Consumer Behavior - cms.phbs.pku.edu.cn

Consumer Behavior

Marshall Urias

Marshall Urias Consumer Behavior 1 / 62

Part 1: Consumer Preferences


Consumer Behavior

Theory of consumer behavior describes how consumers allocate incomeamong different goods and services to maximize their well-being.

Consumer behavior is best understood in three distinct steps:

1 Consumer Preferences

2 Budget Constraints

3 Consumer Choices


Consumer Preferences

Everyone decides which goods and services to spend their money on.

How do you choose?

Economists represent this decision process by describing preferencesover different bundles of goods

If someone decides to buy a Huwaei phone instead of Xiaomi, it mustbecause they get more satisfaction from owning the Huwaei phone.

If someone decides to purchase a hamburger now instead of later, itmust be because they are hungry



We consider possible consumption bundles in some set X ∈ Rk+, the

consumption set, which is closed and convex.

The consumer is assumed to have preferences over the consumptionbundles in X . These preferences are described by a preference relation,where we write x � y to say that the consumer likes bundle x at least asmuch as bundle y .

Our goal is to be able to order the set of bundles, so we need thepreference relation to satisfy some standard properties:

1 Complete. For all x and y in X , either x � y or y � x or both.

2 Transitive. For all x , y , and z in X , if x � y and y � z , then x � z .



We can now describe agents behavior over different consumption bundles.We say that the consumer...

... strictly prefers bundle x to bundle y when x � y

... weakly prefers bundle x to bundle y when x � y

... is indifferent between bundle x to bundle y when x ∼ y

Do you think these preferences are reasonable and/or logical?



When economists describe agents as being rational, they mean thatpreferences are complete and transitive:

Completeness implies that consumers are able to make a choicebetween any two given bundles.

Transitivity guarantees that consumers can find a best choice.

Assuming that agents can compare alternative choices, and rank thesechoices clearly, is not obvious for all experiences. However, it seemsreasonable for many problems encountered in economics.


Indifference Curves

If we want to describe preferences and choice in general, then we need toestablish two more assumptions that will result in well-behaved indifferencesets.

1 Monotonicity. If x ≥ y and x 6= y , then x � y

2 Convexity. Given x , y , and z in X such that x � y and y � z , thenit follows that tx + (1− t)y � z for all 0 ≤ t ≤ 1.


Indifference Curves

Monotonicity guarantees that the indifference curves are in fact curves(not “thick” sets) and have a negative slope.


Indifference Curves

Convexity concerns the trade-offs that a consumer is willing to makeamong different goods:

Diminishing marginal rate of substitution: From any initialconsumption bundle x , and for any two commodities, it takesincreasingly larger amounts of one commodity to compensate forsuccessive unit losses of the other.Diversification:Averages are preferred to extremes


Indifference Curves

Do you think that monotonicity is a reasonable assumption? What aboutconvexity?


Utility Function

For analytical purposes, it is very helpful to summarize the consumer’spreferences by a utility function u : Rk

+ → R, so that we can solve for theoptimal consumption bundle (we will discuss what constraints theconsumer faces soon).

A utility function assigns a number to every possible consumption bundlesuch that more-preferred bundles get larger values than less-preferredbundles.


Utility Function

The word “utility” was originally used to define an indicator a person’soverall happiness, but there are several conceptual problems with this:

How are we supposed to quantify the “amount” of utility associatedwith different choices?

What would it mean to say that an extra candy bar gives me twice asmuch utility as an extra carrot?

Is utility comparable across individuals?

Does the concept of utility have any meaning independent of anindividual’s subjective appraisal?

Because of these issues, economists have abandoned the view of utility ofbeing a measure of happiness, and instead see utility as only a way todescribe preferences.


Utility Function

The only property of a utility assignment that is important is how it ranksdifferent consumption bundles; the size of the utility difference betweenany two bundles doesn’t matter.

The utility function u(·) that represents the preference relation � is notunique: any monotonic transformation of u(·) will do.


Utility Function

Geometrically, a utility function is a way to label indifference curves. Sinceevery bundle on an indifference curve must have the same utility, a utilityfunction is a way of assigning numbers to the different indifference curvesin a way that higher indifference curves get assigned larger numbers.


Utility Function

Are we assured that any preference relation can be represented by a utilityfunction?

