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Consumer Theory^ Lecture 2^ Reading: Perlo¤ Chapter 3^ August 2015
Introduction 1 / 70
In this chapter, we formalize the concept of consumer choice. We see how an individual, with certain preferences, makes decisionswhen faced with a constraint. It is the foundation of the demand curve we saw yesterday.
Outline^ Preferences
- Assumptions we make about how individuals rankdi¤erent options. Utility - A convenient way to summarize preferences. Budget Constraint^ - What can people a¤ord? Constrained Consumer Choice
- Given income, prices and preferences, what is the best choice I can make? Behavioral Economics - Do individuals behave in reality how we have assumed?
Preferences 3 / 70
Individuals prefer certain sets of goods to others In order to formalize this abstract concept in a useful way, we need tomake a few assumptions about these preferences
Preferences^ But
rst lets go over some notation.^ a^ bundle^ (or basket) is simply of a combination of goods and services.^ bundle^ a^ is two haircuts and a can of Pepsi.^ bundle^ b^ is four cans of Pepsi and no haircuts.
Preferences 5 / 70
Consumers rank bundles in terms of desirability. a^ %^ b.^ I like bundle
a at least^ as much as
b.^ I^ weakly prefer a
to^ b
a^ ^ b. I de nitely like
a^ more than^ b.^ I
strictly prefer a^ to
b
a^ ^ b.^ I like the bundles equally.
I am^ indi¤erent^
between^ a^ and^ b.^ 6 / 70
Preferences^ We make three assumptions about preferences.
If the rst two are met, an individual is considered to be rational.
Completeness When facing bundles
a^ and^ b, the consumer must be able to say either^ a^ %^ b,^ b^ %
a,^ or^ a^ ^ b. In other words, the consumer can rank any two bundles.
Rules out
the possibility that the consumer cant make a decision.
Preferences 7 / 70
Transitivity Here we just mean that preferences are logically consistent. If^ a^ %^ b^ and^ b^ %^ c, by the transitive property
a^ %^ c
8 / 70
Utility^ For ordinal, the ordering is all that matters.^ For cardinal, the order AND the magnitude matter.^ Bob did better on Sam on the test is an ordinal ranking^ Bob got a 97% and Sam got a 15% is a cardinal ranking.
Utility 13 / 70
Lets graph^ U^ = (q
(^12) q) (^12) This is not incredibly useful.
But we can turn this into two dimensions.
14 / 70
Utility^ An^ indi¤erence curve
shows all the bundles that the consumer sees as equally desirable. Lets x a level of utility at
U^ and draw all the combinations of of
q^1
and^ qthat give the same level of utility.^2 For^ U^ =^10 ,^ the equation for the indi¤erence curve would be^100 q=^.^2 q^1
Utility 15 / 70
Any combination on this indi¤erence curve will give us a utility of 10
We are indi¤erent between any basket on this curve. There is an indi¤erence curve associated with each level of utility. An^ indi¤erence map
shows all the indi¤erence curves. As we move from the origin, utility increases.
16 / 70
Utility^ Here, baskets A and B give us the same level of utility because theyare on the same curve.^ Basket C gives us a higher utility because it is on a higher indi¤erencecurve.^ c^ ^ a,^ c^ ^ b
,^ b^ ^ a.
Utility 17 / 70
EXAMPLE
Suppose our utility function is
U^ =^2 q+^3 q^1 Lets draw an indi¤erence curve for
U^ =^ 2 and^ U^ =^ 3.
Utility
Properties of Indi¤erence Curves 1. Bundles on indi¤erent curves farther from the origin are preferred This comes from the more is better principle
Utility 19 / 70
Properties of Indi¤erence Curves
2.^ There is an indi¤erence curve through every possible bundle^ This comes from the completeness property^ The consumer is able to rank any combination of bundles
Utility
EXAMPLE
What would an indi¤erence curve look like for two goods where youget satiated?^ That is, you only want to consume so much of thegood then you start getting sick of it.
