3 - A Consumer's Constrained Choice Flashcards
How can we use information about consumers’ allocation of their budget across various goods in the past to predict how a price change will affect their demands for goods today? Are consumers better off receiving cash or a comparable amount in food stamps? Should people buy insurance or save their money? work at home or in the marketplace? Have children? Invest in bonds or in stocks? How do we answer such questions?
To answer these questions and other questions about how consumers allocate their income over many goods, we use a model that lets us look at an individual’s decision making when faced with limited income and market-determined prices.
What does the model we develop throughout this chapter allow us to do?
This model allows us to derive the market demand curve that we used in our supply-and-demand model and to make a variety of predictions about consumers’ responses to changes in prices and income.
Our model of consumer behavior is based on which premises? (3)
- Individual tastes or preferences determine the amount of pleasure people derive from the goods and services they consume.
- Consumers face constraints, or limits, on their choices.
- Consumers maximize their well-being or pleasure from consumption subject to the budget and other constraints they face.
How do consumers consume?
Consumers buy the goods that give them the most pleasure, subject tot he constraints that they cannot spend more money than they have nor can they spend it in ways forbidden by the government.
How do we summarize a consumer’s preference ranking for weak preference?
Using a preference relation ≥. If the consumer likes Bundle a at least as much as Bundle b, we say that the consumer weakly prefers a to b, which we write a≥b
How do we summarize a consumer’s preference ranking for strict preference?
If the consumer weakly prefers Bundle a to b, a≥b, but the consumer does not weakly prefer b to a, then we say that the consumer strictly prefers a to b, which we write a>b.
How do we summarize a consumer’s preference ranking for indifference?
If the consumer weakly prefers Bundle a to b and b to a, that is a≥b and b≥a, then we say that the consumer is indifferent between the bundles a and b, or likes the two bundles equally, which we write a~b.
What is the completeness property?
The completeness property holds that, when facing a choice between any two bundles of goods, Bundle a and b, a consumer can rank them so that one and only one of the following relationships is true: a≥b, b≥a or both relationships hold so that a~b
What does the completeness axiom rule out?
It rules out the possibility that the consumer cannot decide which bundle is preferable.
What is the transitivity property?
According to this property, a consumer’s preferences over bundles is consistent in the sense that, if the consumer weakly prefers a to b, a≥b, and weakly prefers b to c, b≥c, then the consumer also weakly prefers a to c, a≥c.
What is the preference relation ≥ said to be if the completeness and transitivity axioms hold?
Rational. That is, the consumer has well-defined preferences between any pair of alternatives.
What is the more is better property?
The more-is-better property states that, all else the same, more of a commodity is better than less of it.
How do economists regularly use the more-is-better principle it their language?
Economists define a good as a commodity for which more is preferred to less, at least at some levels of consumption. In contrast, a bad is something for which less is preferred to more, such as pollution.
What is one of the simplest ways to summarize information about a consumer’s preferences?
To create a preference map - a graphical interpretation - of them. For simplicity, we concentrate on choices between only two goods, but the model can be generalized to handle any number of goods.
What are indifference curves?
The set of all bundles of goods that a consumer views as being equally desirable.
What is a preference map?
A complete set of indifference curves that summarize a consumer’s tastes.
Why do we call preference maps “maps”?
Because it uses the same principle as a topographical or contour map, in which each line shows all points with the same height or elevation.
What does each indifference curve in an indifference map consists of?
Each indifference curve in an indifference map consists of bundles of goods that provide the same utility or well-being for a consumer, but the level of well-being differs from one curve to another.
Given our assumptions, all indifference curve maps must have which five important properties?
- Bundles on indifference curves farther from the origin are preferred to those on indifference curves closer to the origin.
- Every bundle lies on an indifference curve.
- Indifference curves cannot cross.
- Indifference curves slope downward.
- Indifference curves cannot be thick.
Why are bundles on indifference curves farther from the origin are preferred to those on indifference curves closer to the origin?
Because of the more-is-better property.
Why does every bundle lie on an indifference curve?
Because of the completeness property.
Why can’t two indifference curves cross?
Because preferences are transitive and consumers prefer more to less.
Why do indifference curves slope downward?
Because of the more-is-better property.
