Chapter Four Consumer Choice Chapter Four

PXX + PYY ≤ I Assume only two goods available: X and Y Price of x: Px ; Price of y: Py Income: I Total expenditure on basket (X,Y): PxX + PyY The Basket is Affordable if total expenditure does not exceed total Income: PXX + PYY ≤ I Chapter Four

Key Definitions Budget Set: The set of baskets that are affordable Budget Constraint: The set of baskets that the consumer may purchase given the limits of the available income. Budget Line: The set of baskets that one can purchase when spending all available income. Chapter Four

The Budget Line The Y-intercept of the budget line shows the amount of good Y that can be purchased when all income is spent on good Y. The slope of the budget line equals the ratio of the goods’ prices.

A Budget Constraint Example Two goods available: X and Y I = $10 Px = $1 Py = $2 Budget Line 1: 1X + 2Y = 10 Or Y = 5 – X/2 Chapter Four

• B • • A Budget Constraint Example Y I = $10 Px = $2 Py = $2 If the price of X rises, the budget line gets steeper and the horizontal intercept shifts in If the price of X falls, the budget line gets flatter and the horizontal intercept shifts out I = $10 Px = $2 Py = $2 Budget line = BL1 I/PY= 5 A • -PX/PY = -1/2 B • • Budget line = BL1 C X I/PX = 10 I/PX = 5 Chapter Four

A Budget Constraint Example Y Shift of a budget line I = $12 PX = $1 PY = $2 Y = 6 - X/2 …. BL2 If income rises, the budget shifts out in a parallel fashion. If income falls, the budget shifts in in a parallel fashion. 6 5 BL2 BL1 10 12 X Chapter Four

A Budget Constraint Example Y Rotation of a budget line If the price of Y rises, the budget line gets flatter and the vertical intercept shifts in. If the price of y falls, the budget line gets steeper and the vertical intercept shifts out. I = $10 PX = $1 PY = $3 Y = 3.33 - X/3 …. BL2 6 BL1 5 3.33 BL2 10 X Chapter Four

Exercise A person has PhP 120 to spend on two goods (X, Y) whose respective prices are PhP3 and PhP5, draw the budget line; what happens to original budget line If the income falls by 25 percent? If the price of X doubles? If the price of Y falls to 4?

Consumer Choice Assume: Consumer’s Problem: Max U(X,Y) (X,Y)  Only non-negative quantities are of importance to consumers. A rational consumer chooses the basket that maximizes his satisfaction given the constraint that his budget imposes. Rational Choice or Optimal Choice is the optimal amount of each good to purchase; consumer choice of a basket of goods that (1) maximizes satisfaction while allowing to live within the budget constraint. It can be expressed as: Consumer’s Problem: Max U(X,Y) (X,Y) Subject to: PxX + PyY < I Chapter Four

Therefore, how do we determine OPTIMAL CHOICE or CONSUMER CHOICE?

Interior Optimum A graphical representation first… Interior Optimum: The optimal consumption basket is at a point where the indifference curve is just tangent to the budget line. (optimal basket at which consumer will be purchasing positive amounts of all commodities) A tangent to a function is a straight line that has the same slope as the function…therefore…. “The rate at which the consumer would be willing to exchange X for Y is the same as the rate at which they are exchanged in the marketplace.” Chapter Four

• • • Interior Consumer Optimum Y B Preference Direction A Optimal Choice (interior solution) IC • C BL X Chapter Four

Interior Consumer Optimum Assumptions U (X,Y) = XY and MU =Y while MU = X I = $1,000 P = $50 and P = $200 Basket A contains (X=4, Y=4) Basket B contains (X=10, Y=2.5) Question: Is either basket the optimal choice for the consumer? X Y Basket A: MRSx,y = MUx/MUy = Y/X = 4/4 = 1 Slope of budget line = -Px/Py = -1/4 Basket B: MRSx,y = MUx/MUy = Y/X = 1/4 Basket B would be the optimal choice for the consumer. Chapter Four

• Interior Consumer Optimum Example Y 50X + 200Y = E 25 = XY Y/X = 1/4 (tangency condition) B • U = 25 2.5 X 10 Chapter Four

Contained Optimization What are the equations that the optimal consumption basket must fulfill if we want to represent the consumer’s choice between two goods? Now, we have two equations to solve for two unknowns (quantities of X and Y in the optimal basket): 1. MUx/Px = MUY/PY 2. PxX + PyY = I Chapter Four

Example: Finding an Interior Solution Eric purchases food (X) and clothing (Y) and has the utility function U(X, Y) = XY. His marginal utilities are MUx = y and MUy = x. He has a monthly income of PhP800. The price of food is Px = PhP20, and the price of clothing is Py = PhP40. Find Eric’s optimal consumption bundle.

