Microeconomics MSc
Utility
Todd R. Kaplan
University of Haifa
November 2010
Kaplan (Haifa)
micro
November 2010
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My favorite utility functions
Perfect substitutes: u (x1 , x2 ) = x1 + x2
Perfect complements: u (x1 , x2 ) = minfx1 , x2 g
Quasilinear: u (x1 , x2 ) = g (x1 ) + x2
Cobb-Douglas: u (x1 , x2 ) = x1a x2b where a > 0 and b > 0.
CES: u (x1 , x2 ) = (x1r + x2r )/r .
When r = ∞, 0, 1, ∞, CES equals which of the following functions:
min, max, perfect substitutes, Cobb-Douglas? Guess.
What do the indi¤erence curves look like.
Find a function that changes C-D into x1c x21
Kaplan (Haifa)
micro
c
November 2010
2 / 30
Marginal Utility
The marginal utility of commodity i is the rate-of-change of total
utility as the quantity of commodity i consumed changes.
MUi =
∂U
∂xi .
What are the marginal utilities of perfect substitutes, perfect
complements, quasi-linear, and Cobb-Douglas, CES?
Kaplan (Haifa)
micro
November 2010
3 / 30
Marginal Rate of Substitution.
MRS =
MU1 /MU2 (note some use negative of this).
This is also the slope of the indi¤erence curve. Why?
MRS is not a¤ected by a transformation V (x ) = f (U (x )). Why?
(hint: chain rule).
What is MRS of Cobb-Douglas, Quasi-Linear, Perfect Substitutes,
Perfect Complements, CES?
Kaplan (Haifa)
micro
November 2010
4 / 30
Choice
Draw the budget set. Draw the indi¤erence curves. Best choice is at
the highest indi¤erence curve that is still in the budget set.
This is equivalent to solve the problem:
maxx1 ,x2 U (x1 , x2 )
s.t. p1 x1 + p2 x2 m, and x1 , x2 0
We maximize utility subject to the budget constraint.
Kaplan (Haifa)
micro
November 2010
5 / 30
Rational Constrained Choice
The most preferred a¤ordable bundle is called the consumer’s
MARSHALLIAN DEMAND at the given prices and budget.
Marshallian demands will be denoted by x1 (p1 , p2 , m ) and
x2 (p1 , p2 , m ). This is the solution to the previous problem.
Note that xi (t p1 , t p2 , t m ) = xi (p1 , p2 , m ).
Indirect utility v (p1 , p2 , m ) is the utility achieved in the previous
problem: v (p1 , p2 , m ) = u (x1 (p1 , p2 , m ), x2 (p1 , p2 , m )).
Note that v (t p1 , t p2 , t m ) = v (p1 , p2 , m ).
Kaplan (Haifa)
micro
November 2010
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To solve the consumer problem
To solve the consumer problem
Check to see what type of preferences. “Smooth” preferences such as
Cobb-Douglas can be solved in one of 3 ways.
1
Substitution.
2
MRS=Slope of Budget constraint.
3
Lagrangian.
Kaplan (Haifa)
micro
November 2010
7 / 30
Substitution method
.
Substitution method.
1
Solve b.c. for one var. p1 x1 + p2 x2 = m
2
Plug into utility u (x1 , m/p2
3
Take the derivative with respect to x1 and set this equal to zero.
4
Use this and original b.c to solve for x1 and x2
p1 x1 /p2 )
Try this for x1 x2
Kaplan (Haifa)
micro
November 2010
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MRS method.
MRS method.
(x1 , x2 ) satis…es two conditions:
1
The budget is exhausted; p1 x1 + p2 x2 = m
2
The slope of the budget constraint, p1 /p2 , and the slope of the
indi¤erence curve containing (x1 , x2 ) are equal at (x1 , x2 ).
Try this for CES
What is solution when r ! 0.
Kaplan (Haifa)
micro
November 2010
9 / 30
Homework 4.
1
For u (x1 , x2 ) = (x1r + x2r ) /r . Show how when r ! ∞, r ! 1, the
Marshallian demand goes to that of perfect complements,
xi = m/(p1 + p2 ), and perfect substitutes, xi = m/pi if pi < pj ,
respectively..
Solve for Marshallian Demand and Indirect Utility for the following utility
functions and methods.
p
2. u (x1 , x2 ) = x1 + x2 by substitution.
3. u (x1 , x2 ) =
Kaplan (Haifa)
e
x1
e
2x2
by MRS.
micro
November 2010
10 / 30
Lagrangian method.
Lagrangian method.
Steps.
1
Set up Langrangian:L = U (x1 , x2 ) + λ(m
2
Take derivatives w.r.t. x1 , x2 , and λ.
3
Set them equal to zero and solve.
p1 x1
p2 x2 )
Note: All methods basically are the same.
Solve for Cobb-Douglas x1a x2b .
Kaplan (Haifa)
micro
November 2010
11 / 30
Cobb Douglas
Solve for Marshallian Demand and Indirect Utility for the following utility
functions and methods.
