqxwqC1

11

)...|()(

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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) = $20c(2) = $30c(3) = $35c(4) = $45c(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(q+1) – C(q)]/1

Actually rate of change, however, so MC(q) ≈ [C(q+Δq) – C(q)]/Δq

And taking the limit as Δq 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

q

qCqMC

)(

)(

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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) < 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) > 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)

q* q

$MC(q)

AVC(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) < AC(q) over some range of q, then AC(q)

must be decreasing over that range.

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)

q* q** q

$

F

MC(q)

AVC(q)AC(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?