Example: Finding an Interior Solution To find an optimal basket, two conditions must be satisfied at an optimum. Optimal basket must lie on the budget line. This means that PxX + PyY = I or 20X + 40Y = 800. Since the optimum is interior, the indifference curve must be tangent to the budget line. We know that a tangency requires that MUx/MUy = Px/Py, or with the given information, y/x = 20/40, or x = 2y.

Example: Finding an Interior Solution So we have two equations with two unknowns. 20x + 40y = 800 and x = 2y Solving simultaneous equations, Substitute x = 2y into the equation 20x + 40y = 800. 20(2y) + 40y = 800 40y + 40y = 800 80y = 800 So y = 10 and x = 2(y) = 2(10) = 20. Eric’s optimal basket involves the purchase of 20 units of food and 10 units of clothing.

In all examples considered, the optimal consumer basket has been INTERIOR (consumer purchase positive amount of both goods). In reality, a consumer might not purchase positive amounts of all available goods. Some consumers may not own a car or a house. Some consumers may not spend money on cigarette or alcohol.

In this case, consumer cannot find an interior basket, then consumer finds an optimal basket at a corner point – a solution to the consumer’s optimal choice problem at which some good is not being consumed at all, in which case the optimal basket lies on an axis.

Corner Solution A corner solution occurs when the optimal bundle contains none of one of the goods. The tangency condition may not hold at a corner solution. How do you know whether the optimal bundle is interior or at a corner? Graph the indifference curves Check to see whether tangency condition ever holds at positive quantities of X and Y Chapter Four

Example: Finding a Corner Point Solution David is considering his purchases of food (x) and clothing (y). He has the utility function U(x,y) = xy + 10x, with marginal utilities MUx = y + 10 and MUy = x. His income is I = 10. He faces a price of food Px = 1 and a price of clothing Py = 2. What is David’s optimal basket?

Corner Point Solution Y 5 X 10 The budget line has a slope of –(Px/Py) = -1/2, as shown in this figure. The equation of the budget line is PxX + PyY = I or x + 2y = 10. 5 X 10 Chapter Four

Solution To find an Optimum, we must make sure that we understand what the indifference curves look like. Both marginal utilities are positive, MUx = y + 10 and MUy = x, so indifference curves are negatively sloped. The MRSx,y, (MUx/MUy) = (y +10)/x, diminishes as we increase x and decrease y along the curve, thus, indifference curves are bowed in toward the origin (convex to origin).

Solution Finally, ICs do intersect x – axis because it is possible to achieve a positive level of utility with positive quantities of x (food) but 0 unit of y (clothing). U(x,y) = xy + 10x This means that the consumer’s optimal basket may be at a corner point along the x – axis. We have plotted three of David’s indifference curves in this figure.

• Corner Point Solution Y 5 X 10 The indifference map: indifference curves are drawn for three levels of utility, U = 80, U = 100, and U = 120. 5 U = 80 U = 100 U = 120 • X 10 Chapter Four

Suppose we (mistakenly) assume that David’s optimal basket is interior (on the budget line and satisfy the tangency condition). If the optimal basket is on budget line, thus, x + 2y = 10. If the basket is at a point of tangency, (MUx/Muy) = (Px/Py) = (y +10)/x = ½ which simplifies to x = 2y + 20. Solving for the unknowns x, y, x = 15 and y = -2.5.

However, the solution does not make sense because neither x nor y can be negative. This tells us that optimal basket is, therefore, not interior, and the optimum will be at a corner point. Where is the optimal basket?

• Corner Point Solution Y 5 R X 10 The optimal consumption basket is R ( a corner solution), where the slope of the indifference curve is -1, where David spends all his income on food, so that x = 10 and y = 0. 5 U = 80 U = 100 U = 120 So at R, David would like to purchase more food and less clothing, but he cannot because R is at a corner solution on the x – axis. At R, David reaches the highest IC possible while choosing a basket on the budget line. R • X 10 Chapter Four

Two Ways of Thinking About Optimality What basket should the consumer choose to maximize utility, given a budget constraint limiting expenditures to PhP800 per month? In this case, since the consumer chooses the basket of x and y to maximize utility while spending no more than PhP800 on the two goods, optimality can be described as follows: Max Utility = U(x, y) (x, y) subject to : PxX + PyY ≤ I = 800 So far we have considered is answering the question...

In this example, the endogenous variables are x and y and the level of utility. The exogenous variables are the prices Px and Py and Income I

The optimality can be described as follows: What basket should the consumer choose to minimize his expenditure (PxX + PyY) and also achieve a given level of utility U? The optimality can be described as follows: Min expenditure = PxX + PyY (X,Y) Subject to : U(x, y) = U This is called the expenditure minimization problem. In this problem, the endogenous variables are still x and y and the level of expenditure, but the exogenous variables are the prices Px, Py, and the required level of utility U. There is another way to look at optimality, by asking a different question:

Using this figure, let’s look for a basket that would require the lowest expenditure to reach indifference curve U2 In this figure, we have drawn three different budget lines. All baskets on the budget line BL1 can be purchased if the consumer spends 640 a month. Unfortunately, none of the baskets on BL1 allows him to reach the indifference curve U2, so he will need to spend more than 640 to achieve the required utility.

