Quick Reminder of the Theory of Consumer Choice
• 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
• Rewritepx x + py y = I
asy = I/py – (px/py) x
This is the budget line
Apples (x)
Bananas (y)
O I/px
I/py
Apples (x)
Bananas (y)
O I/px
I/py
Apples (x)
Bananas (y)
Budget Line:
px x + py y = I
(Slope = - px/py)
O I/px
I/py
• If I increases, the new budget line is higher and parallel to the old one.
Apples (x)
Bananas (y)
O I/px
I/py
Apples (x)
Bananas (y)
O I/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)
O I/px
I/py
Apples (x)
Bananas (y)
O I/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 thatU = 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)
O I/px
I/py
Consumer Optimum
Apples (x)
Bananas (y)
O I/px
I/py
x*
y* C
Consumer Optimum
Apples (x)
Bananas (y)
O I/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:
MRSxy = MUx/ MUy
• 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 MUx
• Fall in utility due to lower y consumption = -Δy times MUy
• Utility is the same if MUx Δx = - MUy Δy, i.e. Δy/ Δx = - MUx/ MUy
• For example, with U = xy, MUx = ∂U/∂x = y
MUy = ∂U/∂y = x
and MRSxy = MUx/ MUy = y/x
• Exercise: Find marginal utilities and MRSxy
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:
MUx/ MUy = px/py
Numerical Example
• Let U = xy again, and suppose px = 3, py = 3, and I = 12.
• The budget line is given by3x + 3y = 12
• Optimal choice requires MRSxy = 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 nowx = 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 = 12y/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’’