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Ch. 4

Learning Objectives:

1) Budget constraint1. Effect of price change2. Effect of income change

2) Indifference curve (IC)1. Represents consumer preferences.2. MRS (marginal rate of substitution) = MUx/MUy = (-)slope of the IC = (-) Δy/Δx

3) Consumer maximization problem1. Maximize utility subject to a budget constraint.2. Two types of solutions:

Interior solution – optimum bundle lies on B.C.Corner solution –optimum bundle lies on a corner of B.C.

3. Optimization rule 1. At interior optimum, the slope of IC = slope of the budget constraint.2. MUx/Px = MUy/Py (interior solution only)

At the optimum, the marginal utility of the last dollar spent on good X must equal the marginal utility of the last dollar spent on good Y.

4) Solve for an interior solution.Two methods:

1. Tangency condition plus the budget constraint.2. Lagrange Multiplier method

6) Identify a corner point solution.

5) Optimal consumption basket solves both a utility maximization problem and an expenditure minimization problem.

7) Applications of consumer theory : -Coupons versus Cash subsidies-Joining a club-Borrowing and Lending

-Understand Future value, present value, and the marginal rate of time preference.

-Quantity Discounts

(8) Revealed Preference theory

Consumer Problem:

- Maximize utility subject to an income constraint.

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This chapter discusses the theory behind consumer decision-making. The consumer chooses how much of good X and how much of good Y he/she can consume given their current income. In other words, the consumer tries to pick a consumption bundle of X and Y that maximizes his/her satisfaction (utility) given a budget constraint.

1. The budget constraint

- Consume on or inside B.C.

The budget constraint represents all of the possible combinations of X and Y that the consumer can consume given his/her income I and the market prices Px and Py of goods X and Y. The concept of the budget constraint is similar to the concept of the production possibilities frontier. The consumer can only consume combinations of X and Y that lie either on or inside the budget constraint.

-Depends on Prices and Income

The budget constraint depends upon the current level of (1) income and the (2) prices of X and Y. If one of these items changes, then the budget constraint and, consequently, the

consumption possibilities change. The consumer will need to choose a new bundle of goods X and Y that maximizes satisfaction given the new budget constraint.

-Formula for B.C.

PxX + PyY ≤ I

PxX + PyY = I because maximize U somewhere on the B.C.

Y = (I/Py) +(-Px/Py)X point –slope form

Y-intercept (I/Py)

Slope = -Px/Py relative market prices

X-intercept (I/Px)

Example 1: affordable consumption bundles

A consumer has an income of $1,000 per month to spend on pizza and Pepsi. The price of a pizza  is $10 and the price of a litre of Pepsi is $2. 

1)  If the consumer spends all of his income on pizza, he can buy 100 pizzas per month. If the  consumer spends all of his income on litres of Pepsi, he can buy 500 litres per month.

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Example 2: Change in Px

If the price of pizza increases to $20, then he can only buy 50 pizzas if he spends all of his income on pizza. The budget constraint pivots inward.

Example 3: Change in Income

When the consumer’s income doubles to $2000 and the prices stay the same, he can consume double the amount of pizza if he spends all of his income on pizza or double the amount of Pepsi if he spends all of his income on Pepsi. The budget constraint shifts outward in a parallel fashion. The slope doesn’t change because the relative price (-Px/Py) is unchanged.

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-Slope of the budget line = Relative MARKET prices

=how many units of the good on the vertical axis the consumer must give up to purchase one unit of the good on the x-axis given the current market prices, Px and Py.

-Question 1: If the slope of the B.C. = -5/8, how much good Y must be traded at the current market prices to get one unit of good X?

-Question 2: What happens to his budget constraint if his income doubles and the prices both double?

2. Indifference Curves

-Shape represents consumer preferences

-Bundles on same IC are equally preferred

-Higher IC represent higher levels of Utility

-MRS = (-) slope of the IC

= rate at which a consumer is willing to trade one good for another.

3. Optimum

-Maximize utility subject to a budget constraint.

The consumer will choose the consumption bundle (X,Y) that maximizes satisfaction given the budget constraint. This means that he/she will choose a consumption bundle on the highest possible indifference curve that still touches the budget constraint (i.e. is still affordable).

Can be expressed as:

-Two solutions possible:

1. Interior solution

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-An interior solution means that the consumer consumes positive amounts of both goods at the optimum.

-An optimal interior solution satisfies the condition that slope of B.C. = slope of IC.

MUxPx

=MUyPy

Meaning: This means that marginal satisfaction from the last dollar spent on each good in the consumption basket must be equal across all

goods consumed.

- If this were not the case andMUxMUy

>PxPy

∨¿ MUxPx

>MUyPy , then

the consumer is not at an interior optimum. At this basket the consumer gets greater utility for every dollar spent on good x than on good y. The consumer can increase utility by taking the last dollar spent on good y and spending it on another unit of good x.

2. Corner solution

- At a corner solution, the consumer consumes 0 units of one good and a positive amount of the other good.