What is a utility function?
The relationship between utility measures and every possible bundle of goods.
Why are utility functions useful?
If we know the utility function, we can summarize the information in indifference maps succinctly.
What is a commonly used utility function?
Cobb-Douglas
Why do economists think of utility functions?
The utility function is a concept that economists use to help them think about consumer behavior; utility functions do not exist in any fundamental sense.
What follows from fact that, typically, consumers can easily answer questions about whether they prefer one bundle to another, however, have difficulty answering questions about how much more they prefer one bundle to another because they don’t have a measure to describe how their pleasure from two goods or bundles differs?
We may know a consumer’s rank ordering of bundles, but we are unlikely to know by how much more that consumers prefers one bundle to another.
What is it called when we only know consumers’ relative rankings of bundles but not how much more that prefer one bundle to another?
Our measure of pleasure is an ordinal measure rather than a cardinal measure.
What is an ordinal measure?
An ordinal measure is one that tells us the relative ranking of two things but does not tell us how much more one rank is valued than another.
What is a cardinal measure?
A cardinal measure is one by which absolute comparisons between ranks may be made. Money is a cardinal measure.
How should we go about using an ordinal utility measure?
If we use an ordinal utility measure, we should not put any weights on the absolute differences between the utility number associated with one bundle and that associated with another. We only care about the relative utility or ranking of the two bundles.
What must we remember when we talk about utility numbers?
That these numbers are not unique and that we assign little meaning to the absolute numbers. We care only whether one bundle’s utility value is greater than that of another.
What does an indifference curve consist of?
All those bundles that correspond to a particular utility measure.
What is the expression for a utility function?
U = U(q1,q2)
This expression determines all those bundles of q1 and q2 that give the consumer U units of pleasure.
What does how willing a consumer is to trade one goods for another depend on?
dq2/dq1, at the consumer’s initial bundle of goods.
What do economists call the slope at a point on an indifference curve?
The Marginal Rate of Substitution MRS
Why do they call the MRS what they do?
Because it is the maximum amount of one good that a consumer will sacrifice (trade) to obtain one more unit of another good.
What does the MRS depend on?
The MRS depends on how much extra utility a consumer gets from a little more of each good.
What do we call the extra utility that a consumer gets from consuming the last unit of a good?
The marginal utility.
What is the equation for marginal utility?
mU1 = 'dU/'dq1 mU2 = 'dU/'dq2
How can we determine the MRS along an indifference curve?
We can determine the MRS along an indifference curve by ascertaining the changes in q1 and q2 that leave the consumers utility unchanged, keeping them on their original indifference curve
Let q2(q1) be the implicit function that shows how much q2 it takes to keep a consumer’s utility at U given they consumes q1. We want to know how much q2 must change if we increase q1, dq2/dq1, given that we require their utility to remain constant. To answer this question, we use the chain rule to differentiate U = U(q1, q2(q1)) with respect to q1, noting that because U is constant, dU/dq1 = 0
dU/dq1 = 0 = ‘dU(q1, q2(q1))/’dq1 + ‘dU(q1, q2(q1))/’dq2
= U1 + U2(dq2/dq1)
What is the intuition behind the equation dU/dq1 = 0 = ‘dU(q1, q2(q1))/’dq1 + ‘dU(q1, q2(q1))/’dq2
= U1 + U2(dq2/dq1)?
The intuition behind the equation is that as we move down and to the right along the indifference curve, we increase the amount of q1 slightly, which increases the consumer’s utility by U1, so we must decrease her consumption of q2 to hold her utility constant and keep her on the U indifference curve. Her decrease in utility from reducing q2 in response to the increase in q1 is U2(dq2/dq1), which is negative because dq2/dq1 is negative.
Rearrange the expression dU/dq1 = 0 = ‘dU(q1, q2(q1))/’dq1 + ‘dU(q1, q2(q1))/’dq2
= U1 + U2(dq2/dq1) to isolate the MRS.
MRS = dq2/dq1 = -[(‘dU/’dq1)/(‘dU/’dq2)] = -U1/U2
What does the willingness to trade fewer q1 for one more q2 as we move down and to the right along the indifference curve reflect?
Diminishing marginal rate of substitution: The MRS approaches zero - becomes flatter or less sloped - as we move down and to the right.
