Microeconomics
Corso E
John Hey
Notation
• Intertemporal choice.
• Two periods: 1 and 2.
• Notation:
• m1 and m2: incomes in the two periods.
• c1 and c2: consumption in the two periods.
• r: the rate of interest.
• 10% r = 0.1, 20% r = 0.2.
• Hence the rate of return = (1+r)
The Budget Line
•
c1(1+r) + c2 = m2 + m1(1+r).
• In the space (c1 ,c2) a line with slope
-(1+r).
• The intercept on the horizontal axis =
m1 + m2/(1+r)
... the present value of the income stream (note that
m2 is discounted at the rate r).
• The intercept on the vertical axis =
m1(1+r) + m2
... the future value of the income stream.
Preferences?
• If I offer you a choice between 10 CDs today
and 10 CDs in a year, which do you prefer?
• 10 CDs today and 11 CDs in a year?
• 10 CDs today and 13 CDs in a year?
• 10 CDs today and 16 CDs in a year?
• 10 CDs today and 20 CDs in a year?
• 10 CDs today and 25 CDs in a year?
• Implications? Individuals discount future …
• … and the discount rate varies from
individual to individual.
The Discounted Utility Model
• Consumption c gives utility u(c) and the
utility of a bundle (c1,c2) is given by:
• U(c1,c2)=u(c1) + u(c2)/(1+ρ)
where ρ is the discount rate of the individual.
• u(c2)/(1+ρ) is the discounted value of the income
of period 2 discounted at the rate ρ – which is
individual-dependent.
• (Recall that m2/(1+r) is the discounted value of the income of the
second period...
... Discounted at the rate of interest r.)
The Discounted Utility Model
• U(c1,c2) = u(c1) + u(c2)/(1+ρ)
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There are two components:
The utility function of the individual: u(c)
The individual’s discount factor: ρ
Usually u(c) is concave in the space
(c,u(c)) (Why?)
• Usually ρ > 0 (Why?)
Indifference curves in the space (c1,c2)
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An indifference curve is given by:
Utility = constant ….
… U(c1,c2) = constant …
… u(c1) + u(c2)/(1+ρ) = constant
• Note the difference U(c1,c2), the utility of
the basket (c1,c2), and u(c), the utility of
consumption c.
Indifference curves in the space (c1,c2)
• u(c1) + u(c2)/(1+ρ) = constant
• If u(c) is linear, we have
• c1 + c2/(1+ρ) = constant
• Hence
c2 = constant - c1 (1+ρ)
• A line with slope (1+ρ).
An example: u(c) = √c
and ρ = 0
• U(c1, c2) = u(c1) + u(c2) = √c1 + √c2
• The indifference curve through the point
(9,9) is given by: √c1 + √c2 = 6
• Other points on this curve are:
• (0,36), (1,25), (4,16), (16,4), (25,1), (36,0)
• Note that at every point √c1 + √c2 = 6
• The equation is : c2 = (6 - √c1)2
• See the next graph ….
The indifference curves
• If u(c) is concave the indifference curves
are convex.
• If u(c) is linear the indifference curves are
linear.
• If u(c) is convex the indifference curves
are concave.
• The slope along the equal consumption
line are –(1+ρ)
The Discounted Utility Model
• If u(.) is concave (linear, convex), the
indifference curves in the space (c1,c2)
convex (linear, concave).
• The slope of every indifference curve on
the equal consumption line in (c1,c2) space
is equal to -(1+ ρ).
• There is a proof in the text.
• Let’s go to Maple …
Past Exam Questions
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In the next two questions you will be asked to consider an individual, taking
intertemporal decisions and having Discounted Utility preferences and utility function
u(x) = x^0.5 (that is, the utility of x is the square root of x). Suppose the individual is
faced with a choice of two intertemporal streams of consumption P and Q. Such a
stream is denoted by (c1,c2) where c1 is the consumption in period 1 and c2 the
consumption in period 2. His discount factor is specified below.
The consumption streams are: P = (16,9) Q = (25,4). The individual's discount factor
is 0.
Question 10: Does the individual prefer stream P or stream Q?
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P
The individual is indifferent
We cannot tell from the information given
Q
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Question 11: Suppose the individual could have the same consumption c in
both periods. What would c have to be to make him indifferent to stream P?
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3.50
12.25
29.00
25.00
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Summary
• The budget line has slope = -(1+r)
• The indifference curves given by
the Discounted Utility Model along
the equal consumption line have
slope
= -(1+ρ)
Chapter 21
• Goodbye!