Cobb Douglas x12 x21 using the MRS method.
Cobb Douglas x12 x23 using the substitution method.
Cobb Douglas x1a x2b by Langrangian.
Which prices does x1 depend upon? What does this mean?
Kaplan (Haifa)
micro
November 2010
12 / 30
Constrained Choice Problems
If preferences are well behaved, then we can usually obtain the ordinary
demands are obtained by solving those 3 methods.
Problems (IMPORTANT!!)
1
Preferences are not convex.
2
Corner Solutions. (x1 = 0 or x2 = 0)
3
Kinky I.C’s such as minfax1 , x2 g
To stop 1, one needs 2nd-order conditions.
Puzzle. Does x12 x22 satisfy this?
Try to solve a case of 2 (perfect substitutes) and 3 (perfect
complements).
Kaplan (Haifa)
micro
November 2010
13 / 30
Roy’s Identity
Theorem
Roy’s identity: xi (p, m ) =
∂v (p,m )
∂p i
∂v (p,m )
∂m
.
Example
Show that Roy’s identity holds for u (x1 , x2 ) = x1 x2 .
Kaplan (Haifa)
micro
November 2010
14 / 30
Envelope Theorem
As part of the proof we need to know the handy Envelope Theorem.
Theorem
Envelope Theorem:If M (a) = maxx f (x, a) = f (x (a), a) then
dM (a)
∂f (x, a)
=
da
∂a
x =x (a )
Proof.
dM (a )
da
=
∂f (x ,a )
∂a
x =x (a )
However,
∂f (x ,a )
∂x
x =x (a )
+
∂f (x ,a )
∂x
x =x (a )
x 0 (a ).
= 0.
Why??.
Kaplan (Haifa)
micro
November 2010
15 / 30
Envelope Theorem Example
Example
Try: f (x, a) = a x
x2
What is x (a)?
What is
∂f (x ,a )
∂x
x =x (a )
and
∂f (x ,a )
?
∂a
x =x (a )
What is f (x (a), a)?
What is
d
da f
Kaplan (Haifa)
(x (a ), a )?
micro
November 2010
16 / 30
Roy’s Identity Proof.
xi (p, m ) =
∂v (p,m )
∂p i
∂v (p,m )
∂m
Proof.
v (p, m ) = maxx 1 u (x1 , m
We have
=
du
dx1
since
∂v (p,m )
∂p 1
∂x1
∂p 1 + u2
du
dx1 = 0 by
We have
∂v (p,m )
∂m
Kaplan (Haifa)
=
(
p1 x1
)
p2
d maxx1 u (x1 ,
dp 1
x1
p 2 ) = u2
m p1 x1
p2
(
)
=
m p 1 x1 (p,m )
)
p2
du (x1 (p,m ),
dp 1
=
x1
p2 )
the Envelope Theorem.
=
d maxx1 u (x1 ,
dm
m p1 x1
p2
micro
)
=
du
dx1
∂x1
∂m
+
u2
p2
=
u2
p2
November 2010
17 / 30
Homework 5.
1
People in Smallsville take pictures and eat pizza. They also like to
smile. Rational Ralph has utility u (s, x1 , x2 ) = x1 x2 + sx1 + x2 where
s is the amount of smiling he does and x1 is the amount of pictures
he takes and x2 is the amount of pizza he consumes. Assume he
maximizes the amount of money he spends on pictures and pizza (he
doesn’t have to pay to smile). Assume the price of pictures and pizza
are $1 each and Ralph has $3.
(i) Assume Rational Ralph is rational and hence maximizes his utility
subject to his budget constraint. As a function of s, how much will Ralph
spend on pictures and pizza. Using the envelope theorem, how much
would his utility go up by smiling more (in terms of x1 and x2 )?
(ii) Crazy Eddie also has utility u (s, x1 , x2 ) = x1 x2 + sx1 + x2 . However,
Eddie feels compelled take a picture for each smile he has. This results in
x1 = s for all s 3 and x1 = 3 for all s > 3. How much would his utility
go up by smiling more (in terms of x1 and x2 )?
Kaplan (Haifa)
micro
November 2010
18 / 30
Homework 5.
(m +p +p )2
1
2
2. Indirect utility is given by v (p1 , p2 , m ) =
. What are the
4p 12 p 2
demands for x1 and x2 ? When is the demand for good x1 greater
than that for good x2 ?
Kaplan (Haifa)
micro
November 2010
19 / 30
Problems
Taxes.
How do taxes a¤ect choice?
The budget constraint with taxes is p1 (1 + vat )x1 + p2 (1 + vat )x2
m (1 t )
Should there be a di¤erence between (vat, t ) = (0.25, 0) and
(vat, t ) = (0, 0.2)?
Blumkin and Ru- e found there is.
People work harder if there is a consumption tax rather than an income
tax.
Form of Money Illusion.
Kaplan (Haifa)
micro
November 2010
20 / 30
Mental Accounting and Relativity
Where the money comes from matters: mental accounting.
Have a $200 ticket to the superbowl and lost it. Do you buy another
one?