What are 5 types of utility functions?
- Perfect Substitutes
- Perfect Complements
- Cobb-Douglas
- Constant Elasticity of Substitution (CES)
- Quasilinear
For Perfect Substitutes, What is the utility function U(q1,q2), U1, U2, and MRS?
iq1 + jq2
i
j
-i/j
For the Utility function of the Perfect Complements, What is the utility function U(q1,q2), U1, U2, and MRS?
min(iq1,jq2)
0
0
0
For the Utility function of the Cobb-Douglas, What is the utility function U(q1,q2), U1, U2, and MRS?
q1^(a)q2^(1-a)
aU(q1,q2)/q1
(1-a)U(q1,q2)/q2
-a/(1-a)
For the Utility function of the Constant Elasticity of Substitution (CES), What is the utility function U(q1,q2), U1, U2, and MRS?
(q1^ρ+q2^ρ)^(1/ρ)
(q1^ρ+q2^ρ)^((1-ρ)/ρ) q1^(ρ-1)
(q1^ρ+q2^ρ)^((1-ρ)/ρ) q2^(ρ-1)
-(q1/q2)^(ρ-1)
For the Utility function of Quasilinear, What is the utility function U(q1,q2), U1, U2, and MRS?
u(q1) + q2
du(q1)/dq1
1
-(du(q1)/dq1)
What is a straight-line utility function?
An extreme case of an indifference curve is a straight line, which occurs when two goods are perfect substitutes.
What are perfect substitutes?
Goods that a consumer is completely indifferent as to which to consume.
What is the slope along indifference curves of perfect substitutes?
The slope of indifference curves of perfect substitutes need not always be -1; it can be any constant rate.
What is the other extreme case utility function?
Perfect complements
What are perfect complements?
Goods that a consumer is interested in consuming only in fixed proportions.
Why is the marginal utility for each good which are perfect complements zero?
Because increasing that good while holding the other one constant does not increase the consumer’s utility.
What do convex indifference curves show?
Convex indifference curves show that a consumer views two goods as imperfect substitutes.
What are 3 convex indifference curves?
- Cobb-Douglas
- Constant Elasticity of Substitution CES
- Quasilinear
What is a budget line?
A budget line, or budget constraint, is the bundles of goods that can be bought is a consumer’s entire budget is spent on those goods at given prices.
For a consumer choosing between two goods, what’s the equation for a budget constraint?
Y = p1q1 + p2q2
Given a budget of Y = p1q1 + p2q2, how much q1 can a consumer afford?
q1 = (Y-p2q2) / p1
According to equation q1 = (Y-p2q2) / p2, when can a consumer buy more q1?
He can buy more q1 with
- a higher income (dq1/qY = 1/p1 > 0)
- the purchase of fewer q2 (dq1/dq2 = -p2/p1 < 0)
- a lower price of q1 [dq1/dp1 = -(Y-p2q2)/p1^2 = -q1/p2 < 0, (dq1/dp2-q2/p1 < 0]
What is an opportunity set?
An opportunity set consists of all the bundles a consumer can buy, including all the bundles inside the budget constraint and on the budget constrain.
What do we call the slope of the budget line?
The Marginal Rate of Transformation (MRT)
What is the marginal rate of transformation?
The trade-off the market imposes on the consumer in terms of the amount of one good the consumer must give up to obtain more of the other good.
What’s the difference between the MRT and MRS?
The MRT is the rate at which a consumer is able to trade q2 for q1 in the market place when the prices he pays and his income are fixed. In contrast, the MRS is the trade-off the consumer would want to make regardless of their income.
How do consumers control their welfare?
Consumers maximize their well-being subject to their budget constraints.
What is an interior solution?
An optimal bundle that has positive quantities of both goods so that it lies between the ends of the budget line.
What is a corner solution?
If a consumer only buys one of the goods, the optimal bundle is at the end of the budget line, where the budget line forms a corner with one of the axes.
Why must a consumer’s optimal bundle lie on the budget line?
For any bundle inside the constraint, there is another bundle on the constraint with more of a least one of the goods, and hence he prefers that bundle. Therefore, the optimal bundle - the consumer’s optimum - must lie one the budget line and be on an indifference curve that does not cross it.