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MUxMUy

<PxPy

∨¿ MUxMUy

>PxPy

∨¿

MUxPx

<MUyPy

MUxPx

>MUyPy

More y increases U More x increases U

(0, I/Py) (I/Px, 0)

-Strong preferences for one good versus the other.

How to find the optimal consumption bundle:

1. First, try to solve for an interior solutiona. Algebra – tangency + budget constraintb. Lagrange multiplier method

2. If no interior solution exists, then test the corner solutionsa. Check to see if utility is positive at a corner.b. Compare MRSxy to Px/Py evaluated at each corner.

If MRS>Px/Py, then corner solution is (I/Px,0).If MRS<Px/Py, then corner solution is (0, I/Py).

Example 5: Interior solution

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U(x,y) = xy and I = $40, Px=4, Py=2

Find the optimal consumption bundle.

Max U=xy s.t. PxX + PyY = I

Use algebra…1. Tangency condition says MUx/MUy = Px/Py

MUx = dU/dX= yMUy= dU/dY= xPx =4 and Py=2

y/x = 4/2 (1)

2. Budget Constraint is always PxX + PyY = I

4X + 2Y = 40 (2)

Solve for X and Y using the two equations (1) and (2).

From (1), y=2x

Sub into (2) 4X + 2(2X) = 40X = 5

Sub X=5 into y=2(5) = 10.

Optimum = (5,10).

Example 6: Interior solution for Perfect Complements

U(X,Y) = min(0.5x,y) and I=40, Px=4, Py=2.

Find optimum.

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Example 7: Problem 4.9

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

U(S, C) = SC + 10(S + C) (Quasi-linear utility)

Pc = $9 and Ps = $3 and I = $30

1. Is it possible to have an interior solution?

No, solving for an interior solution results in nonsense.

a. Tangency condition: MRScs = Pc/Ps

MUc = dU/dC = S + 10

MUs = dU/dS = C + 10

(S+10)/(C+10) = 9/3 (1)

b. Budget Constraint:

PcC +PsS = I

9C + 3S = 30 (2)

C. Solve for two equations in 2 unknowns (S AND C):

(S+10)/(C+10) =3 → S = 3C + 20

→ 9C + 3(3C + 20) = 30

→18C = -40

Solution is (C,S) = (-40/18, -100) NO! We cannot have negative amounts of S and or C. So there is no interior solution!

2. Is it possible to have a corner solution? Does U curve intersect the y or x axis?

Yes. It is possible to have 0 consumption of one good and still get positive utility with positive consumption of the other good.

When S = 0 and C>0, U = 10C

When C=0 and S>0, U = 10S

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3. Which corner? Compare MRS to slope of B.C. at each corner.

MRScs = (S+10)/(C+10) and Pc/Ps=3

At (C,S) = (0, I/Ps) = (0, 30/3) → MRScs = (10+10)/(0+10) = 2 < 3

At (C,S) = (I/Pc, 0) = (30/9, 0) → MRScs = (0+10)/(10/3 + 10) = .75 < 3

In both cases, MUc/MUs < Pc/Ps → MUc/Pc < MUs/Ps

So he should increase consumption of S until he runs out of money. Optimum is (0, 10)!

(F) Marketing example of using expenditure data to calculate consumers’ MRS between dinners at home and dinners out.

1. Consumer Expenditure Data 2007 shows that average U.S. household spent $3,465 on food consumed at home and $2,668 on food consumed away from home in 2007.

2. In 1995, U.S. consumer consumed 66% of total calories at home and 34% away from home.

3. Assuming that consumption patterns did not change between 1995 and 2007, what is the MRS of eating at home versus eating out?

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PxX = $3,465PyY = $2,668X / Y = (66/34)

Since (x/y) = (66/34), we can substitute in to get:

Interpret this number.

Utility maximization = Expenditure minimization

We can phrase the consumer optimization problem in two ways:

1. The consumer chooses a level of consumption of two goods X and Y such that the consumption bundle maximizes satisfaction subject to the budget constraint that limits expenditures to a particular income level I*.

Exogenous: Px,Py, and I (an income/expenditure constraint)Endogenous: X , Y, and U

Here we choose the bundle that allows us to reach the highest level of utility U2 subject to an expenditure constraint.

2. Likewise, the consumer can choose a level of consumption of X and Y that minimizes expenditure on the two goods subject to a constraint that utility must be at least as high

as the level U2.

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Exogenous: U=U2, Px, PyEndogenous: X, Y, and Expenditure (I)

Think of shifting the budget constraint inward to lower income levels until the expenditure line is just tangent to the given utility curve U=U*. This is the optimal consumption bundle.

Policy Examples of optimization with different budget constraints

When analyzing public policy it is often easiest to think of the consumer consuming the good of interest and a composite good H with a price of 1 dollar.

Composite good: represents expenditure on all other goods and has a price set at $1.00.

If the composite good is on the vertical axis, then the y-intercept of the budget constraint represents both the quantity of the composite good purchased and the expenditures

on the composite good.

-Coupons versus Cash subsidies

1. Housing v. Composite good, no housing subsidy

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NOTE: with the composite good y on the y axis, the slope of the B.C. = -Ph/Py = -Ph.

2. Income subsidy for housing vs. a voucher for housing

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