Quick Reminder of the Theory of Consumer Choice Professor Roberto Chang Rutgers University January 2007

Reminder of Theory of Consumer Choice, as given by Mankiw, Principles of Economics, chapter 21, and other elementary textbooks.

A Canonical Problem Consider the problem of a consumer that may choose to buy apples (x) or bananas (y) Suppose the price of apples is px and the price of bananas is py. Finally, suppose that he has I dollars to spend.

The Budget Set The budget set is the set of options (here, combinations of x and y) open to the consumer. Given our assumptions, the total expenditure on apples and bananas cannot exceed income, i.e. px x + py y ≤ I

Rewrite px x + py y = I as y = I/py – (px/py) x This is the budget line

Apples (x) Bananas (y) OI/px I/py

Apples (x) Bananas (y) OI/px I/py

Apples (x) Bananas (y) Budget Line: px x + py y = I (Slope = - px/py) OI/px I/py

If I increases, the new budget line is higher and parallel to the old one.

Apples (x) Bananas (y) OI/px I/py

Apples (x) Bananas (y) OI/px I/py I’/px I’/py I’ > I

If px increases, the budget line retains the same vertical intercept, but the horizontal intercept shrinks

Apples (x) Bananas (y) OI/px I/py

Apples (x) Bananas (y) OI/px I/py I/ px’ px’ > px

Preferences Now that we have identified the options open to the consumer, which one will he choose? The choice will depend on his preferences, i.e. his relative taste for apples or bananas. In Economics, preferences are usually assumed to be given by a utility function.

Utility Functions In this case, a utility function is a function U = U(x,y), where U is the level of satisfaction derived from consumption of (x,y). For example, one may assume that U = log x + log y or that U = xy

Indifference Curves It is useful to identify indifference curves. An indifference curve is a set of pairs (x,y) that yield the same level of utility. For example, for U = xy, an indifference curve is given by setting U = 1, i.e. 1 = xy A different indifference curve is given by the pairs (x,y) such that U = 2, i.e. 2 = xy

x y Utility = u0 Three Indifference Curves

x y Utility = u0 Utility = u1 Three Indifference Curves Here u1 > u0

x y Utility = u0 Utility = u1 Utility = u2 Three Indifference Curves Here u2 > u1 > u0

Properties of Indifference Curves Higher indifference curves represent higher levels of utility Indifference curves slope down They do not cross They “bow inward”

Optimal Consumption In Economics we assume that the consumer will pick the best feasible combination of apples and bananas. “Feasible” means that (x*,y*) must be in the budget set “Best” means that (x*,y*) must attain the highest possible indifference curve

Apples (x) Bananas (y) OI/px I/py Consumer Optimum

Apples (x) Bananas (y) OI/px I/py x* y* C Consumer Optimum

Apples (x) Bananas (y) OI/px I/py x* y* C Consumer Optimum

Key Optimality Condition Note that the optimal choice has the property that the indifference curve must be tangent to the budget line. In technical jargon, the slope of the indifference curve at the optimum must be equal to the slope of the budget line.

The Marginal Rate of Substitution The slope of an indifference curve is called the marginal rate of substitution, and is given by the ratio of the marginal utilities of x and y: MRS xy = MU x / MU y Recall that the marginal utility of x is given by ∂U/∂x

Quick derivation: the set of all pairs (x,y) that give the same utility level z must satisfy U(x,y) = z, or U(x,y) – z = 0. This equation defines y implicitly as a function of x (the graph of such implicit function is the indifference curve). The Implicit Function Theorem then implies the rest.

Intuition: suppose that consumption of x increases by Δx and consumption of y falls by Δy. How are Δx and Δy to be related for utility to stay the same? Increase in utility due to higher x consumption is approx. Δx times MU x Fall in utility due to lower y consumption = -Δy times MU y Utility is the same if MU x Δx = - MU y Δy, i.e. Δy/ Δx = - MU x/ MU y

For example, with U = xy, MU x = ∂U/∂x = y MU y = ∂U/∂y = x and MRS xy = MU x / MU y = y/x Exercise: Find marginal utilities and MRS xy if U = log x + log y

Back to our consumer problem, we knew that the slope of the budget line is equal to the ratio of the prices of x and y, px/py. Hence the optimal choice of the consumer must satisfy: MU x / MU y = px/py

Numerical Example Let U = xy again, and suppose px = 3, py = 3, and I = 12. The budget line is given by 3x + 3y = 12 Optimal choice requires MRS xy = px/py, that is, y/x = 3/3 = 1 The solution is, naturally, x = y = 2.

Changes in Income Suppose that income doubles, i.e. I = 24. Then the budget line becomes 3x + 3y = 24 The MRS = px/py condition is the same, so now x = y = 4

x y O C I/px I/py

x y O C I/px I/py An increase in income I’ > I I’/px I’/py

x y O C I/px I/py An increase in income I’ > I I’/px I’/py C’

In the precious slide, both goods are normal. But it is possible that one of the goods be inferior.

x y O C I/px I/py An increase in income, Good y inferior I’ > I I’/px I’/py C’

Changes in Prices In the previous example, suppose that px falls to 1. The budget line and optimality conditions change to x + 3 y = 12 y/x = 1/3 Solution: x = 6, y = 2.

x y O C I/px I/py

x y O C Effects of a fall in px px > px’ I/px I/px’ I/py

x y O C’ C Effects of a fall in px px > px’ I/px I/px’ I/py

If x is a normal good, a fall in its price will result in an increase in the quantity purchased (this is the Law of Demand) This is because the so called substitution and income effects reinforce each other.

x y O C’ C I/px I/px’ I/py

x y O C’ C Substitution vs Income Effects I/px I/px’ I/py C’’