Where can we find the optimal bundle from a graph of IC’s and a BC?
The optimal bundle is on the highest indifference curve that touches the budget line.
What’s peculiar about the consumer’s optimal bundle? (think graphically)
The budget constraint and the indifference curve have the same slope at the point where they touch.
What follows from the fact that the budget constraint and the indifference curve have the same slope at the point where they touch.?
The consumer’s utility is maximized at the bundle where the rate at which he is willing to trade one good for another equals the rate at which he can trade in the market.
What follows from the fact that the consumer’s utility is maximized at the bundle where the rate at which he is willing to trade one good for another equals the rate at which he can trade in the market?
MRS = -U1/U2 = -p1/p2 = MRT
How can we rearrange the equation
MRS = -U1/U2 = -p1/p2 = MRT?
U1/p1 = U2/p2
Which two equivalent conditions can we use to find the interior solution from a graph?
- Highest indifference curve rule
- Tangency rule
What is the highest indifference curve rule?
The optimal bundle is on the highest indifference curve that touches the constraint.
What is the tangency rule?
The optimal bundle is the point where an indifference curve is tangent to the budget line. Equivalently, MRS = MRT and U1/p1 = U2/p2.
When can the highest indifference curve rule be used?
The highest indifference curve rule can always be used to find either interior or corner solutions.
When can the tangency rule be used?
The tangency rule only applies for interior solutions where the indifference curve has the usual shape: it s a downward sloping, smooth curve that is convex to the origin.
A consumer’s objective is to maximize their utility. What does that amount to mathematically?
Maxq1,q2 U(q1,q2) s.t. Y = p1q1 + p2q2
Given Maxq1,q2 U(q1,q2) s.t. Y = p1q1 + p2q2
which two different methods allows us to solve this?
- The Substitution Method
- The Langrangian Method
Describe the substitution method of maximizing utility.
First, we can substitute the budget constraint into the utility function. Using algebra, we can write the budget constraint as q1 = (Y-p2q2) / p1. Then we can write Maxq2 U( (Y-p2q2)/p1 , q2) This becomes an unconstrained problem, so we can use standard maximization techniques to solve it - by setting the derivative of U( (Y-p2q2)/p1 , q2) equal to 0 and solving for q2 in terms of price and income. then substituting q2 back into the expression to solve for q1 in terms of prices and income.
What is the Langrangian expression for maximizing utility.?
ℒ = U(q1,q2) + λ(Y - p1q1 - p2q2)
Describe the Lagrangian method of maximizing utility given
ℒ = U(q1,q2) + λ(Y - p1q1 - p2q2).
Write the three conditions ∂ℒ/∂q1 = ∂U/∂q1 - λp1 = U1 - λp1 = 0 ∂ℒ/∂q2 = ∂U/∂q2 - λp2 = U2 - λp2 = 0 ∂ℒ/∂λ = Y - p1q1 - p2q2 = 0 These three first order conditions can be solved for the optimal value or q1, q2, and λ.
Given the Langrangian method, what is λ?
λ = U1/p1 = U2/p2
For perfect complements, should we look for an interior solution or a corner solution?
Interior
For perfect substitutes, should we look for an interior solution or a corner solution?
Interior or Corner
For Cobb-Douglas functions, should we look for an interior solution or a corner solution?
Interior
For CES functions, should we look for an interior solution or a corner solution?
Interior
For quasilinear functions, should we look for an interior solution or a corner solution?
Interior or Corner
When do corner solutions occur and when do interior solutions occur?
If a utility function’s indifference curves do not hit the axes, a consumer’s optimal bundle must be in the interior of the budget constraint. If a consumer has a perfect complements utility function or Cobb-Douglas utility function, the indifference curves do not hit the axes, so the optimal bundle lies in the interior.
What happens if the ρ in the CES function became a 1?
Given U(q1,q2) = (q1^ρ+q2^ρ)^(1/ρ), setting ρ=1 gives us a perfect substitution function U(q1,q2) = q1 + q2
How come perfect substitutes could have an interior solution or a corner solution?
Because a perfect substitutes utility function has straight line indifference curves that hit the axes.
What determines whether a perfect substitutes function has an interior solution or a corner solution?
