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Theory of consumer in describes preferences, utility, budget constraint, constrained consumer choice and given examples and diagrams.
Typology: Lecture notes
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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.
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
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.
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
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.
Transitivity Here we just mean that preferences are logically consistent. If^ a^ %^ b^ and^ b^ %^ c, by the transitive property
a^ %^ c
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Lets graph^ U^ = (q
(^12) q) (^12) This is not incredibly useful.
But we can turn this into two dimensions.
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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
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.
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,^ b^ ^ a.
Suppose our utility function is
U^ =^2 q+^3 q^1 Lets draw an indi¤erence curve for
U^ =^ 2 and^ U^ =^ 3.
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
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
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.
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
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.
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.
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?
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
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.
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
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 line. The budget line tells us all the combinations of goods a consumer canbuy if she spends all her money.
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.
the set of all goods the consumer can possibly buy.
Y^ ^ pq+^ pq^11 The budget line is the frontier of the opportunity set.
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
MRT^ is not constant?
You make^ $^ 100,000 a year The price of pickles is
$^ 5 and the price of cell phones is
Draw the budget set, what is the marginal rate of transformation?
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
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
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
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.
in this case, and the consumer will only buy one of the goods and none of the other.
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.
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
while not violating the constraint
If we know that there is an interior solution, we can use two di¤erentmethods. the^ substitution method
and the^ Lagrangian method.
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
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
∂ L ∂ U^ =^ ^ λ ∂ q^ ∂ q^22
p=^0
∂ L=^ Y^ ^ qp^1 ∂λ^
+^ qp=^0 1
λ∂ U λ^ =^ ^ ∂ q^1
(^1) p^1 Plug this into equation two
∂ U ∂ U^1 ^ ^ p^2 ∂ q^ ∂ q^ p^211
If we rearrange terms we will get the exact same solution
∂ U^ ∂ U ∂ q^ ∂ q^12 = p^ p^12 This is just the^ MRS
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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)
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
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
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