Utility 25 / 70
It is useful to know how much of one good the consumer is willing totrade for another. This is captured by the slope of the indi¤erence curve... and this iscalled the^ marginal rate of substitution
(MRS^ )
Utility^ The^ MRS^ tells us the maximum amount of one good a consumer willsacri
ce to get one more of the other^ If your^ MRS^
is -2 you would be willing to give up two burritos for onepiece of pizza. It is simply the slope of the indi¤erence curve.
Utility 27 / 70
Utility^ To
nd the MRS, we need to understand
marginal utility. The marginal utility is the extra utility you get from increasingsomething by an in nitesimally small amount.
Utility 29 / 70
The marginal utility of
∂ U qis just. 1 ∂ q^1 The marginal rate of substitution is the negative ratio of marginalutilities
∂ U ∂ q^1 MRS = ∂ U ∂ q^2 What is the intuition behind this?
Utility
EXAMPLE
What is the^ MRS^
for this utility function?^ U^ = (qq^12
What does this mean? If you eat 6 units of
qand 3 units of^1
q, how many^ q^21
would you
give up for a^ q?^2
Utility 31 / 70
Most indi¤erence curves are convex to the origin. This means that people prefer averages to extremes For example...^ if you have 100 pairs of trousers and no shirts, you would be willing togive up a lot of trousers for one shirt.^ If you have 100 shirts and no trousers, you would be willing to give upa lot of shirts for one pair of trousers.
Utility
EXAMPLE
Draw the indi¤erence curves for the following utility functions
U^ =^ x+^ x^1 2 U^ =^ xx^12 U^ =^ minfx,^31 xg^2 U^ =^ maxfx,^ x^1
g 2
Budget Constraint 37 / 70
Preferences make zero reference to income constraints. Given our assumptions on preferences, consumers would like anin nite amount of both goods. Obviously, consumers are constrained by the amount of money theyhave.
Budget Constraint^ We can depict this constraint using a
budget line. The budget line tells us all the combinations of goods a consumer canbuy if she spends all her money.
Budget Constraint 39 / 70
If^ Y^ is the consumers income and
pand^ pare the prices of^1
qand^1
qrespectively, the budget line will take the form.^2
Y^ =^ pq+^ p^11
q^2 Y^ ^ pq^11 q= 2 p^2 Of course people save and borrow, but we can easily incorporate thisby talking about consumption today and consumption tomorrow astwo di¤erent goods.
Budget Constraint^ We call the^ opportunity set
the set of all goods the consumer can possibly buy.
Y^ ^ pq+^ pq^11 The budget line is the frontier of the opportunity set.
Budget Constraint 41 / 70
The slope of the budget line is the
marginal rate of transformation (MRT^ )^ This is how many units of one good the consumer must give up inorder to purchase on more of the other.^ This is just equal to the price ratio
dq^2 MRT = =^ dq^1 p^1 p^2
Budget Constraint^ Can you think of a case in which the
MRT^ is not constant?
Budget Constraint 43 / 70
EXAMPLE
You make^ $^ 100,000 a year The price of pickles is
$^ 5 and the price of cell phones is
$^ 100.
Draw the budget set, what is the marginal rate of transformation?
Constrained Consumer Choice^ It is possible to have an
interior solution
or a^ corner solution An interior solution is one in which the optimal bundle has positivequantities of both goods A corner solution is when the optimal bundle contains none of one ofthe goods
Constrained Consumer Choice 49 / 70
If there is an interior solution, the indi¤erence curve is
tangent^ to the
budget constraint at the optimal bundle. This means that the indi¤erence curve and the budget line have thesame slope. The slope of the budget constraint is the price ratio, and the slope ofthe indi¤erence curve is the marginal rate of substitution. The optimal bundle is the one such that.^ MRS
MUp^1 = =^ ^ MU^2 1 =^ MRT p 2
Constrained Consumer Choice^ To get some intuition for why this is the optimal bundle, letsrearrange the thing
MUp^11 =^ ^ MU^ p^22 MUMU^1 =^ p^1 (^2) p 2 Think of this as "bang for your buck." The utility you get per dollar spent is the same for both goods
Constrained Consumer Choice 51 / 70
Suppose the marginal utility from pizza is 5 and pizza costs $
.^ The
marginal utility from burritos is 6 and the price of a burrito is $3 We are not at an optimum because
MUMU^12 >^.^ If I buy 1 less^ p^ p^1 burrito and 3 more pizzas, I will lose 6 units of utility, but I will gain15 units.