We need one additional assumption to guarantee the existence of a utilityrepresentation.

1 Continuous. For any sequence of pairs {(xn, yn)}∞n=1 with xn � yn

for all n, x = limn→∞ xn and y = limn→∞ yn, we have x � y .

This assumption rulers out certain discontinuous behavior. For example, ify is strictly preferred to z , and if x is a bundle that is close enough to y ,then x must be preferred to y .


Utility Function

Theorem 1.1 Existence of a Utility Function. Suppose that preferencesare complete, transitive, and continuous. Then there exists a utilityfunction u : Rk

+ → R which represents those preferences.


Utility Function

Let’s begin with a geometric-quasi-proof for the case of monotonicpreferences.

Suppose we are given an indifference map that shows indifference curves,upper contour set, and lower contour set.

1 Draw a diagonal line through the indifference map

2 Label each indifference curve with its distance from the origin

3 Since preference are monotonic, the diagonal line intersects everyindifference curve exactly once, and those bundles on higherindifference curves are getting larger labels

4 Thus, we have a utility function


Utility Function


Utility Function

Now we can do an actual proof for monotonic preferences:

1 Let e denote the k-vector whose elements are all equal to 1, and let ube a nonnegative scalar.

2 By continuity, the upper contour set A+ = {u ∈ R+ : ue � x} andlower contour set A− = {u ∈ R+ : x � ue} are nonempty and closed.

3 The nonempitness and closedness of A+ and A− imply thatA+ ∩ A− = �, thus there exists a scalar u such that ue ∼ x

4 By monotonicity, u1e � u2e whenever u1 > u2, hence, there can be atmost one scalar satisfying ue ∼ x

5 We can take u(x) as our utility function

(Showing that u(x) is continuous is omitted)


Marginal Rate of Substitution

One important measure of a consumer’s preferences, is how much theyneed to be compensated to sacrifice some good.

Let u(x1, ..., xk) be a utility function Suppose that we decrease the amountof good i ; how does the consumer have to change his consumption ofgood j in order to keep utility constant?

By assumption, the change in utility must be zero, so

∂u(x)

∂xidxi +

∂u(x)

∂xjdxj = 0



The marginal rate of substitution measures the rate at which aconsumer is willing to substitute one good for another:

MRS ≡dxjdxi

= −∂u(x)∂xi

∂u(x)∂xj



Let’s try to build some intuition for the trade-offs a consumer faces whendeciding what bundle to purchase.

Suppose that the consumer is currently enjoying bundle (x1, x2).

We offer him a chance to trade: he can exchange good 1 for good 2,or good 2 for good 1, in any amount at a “rate of exchange” E .

If the consumer gives up dx1, then he can get E · dx1 units of good 2;if the consumer gives up dx2, then he can get dx2/E units of good 1.

What would the rate of exchange have to be in order for theconsumer to want to stay put at (x1, x2)?



Any time the exchange line crosses the indifference curve, there will besome points on that line that are preferred to (x1, x2) that lie above theindifference curve. Thus, in order to the consumer to be satisfied stayingat (x1, x2), the exchange line must be tangent to the indifference curve.The exchange rate must be equal to the marginal rate of substitution.



Notice that the marginal rate of substitution falls as we move down theindifference curve. The declining MRS reflects our convexity assumptionabout preferences, which economically captures the concept ofdiminishing marginal rate of substitution.


Summary of Key Points

Key points so far...

Economists make assumptions about consumers’ preferences, suchthat their behavior is rational.

A utility function is how we mathematically represent preferencesinsofar as we can rank better or worse alternatives.

Indifference curves (the level curves of our utility function) show aconsumer’s willingness to substitute one good for another.


Part 2: Budget Constraints


Budget Constraint

The economic theory of the consumer is simple: consumers choose thebest bundle of goods that they can afford.

We’ve already discussed what it means by the “best” bundle, now we needto define what is mean by “can afford”.


Budget Constraint

The budget constraint shows the feasible set of consumption bundlesgiven some fixed amount of income.

Let x denote a vector of consumption goods and p be the vector of pricesfor those goods, and let m denote the amount of income the consumer hasto spend. The budget constraint is given by

x · p ≤ m

For given prices p and income m, the budget constraint shows all possibleconsumption bundles x that the consumer can afford to buy.


