Lecture Notes on Elasticity of Substitution

Ted Bergstrom, UCSB Economics 210A

March 3, 2011

Today's featured guest is \the elasticity of substitution."

Elasticity of a function of a single variable

Before we meet this guest, let us spend a bit of time with a slightly simpler notion, the elasticity of a a function of a single variable. Wherefis a dierentiable real-valued function of a single variable, we dene the elasticity off(x) with respect tox(at the pointx) to be (x) =xf0(x)f(x):(1)

Another way of writing the same expression 1 is

(x) =xdf(x)dx f(x)=df(x)f(x)dx x :(2) From Expression 2, we see that the elasticity of off(x) with respect toxis the ratio of the percent change inf(x) to the corresponding percent change inx. Measuring the responsiveness of a dependent variable to an independent variable in percentage terms rather than simply as the derivative of the func- tion has the attractive feature that this measure is invariant to the units in which the independent and the dependent variable are measured. For exam- ple, economists typically express responsiveness of demand for a good to its price by an elasticity.

1In this case, the percentage change in quantity is the1

Some economists nd it tiresome to talk about negative elasticities and choose to dene the price-elasticity as the absolute value of the percentage responsiveness of quantity to price. 1 same whether quantity is measured in tons or in ounces and the percentage change in price is the same whether price is measured in dollars, Euros, or farthings. Thus the price elasticity is a \unit-free" measure. For similar rea- sons, engineers measure the stretchability of a material by an \elasticity" of the length of the material with respect to the force exerted on it. The elasticity of the functionfat a point ofxcan also be thought of as the slope of a graph that plots lnxon the horizontal axis and lnf(x) on the vertical axis. That is, suppose that we make the change of variablesu= lnx andv= lnyand we rewrite the equationy=f(x) asev=f(eu). Taking derivatives of both sides of this equation with respect touand applying the chain rule, we have e vdvdu =euf0(eu) (3) and hence dvdu =euf0(eu)e v=xf0(x)f(x)=(x);(4) where the second equality in Expression 4 is true becauseeu=xandev= f(x). Thusdvdu is the derivative of lnf(x) with respect to lnx. We sometimes express this by saying that (x) =dlnf(x)dlnx:(5) It is interesting to consider the special case where the elasticity off(x) with respect toxis a constant,that does not dependent onx. In this case, integrating both sides of Equation 5, we have lnf(x) =lnx+a(6) for some constanta. Exponentiating both sides of Equation 6, we have f(x) =cx(7) wherec=ea. Thus we see thatfhas constant elasticityif and only iffis a \power function" of the form 7. In general, the elasticity offwith respect toxdepends on the value of x. For example iff(x) =abx, then(x) =bxabx. In this case, asxranges from 0 toa=b,(x) ranges from 0 to1. 2

Elasticity of inverse functions

Another useful fact about elasticities is the following. Suppose that the functionfis either strictly increasing or strictly decreasing. Then there is a well dened inverse function, dened so that such that(y) =xif and only iff(x) =y. It turns out that if(x) is the elasticity off(x) with respect to x, then 1=(x) is the elasticity of(y) with respect toy. Proof.Note that since the functionis the inverse off, we must have (f(x)) =x. Using the chain rule to dierentiate both sides of this equation with respect tox, we see that ify=f(x), then0(y)f0(x) = 1 and hence

0(y) = 1=f0(x). Therefore wheny=f(x), the elasticity of(y) with respect

toyis y0(y)(y)=f(x)(y)f0(x): But sincey=f(x), it must be that(y) =x, and so we have y

0(y)(y)=f(x)xf

0(x)=1(x):Application to monopolist's revenue function

One of the most common applications of the notion of elasticity of demand is to monopoly theory, where a monopolist is selling a good and the quantity of the good that is demanded is a functionD(p) of the monopolist's pricep. The monopolist's revenue isR(p) =pD(p). Does the monopolist's revenue increase of decrease if he increases his price and how is this related to the price elasticity? We note thatR(p) is increasing (decreasing) in price if and only if lnR(p) increases (decreases) as the log of price increases. But lnR(p) = lnp+ lnD(p). Then dlnR(p)dlnp= 1 +dlnD(p)dlnp= 1 +(p) where(p) is the price elasticity of demand. So revenue is an increasing function ofpif(p)>1 and a decreasing function ofpif <1. In the former case we say demand is inelastic and in the latter case we say demand is elastic. 3

Elasticity of substitution

Now we introduce today's main event{ the elasticity of substitution for a func- tion of two variables. The elasticity of substitution is most often discussed in the context of production functions, but is also very useful for describing util- ity functions. A rm uses two inputs (aka factors of production) to produce a single output. Total outputyis given by a concave, twice dierentiable functiony=f(x1;x2). Letfi(x1;x2) denote the partial derivative (marginal product) offwith respect toxi. While the elasticity of a function of a sin- gle variable measures the percentage response of a dependent variable to a percentage change in the independent variable, the elasticity of substitution between two factor inputs measures the percentage response of the relative marginal products of the two factors to a percentage change in the ratio of their quantities. The elasticity of substitution between any two factors can be dened for any concave production function of several variables. But for our rst crack at the story it is helpful to consider the case where there are just two inputs and the production function is homogeneous of some degreek >0. We also assume that the production function is dierentiable and strictly quasi-concave. Fact 1.Iff(x1;x2)is homogeneous of some degreekand strictly quasi- concave, then the ratio of the marginal products of the two factors is deter- mined by the ratiox1=x2andf1(x1;x2)=f2(x1;x2)is a decreasing function of x 1=x2. Proof.Iffis homogeneous of degree k, then the partial derivatives offare homogeneous of degreek1.2Therefore f

1(x1;x2) =xk12f1x1x

2;1 and f

2(x1;x2) =xk12f2x1x

2;1

It follows that

1(x1;x2)f

2(x1;x2)=xk12f1x1x

2;1x k12f2x1x

2;1(8)2

To prove this, note that iffis homogeneous of degreek, thenf(x) =kf(x). Dierentiate both sides of this equation with respect toxiand arrange terms to show that f i(x) =k1fi(x). 4 f 1x1x 2;1f 2x1x

2;1:(9)

Therefore the ratio of marginal products is determined by the ratiox1=x2.

Let us dene this ratio as

g x1x 2 =f1(x1;x2)f

2(x1;x2):

Since strict quasi-concavity implies diminishing marginal rate of substitution,

it must be thatgis a strictly decreasing function ofx1=x2.Sincegis strictly decreasing, it must be that the functionghas a well-

dened inverse function. Let's call this inverse functionh. Let prices of the two inputs be given by the vectorp= (p1;p2). Suppose that the rm always chooses factors so as to minimize its costs, conditional on its output level. Then it must be that at pricesp, the rm uses factors in the ratiox1=x2such that g(x1=x2) =f 1x1x 2;1f 2x1x

2;1=p1p

2 or equivalently such thatquotesdbs_dbs2.pdfusesText_2

[PDF] Lecture Notes on Elasticity of Substitution - New York University

Graphique 412 : l’élasticité de substitution

Modèle d’estimation de l’élasticité de substitution et du