ECO 305 — Fall 2001 Precepts Week 3 — Questions This week we will see how economic statements about choices translate into the mathematics of budget constraints. We will also develop precise calculus formulations for the concept of (point) elasticity that was heuristically introduced in ECO 102, and a particular application of the concept. Once again, a solution handout will be available at the end of the precept. Question 1:

Write the mathematical formula, and draw a graph, for the budget constraints in the following two situations. (a) You get an allowance of $100 per week. You can spend it on two things: pizza, and talking to your friends back home on the phone. Each pizza costs $10. Your phone company offers two plans. The first has no fixed fee, and charges 5 cents per minute of use. The second has a fixed fee of $20 per month, plus 3 cents per minute of use. (b) You have the following campus job opportunities available. You can work as a research assistant for a professor; this job requires 10 hours of work per week and pays $15 per hour. You can also work in the campus dining hall at $10 per hour, and can choose your number of hours there. You are not allowed to work for money more than 20 hours per week total, although you may work more as a volunteer for some causes. Question 2:

In economics we often want to compare proportional changes in two variables. If y = f (x), and a change ∆x in x leads to a change ∆y in y, that is, y + ∆y = f (x + ∆x), then the elasticity of y with respect to x is defined as ey = lim ∆y / y = x lim ∆y = x dy ex ∆x→0 ∆x / x y ∆x→0 ∆x y dx Define Y = ln(y) and X = ln(x), then

Y = ln(f (eX ))

and, differentiating using the Chain Rule, dY = 1 f  (eX ) eX = 1 f  (x) x = x dy dX f (eX ) f (x ) y dx so the elasticity can be thought of as a “double-log derivative” ey d ln(y) ex = d ln(x) The purpose here is to formulate a “calculus of elasticities”: (1) If y = A xk where A > 0 and k are constants, what is the elasticity of y with respect to x? What class of functions has a constant elasticity? (2) Show that the elasticity of the product of two functions is the sum of their elasticities. (3) Find an expression for the elasticity of the sum of two functions. (4) If z is a function of y and y is a function of x, prove the “chain rule”:

ez ez ey ex = ey ex .

1

Question 3:

Suppose a consumer’s utility function is 
U (X, Y ) = kx (X )β + ky (Y )β 1/β

where kx, ky > 0, β < 1 are algebraic constants (parameters). He maximizes this subject to the budget constraint Px X + P y Y ≤ M . Show that the optimum quantities satisfy the equation

X =  kx Py 1/(1−β) Y ky P x

The elasticity of the quantity-ratio on the left hand side, when regarded as a function of the price ratio, is called the elasticity of substitution in consumption, and often denoted by σ . For this function, find an expression for σ. Show that as Px /Py increases, the ratio of the expenditures on the two goods, namely (Px X )/(Py Y ), increases if σ < 1 and decreases if σ > 1.

2

ECO 305 — Fall 2003 Precepts Week 3 — Solutions Questsion 1: Examples of Budget Constraints

It is easier to draw the picture first and then figure out the formula. (a) Here you should draw the budget line corresponding to each plan separately first, and then take the better of the two possibilities. The picture is shown below. Write M for the number of minutes on the phone and P for the number of pizzas. (The picture allows fractional pizzas; you might be sharing them with friends, or averaging over several months.) The two lines intersect at M = 1000, P = 5. Therefore the algebraic statement of the budget constaint is 0.05 M + 10 P = 100 0.03 M + 10 P = 80

if M < 1000 and P > 5 ; if M ≥ 1000 and P ≤ 5 .

