Advanced Microeconomics - University of Cagliari

Advanced Microeconomics

Ivan Etzo

University of Cagliari

[email protected] in Scienze Economiche e Aziendali, XXXIII ciclo

Ivan Etzo (UNICA) Lecture 4: Cost Functions 1 / 22

Overview

1 Short-run Cost functionsTotal CostsAverage CostsMarginal costs

2 Long-run costsLong Average CostsLong Marginal Costs


Short-run Cost functions

The cost function measures the minimum cost of producing a givenlevel of output for some fixed factor prices.

The cost function describes the economic possibilities of a firm.

Type of Short-run cost functions:

Average (total) costsAverage fixed costsAverage variable costsMarginal costs

Cost functions are important in studying the determination of optimaloutput choices.

How are these cost functions related to each other?

How are a firm’s long-run and short-run cost functions related?


Total Costs

Consider the cost function c(w1,w2, y) and let’s consider the factorprices fixed, so that the cost function is a function of the outputalone, c(y).

In the short-run there are some costs that are fixed, that is they donot depend on the output level (e.g. plant size).

Contrarily, the variable costs change when output changes (e.g.labor).

Accordingly, total short-run costs can be written as the sum of thevariable costs, cv (y) and the fixed costs, FC :

c(y) = cv (y) + FC


Average Costs

The average cost function measures the cost per unit of outputand can be written as follows:

AC (y) =c(y)

y=

cv (y)

y+

FC

y= AVC (y) + AFC (y)

where AVC (y) stands for average variable costs and AFC (y)stands for average fixed costs.

What do these functions look like?

Let’s consider the AFC (y) = FCy :

when y → 0 then AFC (y)→∞when y →∞ then AFC (y)→ 0

Thus, AFC (y) is a rectangular hyperbola


Average Fixed Costs Curve


Average variable costs

The average variable costs, AVC (y) = cv (y)y

As the output increases the average variable costs increase.

This is due to the presence of fixed factors which eventually bringinefficiencies in production.


Total average costs

The (total) average cost curve is the sum of the two previous curves:

Initially the average cost curve decreases because AFC are decreasing.

But as y increases AFC become smaller while AVC ′ increases, thuseventually AC increases as well

the level of output which minimizes the AC is called minimalefficient scale.Ivan Etzo (UNICA) Lecture 4: Cost Functions 8 / 22

Marginal costs

The marginal cost curve measures the change in costs for a givenchange in output.

MC (y) =dc(y)

dy=

c(y + dy)− c(y)

dy.

or equivalently,

MC (y) =dcv (y)

dy=

cv (y + dy)− cv (y)

dy

.because c(y) = cv (y) + FC

What is the relationship between the MC and the AVC curve?

The MC curve must lie below (above) the AVC curve when this oneis decreasing (increasing).


Marginal costs and average cost curveFormally

If the the AVC curve is decreasing then we must have that

d

dy

(cv (y)

y

)< 0

By applying the quotient rule for derivates,

d

dy

(cv (y)

y

)=

yc ′v (y)− cv (y)

y2< 0

yc ′v (y)− cv (y) < 0

c ′v (y) <cv (y)

y

This also implies that the MC curve must intersect the AVC at itsminimum point.


Marginal costs and average cost curves

Important points:The AVC curve may initially slope down due to increasing averageproducts of the variable factor for small output levels.the AC curve will initially slope down due to decreasing average fixedcosts.The AFC curve can be obtained as a difference between AC curve andAVC .The MC and AVC are the same at the first unit of output.The MC equals both the AC and the AVC at their minimum point.


Marginal costs and variable costs

The area underneath the MC curve until y is the variable cost ofproducing y units of output.

cv (y) =

∫ y ′

0

dcv (x)

dxdx = cv (y ′)− cv (0) = cv (y)


Cost curves: Examples with the Cobb-Douglas technology(Short-run)

Suppose that, in the short-run, the factor 2 is fixed at x̄2, then thecost minimization problem is the following:

minx1,x2

w1x1 + w2x̄2

such thatf (x1, x2) = xa1 x̄

b2

Solve the constraint for x1 and get

x1 = y1a x̄

ab

2

Thusc(w1,w2, x̄2) = y

1a x̄

ab

2 w1 + w2x̄2


Cost curves: Examples with the Cobb-Douglas technology(Short-run)

The short-run Cobb-Douglas cost functions are the following:

The Short-run costs: cs(y ,w1,w2, x̄2) = y1a x̄

ab

2 w1 + w2x̄2

The Short-run Average Costs (SAC): SAC (y) = y1−aa x̄

ab

2 w1 + w2x̄2

y

The Short-run Average Variable Costs (SAVC): SAVC (y) = y1−aa x̄

ab

2 w1

The Short-run Average Fixed Costs (SAFC): SAFC (y) = w2x̄2

y

The Short-run Marginal Costs SMC (y) = y1−aa x̄

ab

2w1

a


Long-run costs

By definition in the long-run all factors are variable, thus it will bealways possible to produce zero units of output at a zero costs.

Let’s consider x̄2 the optimal plant size for a firm that produces acertain level of output.

Accordingly, and keeping the factor prices fixed, the short-run costfunction is cs(y , x̄2).

Think of the firm adjusting the plant size to a different level ofoutput, it turns out that when the fixed factor becomes variable, thatis in the long run, it is a function of y.

Then, the Long-run cost function can be written as follows:

c(y) = cs(y , x̄2(y))

In other words, the minimum cost when all factors are variable (i.e.the long run cost function) is just the minimum cost when factor 2 isfixed at the level that minimizes the long-run costs.


Long-run and short-run costs

Let’s consider some level of output y∗, the optimal plant size toproduce y∗ is x̄∗2 = x̄2(y∗)

We know thatc(y∗) = cs(y∗, x̄∗2 )

Thus, in y∗ the long-run costs are equal to the short-run costs.

It turns out that the long-run cost to produce y cannot be greaterthan the short-run to produce the same level of output when factor 2is fixed, that is:

c(y) ≤ cs(y , x̄∗2 )

And, as a consequence it must be that:

AC (y) ≤ ACs(y , x̄∗2 )


Long-run and short-run costsgraphically


Long-run and short-run costsgraphically

The LAC curve is the lower envelope of the SAC curves.


The Long-run marginal costsThe economic intuition

The marginal cost measures the change in cost of production whenoutput changes

The long run marginal costs will consist of two components, namely:1 how costs change when the plant size (i.e. factor 2) is fixed2 how costs change when the plant size (i.e. factor 2) can be adjusted

Obviously, if the plant size is chosen optimally (i.e. if x̄2 = x̄∗2 ) thenthe second component is equal to zero!

Thus, at the optimal choice the long-run marginal costs and theshort-run marginal costs are equal.


The Long-run marginal costs = short-run marginal costsMathematical proof

We know that, by definition:

c(y) ≡ cs(y , x̄2(y))

Let’s indicate x̄2 with k . Differentiating with respect to y gives

dc(y)

y=

∂cs(y , k)

∂y+

∂cs(y , k)

∂k

∂k(y)

∂y.

When we evaluate this expression for the output level y∗ and the associatedoptimal level of k, that is k∗, then we know that :

∂cs(y∗, k∗)

∂k= 0

In fact, this is the necessary condition for k∗ to be the optimal level whichminimizes the costs when y = y∗. Thus, we are left with:

dc(y)

y=

∂cs(y , k)

∂y=⇒ LMC (y) ≡ SMC (y)


The Long-run marginal costs


The Long-run marginal costs


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Advanced Microeconomics - University of Cagliari