The slope of their indifference curves relative to the budget constraint.
When does a perfect substitutes function have a corner solution?
When MRS ≠ MRT
When does a perfect substitutes function have an interior solution?
When MRS = MRT
Here, the curves lie on top of each other so the consumer is indifferent to any point on his budget constraint.
When does a perfect substitutes function have a corner solution on the y-axis?
When MRS < MRT
When does a perfect substitutes function have a corner solution on the x-axis?
When MRS > MRT
How do we determine if a quasilinear function has an interior or corner solution?
We first check to see if there is an interior solution using the tangency condition and the budget constraint. If we find that these conditions imply that the consumer wants to buy positive quantities of both goods, we can found an interior solution. Otherwise, we have to determine a corner solution as a second step.
For quasilinear functions, if we don’t have an interior solution, we must have a corner solution. How do we know which good the consumer buys?
If his their income is low, they spend all their money on q1, buying q1 = Yp1, which is a corner solution. If he has enough income for an interior solution, he buys a fixed amount of q1 and spends all his extra money on q2 as his income rises. Loosely speaking, the consumers views q1 as a necessity and q2 as a luxury.
Earlier, based on introspection, we argued that most indifference curves are convex to the origin. Now that we know how to determine a consumer’s optimal bundle, we can give a more compelling explanation as to why we assume that indifference curves are convex. We can show that if indifference curves are smooth, optimal bundles lie either on convex sections of indifference curves or at the point where the budget constraint hits an axis. Explain.
Suppose that indifference curves were strictly concave to the origin. There is an IC that is tangent tot he budget line, but that bundle is not optimal. There is a bundle on a higher indifference curve at some corner solution. If a consumer had strictly concave indifference curves, the consumer would buy only one good. Thus, if consumers are to buy more than a single good, indifference curves must have convex sections.
If indifference curves have both concave and convex sections, where does the optimal bundle lie?
In a convex section, where the MRS = MRT, or at a corner. Thus, if a consumer buys positive quantities of two goods, the IC is convex and tanget to the budget line at the optimal bundle.
If a bundle were on the concave part, it can’t be optimal for the same reasons that concave ICs can’t be optimal for two goods.
What is a related dual constraint problem related to utility maximization?
Minimizing Expenditure.
What is a Minimizing Expenditure problem?
Where a consumer wants to find a combination of goods that achieves a particular level of utility for the least expenditure. We ask: How can the consumer make the lowest possible expenditure to maintain their level of utility at a particular level, here denoted U’, really it’s U overline.
What is the rule for minimizing expenditure while achieving a given level of utility?
To choose the lowest expenditure such that the budget lines touch - is tangent to - the relevant indifference curve.
What’s the thinking behind minimizing expenditure?
The slope of all the expenditure or budget lines is -p2/p1, which depends only on the market prices and not on income or expenditure. Thus, solving either of the two problems - maximizing utility subject to a budget constraint or minimizing expenditure subject to maintaining a given level of utility - yields the same optimal values for this problem.
Why is it sometimes more useful to the expenditure-minimizing approach?
Because expenditures are observable and utility level are not.
How can we use calculus to solve an expenditure-minimizing problem?
minq1,q2 E = p1q1 + p2q2 s.t. U’ = U(q1,q2)
The solution of this problem is an expression of the minimum expenditure as a function of the prices and the specified utility level:
E = E(p1,p2,U’)
What is E = E(p1,p2,U’)?
We call this expression the expenditure function: the relationship showing the minimal expenditures necessary to achieve a specific utility level for a given set of prices.
What insight do we gain from behavioral economics?
Behavioral economics adds insight from psychology and empirical research on human cognition and emotional biases to the rational economic model to better predict economic decision making.
Which three aspects of behavioral economics do we cover in this chapter?
- Test of Transitivity
- The Endowment Effect
- Salience
In our presentation of the basic consumer choice model at the beginning of this chapter, we assumed that consumers make transitive choices. But do consumers actually make transitive choices?
A number of studies show that adults are transitive over 90% of the time which children less so.
What did psychologists Bradbury and Ross show about transitivity?
That novelty is responsible for most intransitive responses, and that this effect is especially strong in children.
What could one conclude from the results of Bradbury and Ross’s tests?