Constrained Consumer Choice^ Sometimes, the highest indi¤erence curve attainable does not occurwhen the budget line and indi¤erence curve are tangent.^ We have a^ corner solution
in this case, and the consumer will only buy one of the goods and none of the other.
Constrained Consumer Choice 53 / 70
EXAMPLES
Graphically, what will happen to your optimal bundle if the price of agood changes or income changes? Graphically show the optimal consumption bundle when the goods areperfect substitutes and when they are perfect complements.
Constrained Consumer Choice^ We can also use math, and we will get the same result.^ Just trying to maximize something subject to a constraint.^ The problem we are trying to solve is.
max^ U(q,^ q)^12 q,q^1 2 s.t. Y = pq+^ pq^11 That is, we need to pick the
qand^ qthat give use the biggest^1
U
while not violating the constraint
Constrained Consumer Choice 55 / 70
If we know that there is an interior solution, we can use two di¤erentmethods. the^ substitution method
and the^ Lagrangian method.
Constrained Consumer Choice^ The
rst step is to multiply the constraint by a constant
λ^ and then
add it or subtract it from the objective functionmaxL^ q,q, λ^1
=^ U(q,^ q)^ ^ λ^^12
qp+^ qp ^ Y^ (^11
If we were minimizing, we would add the constraint rather thansubtract. λ^ is the Lagrangian multiplier - it penalizes us for breaking theconstraint
Constrained Consumer Choice 61 / 70
Take the rst order condition with respect to our controls, and thensolve the system to get the solution^ ∂
L ∂ U^ =^ ^ λ p ∂ q^ ∂ q^11
=^0
∂ L ∂ U^ =^ ^ λ ∂ q^ ∂ q^22
p=^0
∂ L=^ Y^ ^ qp^1 ∂λ^
+^ qp=^0 1
(3)^ 62 / 70
Constrained Consumer Choice^ Solve equation 1 for
λ∂ U λ^ =^ ^ ∂ q^1
(^1) p^1 Plug this into equation two
∂ U ∂ U^1 ^ ^ p^2 ∂ q^ ∂ q^ p^211
=^0
Constrained Consumer Choice 63 / 70
If we rearrange terms we will get the exact same solution
∂ U^ ∂ U ∂ q^ ∂ q^12 = p^ p^12 This is just the^ MRS
=^ MRT
64 / 70
Constrained Consumer Choice
EXAMPLE
Use the Langrangian method to nd the optimal consumption bundlesfor the following utility functions
(^12) U = (qq) (^12) ρ (^) U = (q)+ ( 1 ^1 ρ^ ρ q) 2 (In practice, we know the tangency condition so we can nd it rightaway and plug it into the constraint)
Constrained Consumer Choice 65 / 70
What if we want to minimize expenditure for some xed level ofutility? Rather than xing the budget line and nding the highest indi¤erencecurve, we x the indi¤erence curve and nd the lowest budget line. This is the exact same thing... two sides of the same coin It can be useful, though because we can observe expenditure but notutility
Constrained Consumer Choice
EXAMPLE
Show that expenditure minimization gives you the same solution asutility maximization for
U^ (q,^ q), and income^12
Y^ and prices^ pand^1 p.^2
Behavioral Economics 67 / 70
We made a number of assumptions on how people behave. Behavioral economics looks at some of the departures from ourrationality assumptions. Experimental Economics (how we see people behave) + EconomicTheory = Behavioral Economics