How about you are just about to buy a ticket to the superbowl and
discover that you are missing $200. Do you still buy it?
Do you go across town when you …nd that you can pay £ 985 instead
of £ 999 on a computer?
How about if you can pay £ 14 instead of £ 28 on a lamp?
How about free instead of £ 14?
Kaplan (Haifa)
micro
November 2010
21 / 30
Dan Ariely found price a¤ects utility!
This happened in painkillers.
They also found this with wine and fMRI studies.
Book vouchers.
A. $10 amazon voucher for free.
B. $20 amazon voucher for $7.
C. $10 amazon voucher for $1.
D. $20 amazon voucher for $8.
Does this violate utility maximization?
Say your budget was $8 and there was an $8 reading light you wanted
to get. The remaining money was spent on doughnuts at $1 each.
You might enjoy jeans more if you pay more for them.
Could also be signalling/game theory.
Kaplan (Haifa)
micro
November 2010
22 / 30
Utility over time.
Say there are two time periods t=0 and 1.
We can have a utility function over consuming a good at time 0 or
time 1. u (c0 , c1 )
We like people to have time impatience: u (x + c, x ) > u (x, x + c )
for all x and c.
Kaplan (Haifa)
micro
November 2010
23 / 30
Common form
In economics, the most common form of utility is
∞
V (c ) =
∑ β t U ( ct )
t =0
(where β < 1)
Does this satisfy time impatience?
Stationary means that if (c0 , c1 , c2 , c3 , . . .)
(c1 , c2 , c3 . . .) (d1 , d2 , d3 . . .).
(c0 , d1 , d2 , d3 . . .), then
Is the above utility stationary?
Kaplan (Haifa)
micro
November 2010
24 / 30
Hyperbolic Discounting
Hyperbolic Discounting
“Use whatever means possible to remove a set amount of money from
your bank account each month before you have a chance to spend it.”
–Advice in New York Times:“Your Money” column [1993]
People perhaps do best to commit not to spend money: pension
funds, not having credit cards.
Can our tools model this?
Kaplan (Haifa)
micro
November 2010
25 / 30
Hyperbolic Discounting Question
Do you prefer $100 today vs. $200 two years from now?
Do you prefer $100 six years from now vs. $200 eight years from now?
What does this violate?
Kaplan (Haifa)
micro
November 2010
26 / 30
Hyperbolic Discounting
Our typical discounting the future:
At time 0, u = ∑t∞=0 βt U (ct ). At time 1, u = ∑t∞=1 βt
Instead here we could use:
1
U ( ct )
∞ 1
1
At time 0, u = ∑t∞=0 1 +
t U (ct ) At time 1, u = ∑t =1 t U (ct )
Why does this work in explaining behaviour?
Utility is Ln ct for t = 1, 2. (This is Cobb Douglas).
Interest rate is 0% real (Japan/US).
We receive income of m at time 1.
At time zero we decide between consuming at time 1 and time 2.
Instead say at time 1 we make the same decision
Kaplan (Haifa)
micro
November 2010
27 / 30
Hyperbolic Discounting
At time 0, we want to save
At time 1, we want to save
1
3
1
1
2+3
1
2
1
1
1+2
=
2
5
of income.
=
1
3
of income.
Conclusion: Ripping up credit cards or enrolling in automatic pension
plans (opt out pension plans) is a good idea
Kaplan (Haifa)
micro
November 2010
28 / 30
References
The classic paper that reintroduced hyperbolic discounting into
economics.
D. Laibson, “Golden Eggs and Hyperbolic Discounting,” Quarterly
Journal of Economics 112 (1997), 443-77.
A paper that strongly critiques the use of such utility functions.
A. Rubinstein, “Economics and Psychology? The Case of Hyperbolic
Discounting.” International Economic Review, 44 (2003), 1207-1216.
Kaplan (Haifa)
micro
November 2010
29 / 30
Homework 5b.
1
2
We have utility of ln x for t = 1, 3 and utiliy of 1.5 at time 2
(independent of how much we consume). We receive income of m at
time 1. Interest rate is 0% real (Japan/US). Discounting is hyperbolic
(1/(1 + t )). At time zero we decide between consuming at time 1
and time 3. (Notice the change.) How much income would we save?
Now say instead, at time 1 we make the same decision. How much
would we save then?
Normal discounting is βt for t periods in the future (where
0 < β < 1). The hyperbolic discounting example we had was
1/(1 + t ) for t periods in the future. Let us say that the discount is
quasi-hyperbolic: β δt for t periods in the future where (0 < δ < 1
and β > 0). Assume utility u (xt ) = ln(xt ) and a consumer has
income m and consumes in periods 1 and 2. If the consumer decides
in period 0 what to consume in periods 1 and 2, what is x1 and x2 ? If
the consumer decides in period 1 what to consume in periods 1 and 2,
what is x1 and x2 ? For what range of β and δ does the consumer’s
Kaplan to
(Haifa)
micro
November 2010
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desire
consume early increase?