Could he reach the indifference curve U2 with a monthly expenditure of 1000? All baskets on BL3, such as baskets R and S, can be purchased by spending 1000 a month. But there are other baskets on U2 that would cost the consumer less than 1000. To find the basket that minimizes expenditure, we have to find the budget line that is tangent to the indifference curve U2. That budget line is BL2, which is tangent to BL2 at point A.

Thus, the consumer can reach U2 by purchasing basket A, which costs only 800. Any expenditure less than 800 will not be enough to purchase a basket on indifference curve U2.

Thus, the utility maximization problem and the expenditure minimization problem are said to be dual to one another. The basket that maximizes utility with a given level of income leads the consumer to a level of utility U2. The same basket minimizes the level of expenditure necessary for the consumer to achieve a level of utility U2.

Duality The mirror image of the original (primal) constrained optimization problem is called the dual problem. Min PxX + PyY (X,Y) subject to: U(X,Y) = U* where: U* is a target level of utility. If U* is the level of utility that solves the primal problem, then an interior optimum, if it exists, of the dual problem also solves the primal problem. Chapter Four

Consumer Choice with Composite Goods Although consumers typically purchase many good and services, economists often want to focus on the consumer’s selection of a particular good or service (such as consumer’s choice of housing or level of education). In that case, it is useful to present the consumer choice problem using a two – dimensional graph with the amount of commodity of interest (say, housing) on the horizontal axis, and the amount of all other goods combined on the vertical axis.

Composite Good, Units I Housing (units) I/Ph The good on the vertical axis is called COMPOSITE good – good treated as a group; good that represents the collective expenditures on every good except the commodity being considered – simply because it is the composite of all other goods. Chapter Four

By convention, the price of a unit of the composite good is Py = 1 By convention, the price of a unit of the composite good is Py = 1. Thus, the vertical axis represents not only the number of units y of the composite good, but also the total expenditure on the composite good (PyY). The horizontal axis measures the number of units of housing, h. The price of housing is Ph. If the consumer’s income is I, he could purchase at most I/Ph units of housing (intercept of the budget line on the horizontal axis). The vertical axis measures the number of units of composite goods, y. The price of the composite good is Py = 1.

If the consumer were to spend all his income on the composite good, he could purchase I units of composite good (I/Py = I/1 = I), thus, the vertical intercept is I, which also represents the level of income. BL has a slope of –Ph/Py = -Ph. Given the consumer’s preferences, the optimal basket is A, where the consumer purchases h units of housing and y units of composite goods.

Suppose we do not know the consumer’s indifference map Suppose we do not know the consumer’s indifference map. Can we infer how the consumer ranks baskets by observing his behavior as his budget line changes? In other words, do the consumer’s choices of baskets reveal information about his preferences? Chapter Four

Revealed Preference Analysis that enables us to learn about a consumer’s ordinal ranking of baskets by observing how his or her choices of baskets change as prices and income vary. Chapter Four

Revealed Preference The main idea behind revealed preference is simple. If the consumer chooses basket A when basket B costs just as much, then we know that A is weakly preferred to (i.e., at least as preferred as) B. (We write this as A≥B, meaning that either A>B or A≈B.) Chapter Four

Revealed Preference When he chooses basket C, which is more expensive than basket D, then we know that he must strongly prefer C to D (C>D). Chapter Four

Revealed Preference Given enough observations about his choices as prices and income vary, we can learn much about how he ranks baskets, even though we may not be able to determine the exact shape of his indifference map. Chapter Four

Revealed Preference Revealed preference analysis assumes that the consumer always chooses an optimal basket and that, although prices and income may vary, his underlying preferences do not change. Chapter Four

This figure illustrates how consumer behavior can reveal information about preferences . Scenario: Given an initial level of income and prices for two goods (housing and clothing), the consumer faces budget line BL1 and chooses basket A. Suppose prices and income change so that budget line becomes BL2, and he chooses basket B. What do the consumer’s choices reveal about his preferences?

First, the consumer chooses basket A when he could afford any other basket on or inside BL1, such as basket B. Therefore , A is at least preferred as B (A≥B). But he has revealed even more about how he ranks A and B. Consider basket C. Since the consumer chooses A when he can afford C, we know that A is at least preferred as C (A≥C). And C lies to the northeast of B. This reveals that C must be strongly preferred to B (C > B). Then, by transitivity, A must be strongly preferred to B ( if A≥ C and C > B, then A > B).

Seatwork Helen’s preferences over CDs (C) and sandwiches (S) are given by U(C, S) = SC + 10(S + C). If the price of a CD is $9 and the price of sandwich is $3, and Helen can spend a combined total of $30 each day on these goods, find Helen’s optimal consumption basket.