(a) P

(b)

I 250

10 8

150

5

100

M 1000

2000

H 10

2667

20

(b) This is made a little tricky because of the order of choice. If you want to work less than 10 hours, you have to take the dining hall job. If you want to work exactly 10 hours, you prefer to take the research assistant job. If you want to work between 10 and 20 hours, you do 10 on the research assistant job and the extra on the dining hall job. You are not allowed to earn any more by working more than 20 hours. Once this is clear the rest is easy. The picture is shown above. At H = 10, I have used the open bullet to signify an excluded point (I = 100) and the filled bullet an included point (I = 150). This is not standard notation, so if you use something like this, explain it. Writing H for hours and I for income, the equation is

 10 H  I =  150 + 10 (H − 10) = 50 + 10 H 250

Question 2 — The Calculus of Elasticities

(1) Power functions have constant elasticity — If y = A xk , then

ey = x A k xk−1 = k ex y

or ln(y) = ln(A) + k ln(x) so

d ln(y) d ln(x) = k 1

if H < 10 if 10 ≤ H < 20 if 20 ≤ H

Conversely, if the elasticity ey/ex = k, a constant, what must be the form of the function y = f (x)? Write this out as x dy = k. y dx

Treat this as a differential equation and solve it by separating the variables:

dy = k dx , y x

therefore where c is a constant of integration. Then

ln(y) = k ln(x) + c ,

y = ec x k = A x k , where A = ec . A simpler way is to write Y = ln(y) and X = ln(x), and dY/dX = k. Then Y = k X + c or ln(y) = k ln(x) + c, so y = xk ec . (2) The elasticity of a product of functions is the sum of their elasticities — If y and z are functions of x, then e(yz) = x d(yz ) = x  z dy + y dz 

ex

More simply, ln(yz) = ln(y) + ln(z ), so

yz dx yz dx dx dy x dz ey ez x = y dx + z dx = ex + ex d ln(yz) = d ln(y) + d ln(z ) d ln(x) d ln(x) d ln(x)

(3) The elasticity of a sum of two functions is a share-weighted average of their elasticities: e(y + z ) = x d ( y + z ) ex y + z  dx dy + dz  = y +x z dx dx

x dy + z x dz y dx y + z z dx ey + z ez ex y + z ex

= y +y z = y +y z

(4) The Chain Rule:

ez = x ex z x = z 

dz dx dz dy by chain rule for derivatives dy dx   x dy  y dz = z dy y dx ey ez = ey ex You can also do this by defining X = ln(x) etc. and using the ordinary calculus chain rule. 2

Question 3:

We are going to need to find ∂U/∂X and ∂U/∂Y . This can get messy, but a transformation simplifies it a little. Raise everything to the power β :

U β = k x ( x )β + ky ( Y ) β Now differentiate:

β U β−1 ∂U/∂X = kx β (X )β−1

Similarly with respect to Y . Then we can find the marginal rate of substitution

 k x   X β − 1 ∂U/∂X MRSyx = ∂U/∂Y = k Y y β < 1 ensures: (1) diminishing MRS; (2) no boundary optima, because X → 0 implies ∂U/∂X → ∞ At the optimum, MRSyx = Px /Py . Therefore X =  kx Py 1/(1−β) Y ky P x

Note that as Px /Py goes up, X/Y goes down — the consumer susbstitutes away from the good whose relative price has risen and toward the good whose relative price has fallen. The elasticity of substitution σ gives us a precise quantitative measure of the ease with which such substitution can be made for a given set of preferences. For the example here, by the formula in Question 2 (1) above, we have

σ = 1/(1 − β ) constant. Therefore the utility function is called the CES (constant elasticity of substitution) utility function. Special cases: (1) Perfect substitution — Straight line indifference curves or lLinear utility: β → 1, σ → ∞. (2) Zero substitution; L-shaped indifference curves: β → −∞, σ → 0. It can be shown that the utility function is then of the form U (X, Y ) = min(X, Y ). (3) When β = 0, σ = 1. It can be shown that the above utility function then tends to the Cobb-Douglas form: kx ln(X ) + ky ln(Y ). The ratio of the expenditures is

P x X =  kx P y  σ P x Py Y k P P  kxy σx Px y1−σ = k Py y

If σ < 1, the power 1 − σ on the RHS of the final expression is negative. In that case, as Px /Py goes up, the positive power also goes up. The general idea is that it is not easy to substitute one of the goods for the other; as the relative price of X goes up, its relative quantity cannot go down very fast, and the relative expenditure on it has to go up. If σ > 1, substitution is more easy, and is made to such an extent that when the relative price of a good goes up, the relative expenditure on it goes down.

3