Advanced Microeconomics

Ivan Etzo

University of Cagliari [email protected] Dottorato in Scienze Economiche e Aziendali, XXXIII ciclo

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

1 Short-run Cost functions Total Costs Average Costs Marginal costs

2 Long-run costs Long Average Costs Long Marginal Costs

Ivan Etzo (UNICA) Lecture 4: Cost Functions 2 / 22 Short-run Cost functions

The cost function measures the minimum cost of producing a given level 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) costs Average fixed costs Average variable costs Marginal costs

Cost functions are important in studying the determination of optimal output choices. How are these cost functions related to each other? How are a firm’s long-run and short-run cost functions related?

Ivan Etzo (UNICA) Lecture 4: Cost Functions 3 / 22 Total Costs

Consider the cost function c(w1, w2, y) and let’s consider the factor prices fixed, so that the cost function is a function of the output alone, c(y). In the short-run there are some costs that are fixed, that is they do not 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 the variable costs, cv (y) and the fixed costs, FC:

c(y) = cv (y) + FC

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

The average cost function measures the cost per unit of output and can be written as follows: c(y) c (y) FC AC(y) = = v + = AVC(y) + AFC(y) y y y

where AVC(y) stands for average variable costs and AFC(y) stands for average fixed costs. What do these functions look like?

FC Let’s consider the AFC(y) = y : when y → 0 then AFC(y) → ∞ when y → ∞ then AFC(y) → 0 Thus, AFC(y) is a rectangular hyperbola

Ivan Etzo (UNICA) Lecture 4: Cost Functions 5 / 22 Average Fixed Costs Curve

Ivan Etzo (UNICA) Lecture 4: Cost Functions 6 / 22 Average variable costs

cv (y) The average variable costs, AVC(y) = y As the output increases the average variable costs increase. This is due to the presence of fixed factors which eventually bring inefficiencies in production.

Ivan Etzo (UNICA) Lecture 4: Cost Functions 7 / 22 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 0 increases, thus eventually AC increases as well the level of output which minimizes the AC is called minimal efficient scale.

Ivan Etzo (UNICA) Lecture 4: Cost Functions 8 / 22 Marginal costs

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

dc(y) c(y + dy) − c(y) MC(y) = = . dy dy or equivalently,

dc (y) c (y + dy) − c (y) MC(y) = v = v v dy 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 one is decreasing (increasing).

Ivan Etzo (UNICA) Lecture 4: Cost Functions 9 / 22 Marginal costs and average cost curve Formally

If the the AVC curve is decreasing then we must have that d c (y) v < 0 dy y By applying the quotient rule for derivates, d c (y) yc0 (y) − c (y) v = v v < 0 dy y y 2

0 ycv (y) − cv (y) < 0 c (y) c0 (y) < v v y

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

Ivan Etzo (UNICA) Lecture 4: Cost Functions 10 / 22 Marginal costs and average cost curves

Important points: The AVC curve may initially slope down due to increasing average products of the variable factor for small output levels. the AC curve will initially slope down due to decreasing average fixed costs. The AFC curve can be obtained as a difference between AC curve and AVC. 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. Ivan Etzo (UNICA) Lecture 4: Cost Functions 11 / 22 Marginal costs and variable costs

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

Z y 0 dcv (x) 0 cv (y) = dx = cv (y ) − cv (0) = cv (y) 0 dx

Ivan Etzo (UNICA) Lecture 4: Cost Functions 12 / 22 Cost curves: Examples with the Cobb-Douglas technology (Short-run)

Suppose that, in the short-run, the factor 2 is fixed atx ¯2, then the cost minimization problem is the following:

min w1x1 + w2x¯2 x1,x2

such that a b f (x1, x2) = x1 x¯2

Solve the constraint for x1 and get

1 a a b x1 = y x¯2

Thus 1 a a b c(w1, w2, x¯2) = y x¯2 w1 + w2x¯2

Ivan Etzo (UNICA) Lecture 4: Cost Functions 13 / 22 Cost curves: Examples with the Cobb-Douglas technology (Short-run)

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

1 a b The Short-run costs: cs (y, w1, w2, x¯2) = y a x¯2 w1 + w2x¯2

1−a a a b w2x¯2 The Short-run Average Costs (SAC): SAC(y) = y x¯2 w1 + y

1−a a b The Short-run Average Variable Costs (SAVC): SAVC(y) = y a x¯2 w1

w2x¯2 The Short-run Average Fixed Costs (SAFC): SAFC(y) = y

1−a a a b w1 The Short-run Marginal Costs SMC(y) = y x¯2 a

Ivan Etzo (UNICA) Lecture 4: Cost Functions 14 / 22 Long-run costs

By definition in the long-run all factors are variable, thus it will be always possible to produce zero units of output at a zero costs. Let’s considerx ¯2 the optimal plant size for a firm that produces a certain level of output. Accordingly, and keeping the factor prices fixed, the short-run cost function is cs (y, x¯2). Think of the firm adjusting the plant size to a different level of output, it turns out that when the fixed factor becomes variable, that is 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, x¯2(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 is fixed at the level that minimizes the long-run costs.

Ivan Etzo (UNICA) Lecture 4: Cost Functions 15 / 22 Long-run and short-run costs

Let’s consider some level of output y ∗, the optimal plant size to ∗ ∗ ∗ produce y isx ¯2 =x ¯2(y ) We know that ∗ ∗ ∗ c(y ) = cs (y , x¯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 greater than the short-run to produce the same level of output when factor 2 is fixed, that is: ∗ c(y) ≤ cs (y, x¯2 )

And, as a consequence it must be that:

∗ AC(y) ≤ ACs (y, x¯2 )

Ivan Etzo (UNICA) Lecture 4: Cost Functions 16 / 22 Long-run and short-run costs graphically

Ivan Etzo (UNICA) Lecture 4: Cost Functions 17 / 22 Long-run and short-run costs graphically

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

Ivan Etzo (UNICA) Lecture 4: Cost Functions 18 / 22 The Long-run marginal costs The economic intuition

The marginal cost measures the change in cost of production when output 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 fixed 2 how costs change when the plant size (i.e. factor 2) can be adjusted ∗ Obviously, if the plant size is chosen optimally (i.e. ifx ¯2 =x ¯2 ) then the second component is equal to zero! Thus, at the optimal choice the long-run marginal costs and the short-run marginal costs are equal.

Ivan Etzo (UNICA) Lecture 4: Cost Functions 19 / 22 The Long-run marginal costs = short-run marginal costs Mathematical proof

We know that, by definition:

c(y) ≡ cs (y, x¯2(y))

Let’s indicatex ¯2 with k. Differentiating with respect to y gives dc(y) ∂c (y, k) ∂c (y, k) ∂k(y) = s + s . y ∂y ∂k ∂y

When we evaluate this expression for the output level y ∗ and the associated optimal level of k, that is k∗, then we know that : ∂c (y ∗, k∗) s = 0 ∂k

In fact, this is the necessary condition for k∗ to be the optimal level which minimizes the costs when y = y ∗. Thus, we are left with: dc(y) ∂c (y, k) = s =⇒ LMC(y) ≡ SMC(y) y ∂y Ivan Etzo (UNICA) Lecture 4: Cost Functions 20 / 22 The Long-run marginal costs

Ivan Etzo (UNICA) Lecture 4: Cost Functions 21 / 22 The Long-run marginal costs

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