Chapter 5 CHOICE
5.1 Optimal Choice

Optimal Choice
 the
bundle of goods that
is associated with the
highest indifference
curve that just touches
the budget line.
5.1 Optimal Choice

The first exception: the
indifference curve might
not have a tangent line.
5.1 Optimal Choice

The second exception:
the optimal point occurs
where the consumption
of some good is zero.
5.1 Optimal Choice

In general there may be
more than one optimal
bundle that satisfies the
tangency condition.
5.2 Consumer Demand

Demanded bundle
 The
optimal choice of good 1 and 2 at some set of
prices and income.

Demand function
 the
function that relates the quantities demanded to
the values of prices and incomes.
5.3 Some Examples

Perfect Substitutes
 when p1＜p2, x1=m/p1
 when p1=p2, x1 could be
any
number between 0 and m/p1
 when p1＞p2, x1=0
5.3 Some Examples

Perfect complements
 x1=x2=m/(p1+p2)
5.3 Some Examples

Neutrals and Bads
 The
consumer spends all income on the good she
likes and doesn’t purchase any of the neutral (bad)
good.
 If commodity 1 is a good and commodity 2 is a
bad, then the demand functions will be
x1= m/p1
x2=0
5.3 Some Examples

Concave Preferences
---The optimal choice is
the boundary point, z,
not the interior tangency
point, x, because z lies
on a higher indifference
curve.
5.3 Some Examples

Cobb- Douglas Preferences
u ( x1 , x2 )  x x
c d
1 2
x1=cm/(c+d)p1
x2=dm/(c+d)p2
p1x1/m= p1cm/m(c+d)p1= c/(c+d)
Lagrange multiplier method

The standard problem
max f (x) s.t. g1 (x)  0,

The Lagrangian
g m (x)  0
m
L  f (x)   i gi (x)
i 1

The F.O.C.
f (x) m gi (x)
  i
0
x j
x j
i 1
gi (x)  0, i  0, i gi (x)  0
Lagrange multiplier method
If the constraints behave in a “regular” manner,
any solution must satisfies the F.O.C.
 If the objective function is concave, and the
constraints are convex, then the F.O.C. is also
sufficient for optimum.

Lagrange multiplier method

Consumer’s problem
max u( x1 , x2 ) s.t.

Can be reformulated as
max u( x1 , x2 ) s.t.

p1 x1  p2 x2  m, x1  0, x2  0
p1 x1  p2 x2  m  0,  x1  0,  x2  0
Lagrangian
L  u( x1 , x2 )   ( p1 x1  p2 x2  m)  1 x1  2 x2
Lagrange multiplier method

F.O.C.
u ( x1 , x2 )
  p1  1  0;
x1
u ( x1 , x2 )
  p2  2  0;
x2
p1 x1  p2 x2  m  0,   0,  ( p1 x1  p2 x2  m)  0;
x1  0, 1  0, 1 x1  0;
x2  0, 2  0, 2 x2  0.
Lagrange multiplier method


The budget constraint will be always satisfied for
locally nonsatiated preferences.
The textbook way:
u ( x1 , x2 )
  p1 , with equality if x1  0;
x1
u ( x1 , x2 )
  p2 , with equality if x2  0.
x2
Lagrange multiplier method

What’s the advantage of Kuhn-Tucker F.O.C.?
 It

treats corner solutions.
What happens if x1>0 and x2=0?
u ( x1 , x2 )
  p1 ,
x1
u ( x1 , x2 )

x1
 i.e.,
u ( x1 , x2 )
  p2 ;
x2
u ( x1 , x2 )
 p1 p2 .
x2
the indifference curve is steeper than the budget line.
5.5 Implications of the MRS
Condition



In well-organized markets, it is typical that everyone
faces roughly the same prices for goods. People may
value their total consumption of the two goods very
differently.
Everyone who is consuming the two goods must have
the same marginal rate of substitution.
The fact that prices reflect how people value things
on the margin is one of the most fundamental and
important ideas in economics.
5.6 Choosing Taxes

Quantity tax
a

tax on the amount consumed of good.
Income tax
a
tax on income.
5.6 Choosing Taxes

quantity tax:
p1x1+p2x2=m
 budget
constraint
(p1+t)x1*+p2x2*=m
 revenue

R*=tx1*
budget constraint for an “equivalent” income tax
p1x1+p2x2=m-tx1*
5.6 Choosing Taxes



(x1*, x2*) is
affordable under
income tax.
Budget line under
income tax cuts the
indifference curve at
(x1*,x2*).
income tax is
superior to the
quantity tax.