Daniel l. Rubinfeld

144 Part:2 Producers, Consumers, and Competitive Markets
4 Individual and Market Demand 45
In §2.3, we explain that the
cross-price elasticity of
demand refers to the percentage
change in the quantity
demanded of one good that
results from a 1-percent
increase in the price of
another good.
duality Alternative way of
looking at the consumer's utility
maximization decision:
Rather than choosing the
highest indifference curve,
given a budget constraint, the
consumer chooses the lowest
budget line that touches a
given indifference curve
In this example, the demand for each good depends only on the price of that
good and on income, not on the price of the other good. Thus, the cross-price
elasticities of demand are O.
We can also use this example to review the meaning of Lagrange multipliers.
To do so, let's substitute specific values for each of the parameters in the problem.
Let Il = 1/2, P s $1, Py = $2, and I = $100. In this case, the choices that maximize
utility are X 50 and Y = 25. Also note that A = 1/100. The Lagrange
multiplier tells us that if an additional dollar of income ,vere available to the
consumer, the level of utility achieved would increase by 1/100. This conclusion
is relatively easy to check. With an income of $101, the maximizing choices of the
two goods are X = 50.5 and Y = 25.25. A bit of arithmetic tells us that the originallevel
of utility is 3565 and the new le\-el of utility 3575. As we can see, the
additional dollar of income has indeed increased utility by .01, or 1/100.
Duality in Consumer
There are two different vvays of looking at the consumer's optimization decision.
The optimum choice of X and Y can be analyzed not only as the problem of
choosing the highest indifference curve-the maximum value of Ll( )-that
touches the budget line, but also as the problem of choosing the lowest budget
line-the minimum budget expenditure-that touches a given indifference
curve. We use the term duality to refer to these two perspectives. To see how this
principle works, consider the follmving dual consumer optimization problem:
the problem of minimizing the cost of achieving a particular level of utility:
subject to the conshaint that
Minimize PxX + PyY
Ll(X,Y) = Ll*
The corresponding Lagrangian is given by
(P PxX + PyY - p.(Ll(X,Y) Ll*) (A4.15)
where p. is the Lagrange multiplier. Differentiating
Thus the total change in the quantity demanded of X resulting from a unit
change in P x is
In § .. L2, the effect of a price
change is divided into an
income effect and a substitution
effect
p. = [Ps/MUs(X,Y)] = [Py/MU,(X,Y)] = 1/A
dX/dPs = (Jx/aPSIU~Ll' + (ax/aI)(aI/aPs ) (A4.17)
Because it is also true that
MUs(X,Y)/MUy(X,Y) = MRS.\) = Ps/ Py
the cost-minimizing choice of X and Y must occur at the point of tangency of the
budget line and the indifference curve that generates utility Ll*. Because this is
The first term on the right side of equation (A4.l7) is the substihltion effect (because
utility is fixed); the second term is the income effect (because income increases).
From the consumer's budget conshaint, I = PxX + P,Y, ,,\'e know by differentiation
that
(A4.18)