Chapter Five

Choice

选择

Structure

5.1 The optimal choice of consumers 5.2 Consumer demand

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.

Movies

CD’s

M1

C1

Highest attainable utility is U2

U1

U2

U3

The optimal choice

The most preferred affordable bundle

x1

x2

x1*

x2*

(x1*,x2*) is the mostpreferred affordablebundle.

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CD’s

M1

C1

Note that slopes are equal here!

U1

U2

U3

Equilibrium condition: Geometrically

Rearranging gives Consumer Equilibrium Condition

MUC/PC= MUM/PM

Movie

CD

M1

C1

Equilibrium condition

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?

Equal Marginal Principle

Disequilibrium Point

Suppose you are at M2, C2.

Movies

CDs

M2

C2

U1

U2

C1

M1

Disequilibrium

Equilibrium

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)

(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 Cobb-

Douglas preferences.

U x x x xa b( , )1 2 1 2

Computing Ordinary Demands - a Cobb-Douglas Example.So we have discovered that the mostpreferred affordable bundle for a consumerwith Cobb-Douglas preferences

U x x x xa b( , )1 2 1 2

is( , )

( ),( )

.* * ( )x xam

a b pbm

a b p1 21 2

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

x1

x2

MRS = 1

Example 3: ‘Kinky’ Solutions -- Perfect Complements

x1

x2U(x1,x2) = min{ax1,x2}

x2 = ax1

‘Kinky’ Solutions -- the Perfect Complements Case

x1

x2U(x1,x2) = min{ax1,x2}

x2 = ax1

xm

p ap11 2

*

x

amp ap

2

1 2

*

5.3 Choosing Taxes: Various Taxes Quantity tax: on x: (p+t)x Value tax: on p: (1+t)p

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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