Budget Constraint

For simplicity, let’s look at x ∈ R2+ where good 1 is whatever we are are

interested in studying, and good 2 is “everything else” the consumer mightwant to consume. The budget constraint will take the form

p1x1 + p2x2 ≤ m

where p1x1 is the amount of income spent on good 1, and p2x2 is theamount of income spent on good 2.


Budget Constraint


Budget Constraint

The slope of the budget line measures the rate at which the market iswilling to substitute good 1 for good 2.

dx2dx1

= −p1p2

The slope of the budget line measures the opportunity cost of consuminggood 1.


Budget Constraint

When prices and/or income change, the set of goods that the consumercan afford changes as well.

A change in income m will shift the budget line.

A change in prices (p1, p2) will rotate the budget line.


Budget Constraint


Numeraire

Although the budget line is specified by two prices and one income, one ofthese variables is redundant. The reason is, all the matters for theconsumer is the trade-offs she faces between choosing different bundles. Itis relative prices that affect her decisions, not absolute prices.

The budget linep1x1 + p2x2 = m

is exactly the same budget line as

p1p2

x1 + x2 =m

p2

where we have pegged p2 = 1. When we set one of the prices to 1, weoften refer to that price as the numeraire price and that good as thenumeraire good.


Taxes, Subsidies, and Rationing

Economic policy often uses tools that affect the consumer’s budgetconstraint.

Taxes

Subsidies

Quotas



If the government imposes a quantity tax, the consumer has to pay acertain amount to the government for each unit of the good shepurchases.

Example: In the U.S. consumers pay 18.4 cents per gallon of gasoline.

The budget line changes to

(p1 + τ)x1 + p2x2 ≤ m



Another kind of tax is a ad valorem (value) tax, which is a tax on theprice of the good rather than the quantity.

Example: In the U.S. most states have a sales tax around 6 percent, so agood priced at $1 actually costs $1.06.

The budget line changes to

(1 + τ)p1x1 + p2x2 ≤ m

The consumer has to pay p1 to the firm and τp1 to the government.



A subsidy is the opposite of a tax. For a quantity subsidy, thegovernment gives an amount to the consumer that depends on the amountof good purchased.

(p1 − s)x1 + p2x2 ≤ m

For an ad valorem subsidy, the government gives back a certain amountof money based on the value of the good.

(1− σ)p1x1 + p2x2



Government have also imposed rationing constraints, which specify thatthe level of consumption of some good is fixed to be no larger than someamount.

Example: During World War II, the U.S. government rationed certainfoods like butter and meat.

The budget line faces an extra constraint

p1x1 + p2x2 ≤ m

x1 ≤ x̄1


Summary of key points

Key points so far..

Consumers’ feasible consumption set is called the budget set, and isdetermined by the prices of goods and the income available

The slope of the budget line represents the market rate at which aconsumer and substitute one good for another

Any changes in prices will rotate the budget line, and any changes inincome will shift the budget line


Part 3: Choice


Optimal Choice

We are now prepared to combine theory of preferences and the budget setin order to examine the optimal choices of consumers.

The optimization problem is simple: Consumers choose the most preferredbundle of goods from their budget sets.

The optimization program is simple: Consumers maximize their utilityfunction subject to their budget constraint.


Optimal Choice

The consumer’s problem is given by

maxx

u(x)

s.t. px ≤ m

or if you like

maxx1,x2

u(x1, x2)

s.t. p1x1 + p2x2 ≤ m


Optimal Choice

A few technical notes:

The objective function (utility) is continuous by assumption

The constraint set is closed and bounded

The maximizing choice x∗ is independent of the choice of utilityfunction used to represent preferences

The optimal choice set is homogenous of degree zero in prices andincome

Under the assumption of strong monotonicity (one could use a weakerassumption called local nonsatiation), the preferred x∗ must meet thebudget constraint with equality


Optimal Choice

We can restate the consumer’s problem as

v(p,m) = max u(x)

s.t. px = m

The function v(p,m) gives us the maximum utility achievable at tivenprices and income is called the indirect utility function. The value of x∗

that solves the problem is the consumer’s demanded bundle: it expressedhow much of each good the consumer desires at a given level of prices andincome.

The function that relates p and m to the demanded bundle is called theconsumer’s demand function. We denote the demand function byx(p,m).


