Download x 1 *,x 2

Chapter Five
Choice
选择
Structure
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
5.1 The optimal choice of consumers
5.2 Consumer demand
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Interior solution （内解）
Corner solution （角解）
“Kinky” solution
5.3 Example: Choosing taxes
5.1 The optimal choice of consumers

The goal of consumers: maximizing utility
subject to the budget constraint
The optimal bundle of goods

Must be on the budget line



points to the left and below the budget line are no
equilibrium. Why?
points to the right and above are no equilibrium
either. why?
Must on the highest indifference curve that
touches the budget line.
The optimal choice
Movies
Highest attainable
utility is U2
M1
U2
U1
C1
U3
CD’s
The most preferred affordable bundle
x2
(x1*,x2*) is the most
preferred affordable
bundle.
x2*
x1*
x1
Equilibrium condition: Geometrically
Movies
Note that slopes are equal here!
M1
U2
U1
C1
U3
CD’s
Equilibrium condition
Rearranging gives Consumer
Equilibrium Condition
MUC/PC= MUM/PM
Movie
M1
C1
CD
Equal Marginal Principle

MUC/PC or MUM/PM : Marginal utiltiy per dollar of
expenditure.

Equal marginal principle: Utility is maximized when
the consumer has equalized the marginal utility per
dollar spent on all goods.

Why is this an equilibrium?
Disequilibrium Point
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
Movies
Disequilibrium
Equilibrium
M2
M1
U2
U1
C2 C1
CDs
Suppose you are at M2, C2.
5.2 Consumer demand



The optimal choice ---the consumer’s
ORDINARY DEMAND （一般需求）at the
given prices and income.
The consumer’s demand functions give the
optimal amounts of each of the goods as a
function of the prices and the consumer’s
income, x1*(p1,p2,m) and x2*(p1,p2,m).
How to compute the optimal x?
Case1: Interior solution

When x1* > 0 and x2* > 0 the demanded
bundle is called INTERIOR solution.
Solve for interior solution (method 1)
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(x1*,x2*) satisfies two conditions:
(a) p1x1* + p2x2* = m
(b) tangency
Solve for interior solution (method 2)

The conditions may be obtained by using the
Lagrangian multiplier method, i.e.,
constrained optimization in calculus.
Example 1: Cobb-Douglas preference

Suppose that the consumer has CobbDouglas preferences.
U( x1 , x 2 )  x1axb2
Computing Ordinary Demands - a
Cobb-Douglas Example.
So we have discovered that the most
preferred affordable bundle for a consumer
with Cobb-Douglas preferences
U( x1 , x 2 )  x1axb2
is
( x*1 , x*2 ) 
(
)
am
bm
,
.
( a  b)p1 ( a  b)p2
Corner solution



But what if x1* = 0?
Or if x2* = 0?
If either x1* = 0 or x2* = 0 then maximizing
problem has a corner solution (角解) (x1*,x2*).
Example 2-- Perfect Substitutes
x2
MRS = 1
x1
Example 3: ‘Kinky’ Solutions -Perfect Complements
x2
U(x1,x2) = min{ax1,x2}
x2 = ax1
x1
‘Kinky’ Solutions -- the Perfect
Complements Case
x2
U(x1,x2) = min{ax1,x2}
*
x2 
x2 = ax1
am
p1  ap 2
x*1 
m
p1  ap 2
x1
5.3 Choosing Taxes: Various Taxes

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Quantity tax: on x: (p+t)x
Value tax: on p: (1+t)p


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Also called ad valorem tax
Lump sum tax: T
Income tax:

Can be proportional or lump sum
Income Tax vs. Quantity Tax
Proposition: Suppose the purpose of taxes is to
raise the same revenue, then consumers are
better off with income tax than with quantity
tax on a certain commodity.
Proof:

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