That it is appropriate to assume that adults exhibit transitivity for most economic decisions but that the theory should be modified when applied to children or when novel goods are introduced.
Given the results of Bradbury and Ross’s tests, how do economists who normally argue that rational people should be allowed to make their own consumption choices so as to maximize their well-being sometimes make exceptions to this?
Some people argue that children’s lack of transitivity or rationality provides a justification for political and economic restrictions and protections placed on young people. For example, many governments effectively prevent youth from drinking.
Experiments s how that people have a tendency to stick with t he bundle of goods that they currently possess. What is one important reason for this?
The Endowment Effect
What is the endowment effect?
The endowment effect occurs when people place a higher value on a goods if they own it than they do if they are considering buying it.
Normally we assume that an individual can buy or sell goods at the market price. Rather than rely on income to buy some mix of two goods, an individual who was endowed with several units of one good could sell them and use that money to buy units of another good. What is the implication of these assumptions?
We assume that a consumer’s endowment does not affect the indifference map.
Plott and Zeiler argued that if you take adequate care to train the subjects in the procedures and make sure they understand them, the results of endowment “don’t hold”. What did List find when examining the actual behavior of sports memorabilia collectors?
That amateurs who do not trade frequently exhibit an endowment effect, unlike professionals and amateurs who trade extensively. Thus, experience may minimize or eliminate the endowment effect, and people who buy goods for resale may be less likely to become attached to these goods.
Others accept the results on experience reducing the endowment effect and have considered how to modify the standard model to reflect the endowment effect. What is one implication of these experimental results?
That people will only trade away from their endowments if prices change substantially.
Given that people will only trade away from their endowments if prices change substantially,how could this resistance to trade could be captured?
By having kink in the indifference curve at the endowment bundle. A kinked indifference curve could have an angle greater than a 90° and be curved at points other than at the kink.
Given that people will only trade away from their endowments if prices change substantially, how could a kink in the indifference curve illustrate the endowment effect?
If the indifference curve has a kink, the consumer does not shift to a new bundle in response to a small price change but does shift if the price change is large.
What do behavioral economists mean by salience?
The idea that people are more likely to consider information if ti is presented in a way that grabs their attention or if it takes relatively little thought or calculation to understand.
Tax salience is the awareness of a tax. How does this awareness affect consumer’s behavior?
If a store’s posted price includes the sales tax, consumers observe a change in the price as the tax rises. On the other hand, if a store posts the pre-tax price and collects the tax at the cash register, consumers are less likely to be aware that the post-tax price increases when the tax rate increases.
One explanation for why a tax has no effect on consumer behavior is consumer ignorance: many consumers ignore or are unaware of taxes. What is an alternative explanation for ignoring tax?
Bounded Rationality
What is Bounded Rationality?
The idea that people have a limited capacity to anticipate, solve complex problems, or enumerate all options. To avoid having to perform hundreds of calculations when making purchasing decisions at a grocery store, many people choose not to calculate the tax-inclusive price.
What is one exception to the idea of bounded rationality?
When post-tax price information is easily available to them, consumers use it.
How could we modify the standard model to incorporate this bounded rationality?
To assume that people incur a cost of making calculations - such as the time taken or the mental strain - and that deciding whether to incur this cost is part of their rational decision-making process.
In modifying the standard model to incorporate bounded rationality, we assume that people incur a cost of making calculations and that deciding whether to incur this cost is part of their rational decision-making process. When do people incur this cost?
People incur this cost only if they think the gain from a better choice of goods exceeds the cost. More people pay attention to a tax when the tax rate is high or when their demand for the good is elastic. Similarly, some people are more likely to pay attention to taxes when making large, one-time purchases rather than small, repeated purchases.
How does tax salience have important implications for tax policy, given that, in chapter 2, we showed that the tax incidence on consumers is the same regardless of whether the tax is collected from consumers or sellers?
We implicitly assumed that everyone was aware of the tax. However, if a tax on consumers rises and consumer’s don’t notice, their demand for the good is relatively inelastic, causing consumers to bear more of the tax incidence. In contrast, if the tax is placed on sellers, and the sellers want to pass on at least some of the tax to consumers, they raise their prices, which consumers observe.