Optimal Choice

We can solve the consumer’s problem using the Lagrangian,

L = u(x)− λ(px −m)

Taking first order conditions we get

∂u(x)

∂xi− λpi = 0 fori = 1, ..., k


Optimal Choice

In order to interpret these condition we can divide the ith first-ordercondition by the jth first-order condition to eliminate the Lagrangemultiplier to obtain

∂u(x∗)∂xi

∂u(x∗)∂xj

=pipj

The lefthandside is the marginal rate of substitution between good i and j ,and the righthandside is the market rate of substitution. Maximizationimplies that these two rates are equal.


Optimal Choice


Optimal Choice

Geometrically, the optimal bundle x∗ must be such that the consumer’sindifference curve is tangent to the budget line. If the indifference curveweren’t tangent, it would cross the budget line, and if it crossed thebudget line, there would be some nearby point on the budget line that liesabove the indifference curve.


Optimal Choice

Technical note:. Mathematically, you should check that the second-ordercondition is satisfied: the Hessian matrix is negative semidefinite, which issimilar to saying that the utility function is quasi-concave. We omit thistopic, but if you’re confused ask me.


Implication of the MRS Condition

We now know that consumer’s will always choose their consumption suchthat their marginal rate of substitution is equal to the market price ratio.

In well-organized markets, it is typical that everyone faces the same pricesfor goods. For example, if everyone faces the same prices of butter andmilk, and everyone is optimizing, then everyone must have the same MRSfor butter and milk.

The market is offering everyone the same rate of exchange for butter andmilk, and everyone is adjusting their consumption of the goods until theirown “internal” marginal valuation of the two goods equals the market’s“external” valuation of the two goods.



Some people may be consuming a lot of butter and a little milk, and somemay be doing the reverse. Some wealthy people may be consuming a lotof milk and a lot of butter while other people may be consuming just alittle of each good. But everyone who is consuming the two goods musthave the same marginal rate of substitution.



Suppose, for example, that the price of milk is $1 a quart and the price ofbutter is $2 a pound. Then the marginal rate of substitution for all peoplewho consume milk and butter must be 2: they have to have 2 quarts ofmilk to compensate them for giving up 1 pound of butter. Or conversely,they have to have 1 pound of butter to make it worth their while to giveup 2 quarts of milk.

Now suppose that an inventor discovers a new way of turning milk intobutter: for every 3 quarts of milk poured into this machine, you get out 1pound of butter, and no other useful byproducts.

Question: is there a market for this device?



Answer: The inventor won’t make any money. For everyone is alreadyoperating at a point where they are just willing to trade 2 quarts of milkfor 1 pound of butter; why would they be willing to substitute 3 quarts ofmilk for 1 pound of butter? The answer is they wouldn’t; this inventionisn’t worth anything.

Since prices measure the rate at which people are just willing to substituteone good for another, they can be used to value policy proposals thatinvolve making changes in consumption. The fact that prices are notarbitrary numbers but reflect how people value things on the margin is oneof the most fundamental and important ideas in economics.


Optimal Choice

Recall that v(p,m) is our indirect utility function, and it is strictlyincreasing in m. If you have more income, you must be able to achieve ahigher level of utility!

We can flip the problem upside down and say: for a given level of utility u,what is the minimal amount of income necessary to achieve utility u andprices p?

The function that relates income and utility in this way—the inverse of theindirect utility function—is called the expenditure function and isdenoted by e(p, u).


Optimal Choice

An equivalent definition of the expenditure function is given by the problem

e(p, u) = minpx

s.t. u(x) ≥ u

The function that relates the expenditure minimizing bundle x∗ to thegiven level of prices and utility is called the Hicksian demand function,and is denoted h(p, u).


Optimal Choice

Theorem 1.2 Utility Maximization and Expenditure Minimization. Ifutility function is continuous and monotone increasing, then utilitymaximization implies expenditure minimization, and expenditureminimization implies utility maximization. The x∗ that solves the utilitymaximization problem is the same as the x∗ that solves the expenditureminimization problem.


Summary of key points

Key points so far..

The objective of the consumer is to maximize its utility subject to itsbudget constraint

The optimal consumption bundle must be such that the marginal rateof substitution is equal to the market rate of substitution

Minimizing expenditure is equivalent to maximizing utility


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