LECTURE NOTE 4
THE CONSUMER CHOICE PROBLEM
W & L INTERMEDIATE MICROECONOMICS
PROFESSOR A. JOSEPH GUSE
1. Introduction
In previous notes, we introduced the primary building blocks of the consumer choice problem - budget
sets, preferences, and the represetation of preference by utility functions. In this note we combine these
elements to solve the consumers problem. Recall that informally, the consumer’s problem (as envisioned by
the economist) is to make oneself as well of as one can. Formally we might write the problem as
max u(x1 , x2 )
x1 ,x2
s.t. p1 x1 + p2 x2 ≤ m
This is the consumer’s problem when there are only two goods whose quantities must be decided by
the consumer and where the income and constant prices are given exogenously. Remember that exogenous
variables in the problem like prices and income are parameters to the problem that we imagine the consumer
having no control over. The consumer only has control over x1 and x2 . These are the choice variables and
the target of our search. In other words, the solution to this problem will be a pair of quantities x1 (p1 , p2 , m)
and x2 (p1 , p2 , m) that make u as large as it can be without violating the budget constraint.
Note that the solutions will actually be functions of the parameters. That is, the utility maximizing
choices of x1 and x2 will potentially (and typically) depend on prices and income levels. The the pair of
functions that solves the problem are called variously demand functions or demand equations.
2. Two Approaches for Solving Consumer Problem for Very Nice Preferences
2.1. The Two Equation Approach. This method involves setting up and solving a system of two equations - the budget line equation and the MRS=MRT equation to which we will often refer as the First Order
Condition (FOC).
1
Budget Line Equation. The first equation is simply the budget line equation. Since monotonicity is part
of niceness, the optimal choice for anyone with nice preferences will always be on the budget line. (allowing
us to convert the inequality in the constraint of the problem (≤) to an equality (=). In other words, the
consumer should optimally choose to spend all of of her income on consumption. (Recall that our simple
two-good model does not have time and therefore no opportunity to borrow and no advantage in saving.)
1In general, first order conditions reflect the notion that first derivatives will reach a certain value at optimum points. The
simplest example is that a the first derivative of a real-valued function of one variable will be equal to zero at either a local
minimum or local maximum point. Under the two-equation approach described here, we expect the ratio of two derivatives to
be equal to something (the price ratio).
1
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W & L INTERMEDIATE MICROECONOMICS PROFESSOR A. JOSEPH GUSE
m
pz
(e)
Various Indifference Curves
Pizza
(slices)
(c)
(b)
(a)
Budget Line
(d)
Beer
(pints)
m
pb
Figure 1. Bundles (a), (b), and (c) all satisfy M RS = M RT . Bundles (e), (b) and (d) all satisfy
the budget line equation. Only bundle (b) satisfies both. The red arrows on the budget line indicate
which direction is “up” (the direction of more preferred bundles).
pb x b + pz x z = m
MRS = MRT. The second equation describes all the bundles where the MRS is equal to the MRT (i.e. the
rate at which one good must be traded off for the other, typically determined by the ration of their market
prices). In the most straight-forward cases (e.g. very nice preferences), this condition will take the form
M RSb,z ≡
∂u(xb , xz )/∂xb
pb
=
∂u(xb , xz )/∂xz
pz
The object on the left is the MRS, in this case the (maximum) rate at which the consumer whose preferences
are being represented by u is willing to give up pizza for beer that would leave the consumer equally well
off (otherwise they shouldn’t be willing). Meanwhile the object on the right (MRT) is amount of pizza the
consumer must give up pizza for beer from any point on the interior of the budget line. In other words, when
evaluated at a consumption budle, the MRS function represents the slope of the indifferent going through
that bundle at that bundle.
pb
pz ,
on the other hand, measures the slope of the budget line.
2
2Some people find it easier to interpret this equation by re-arranging it as follows
∂u(xb , xz )/∂xb
∂u(xb , xz )/∂xz
=
pb
pz
When written like this, a natural interpretation is that the optimal choice must be a bundle at which the marginal utility per
dollar of spending is the same for both goods. In other words if the consumer could gain more utility for a dollar of spending
on pizza than they would lose by taking a dollar of spending away from beer, then they should do that.
LECTURE NOTE 4
THE CONSUMER CHOICE PROBLEM
3
Why is this a condition that the optimal choice must meet? To convince your self that it is, imagine
the consumer choosing a bundle on the budget where the condition did not hold. Bundle (e) in Figure 1 is
a point where the indifference curve is “steeper” than the budget line which is another way of saying that
M RSb,z (e) >
pb
pz .
In english words, we interpret the situation at (e) to be “The consumer is willing to give
up more pizza for beer than she has to”. This means that she would be better off taking a trade being
offered in the market in which she get a little more beer and only gives up the pizza required. In Figure 1,
this idea is emphasized by a red arrow at bundle (e) pointing “uphill” along the budget line (the direction
of improvment). Since we have the opposite situation at bundles like (d) (i.e. M RSb,z (d) <
pb
pz ),
the arrow
points in the other direction. On the other hand when the condition is met with equality as it is at bundle
(b) (i.e. M RSb,z (b) =
pb
pz ),
then we are at the highest point on the budget line.
Figure 1 illustrates graphically what it means for a bundle to satisfy one or both of these equations. Note
that in general many bundles can satisfy either equation. For example points (a), (b) and (c) in the figure are
all bundles where the slope of the indifference curve going through them is equal to the slope of the budget
line (even though they may not all lie on the budget line). However, exactly one bundle in the picture bundle (b) - satisfies both equations and that bundle is the soluction to the optimal choice problem. This
will always be the case as long as preferences are very nice.
Example See appendix of Chap 5.
2.2. Single Equation Approach. With the single equation approach we incorporate the budget line equation into the utlity function which converts the CMP from a two-dimensional choice problem with a constraint
to a one-dimensional problem without constraints (aside from non-negative constraints on the quantities of
all goods).
Orignal Problem:
max u(xb , xz ) s.t. pb xb + pz xz ≤ m
xb ,xz
New Problem:
max u(xb ,
xb
m − pb x b
) s.t. (nothing)
pz
Note that we removed xz as a choice. In the new problem we imagine the consumer only choosing, xb the
quantity of beer, while the quantity of the pizza is imiplicitly determine by however much can be afforded
after the beer decision is made. Rearranging the budget line equation gives us xz =
m−pb xb
.
pz
This method
just takes the right-hand of that re-arranged equation and plugs it into the utility function in place of xz .
Figure 2
3. Very Nice, Nice and Not so Nice
• Very Nice Under what conditions can one set up the equations described above and expect to find
a unique solution and call that solution the optimal choice? The answer is under the condition of
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W & L INTERMEDIATE MICROECONOMICS PROFESSOR A. JOSEPH GUSE
”Utility”
u∗ = u(b)
u(d) = u(e)
0
(e)
m
pz
(b)
Beer
Pizza
(d)
m
pb
0
”Utility”
v ∗ = f (u∗) = v(b)
v(e) = v(d)
0
m
pz
(e)
(b)
Beer
Pizza
(d)
m
pb
0
Figure 2. Slice of Utility Surfaces Along the Budget Line. This illustrate the one-equation
approach. Beer increases along the horizontal axis while pizza decreases. Expenditure level is fixed
at m. Upper panel shows utilities levels as represented by u. Lower panels shows utility levels as
represented by v - a monotonic transformation of u.
very nice preferences. Recall that, in addition to being rational, “nice” preferences are continuous3,
monotonic and convex. This level of niceness is enough to guarantee the existence of a unique
solution to the CMP, but it is not enough to guarantee that the first order condition (in either the
two or one-equation approaches) will indentify it. In order to be assured that the F.O.C. identifies
the unique solution we need an additional set of conditions.
lim M RSb,z = 0 ⇐⇒ lim M RSz,b = ∞
xz →0
xz →0
lim M RSz,b = 0 ⇐⇒ lim M RSb,z = ∞
xb →0
xb →0
The first one says that the rate at which the consumer is willing to give up pizza for additional
beer goes to zero as the quantity of pizza goes to zero. In other words, as one evaluate bundles
3Continuity means that all the upper contour sets generated by a preference relation are closed and it ensures the existence
of a utility function that represents it.
LECTURE NOTE 4
THE CONSUMER CHOICE PROBLEM
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with less and less pizza the consumer is not only willing to give up less for an additional beer (that
along would be diminishing MRS, eventually we wouldn’t agree to give up pizza for any amount of
beer. The condition on the second line says the same thing but with the roles of beer and pizza
reversed. Graphically, the first of these additional “extra nice” conditions on preferences mean
that the consumer’s indifference curves never cross beer axis, while the second one implies that her
indifference curves never cross the pizza axis.
Another extra nice condition that must hold in order for the the F.O.C. to uniquely identify the
solution is that the F.O.C. itself exists. This requires that u is everywhere differentiable w.r.t. both
goods.
• Nice with Corners Corner Solutions are possible if preference are merely nice but do not satisfy the
MRS limit conditions (above).
• Nice but Kinky Can’t set up F.O.C. Need to be clever.
• Non-convexity The FOC may have many solutions or FOX may identify a utility-minimizing bundle.
• Violations of Monotonicity Solution may not lie on the budget line or can’t frequently lead to corner
solutions.