Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility...

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Rational Choice

CHOICE

1. Scarcity (income constraint)

2. Tastes (indifference map/utility function)

ECONOMIC RATIONALITY

The principal behavioral postulate is that a decision-maker chooses its most preferred alternative from those available to it.

The available choices constitute the choice set.

How is the most preferred bundle in the choice set located/found?

RATIONAL CONSTRAINED CHOICE

Affordablebundles

More preferredbundles

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

MRS=x2/x1 = p1/p2

Slope of the indifference curve

Slope of the budget constraint

Individual’s willingness to trade

Society’s willingness to trade

At Equilibrium E

The most preferred affordable bundle is called the consumer’s ORDINARY DEMAND at the given prices and income.

Ordinary demands will be denoted byx1*(p1,p2,m) and x2*(p1,p2,m).

The slope of the indifference curve at (x1*,x2*) equals the slope of the budgetconstraint.

(x1*,x2*) satisfies two conditions: (i) the budget is exhausted, i.e.

p1x1* + p2x2* = m; and (ii) the slope of the budget constraint,

(-) p1/p2, and the slope of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*).

COMPUTING DEMAND

How can this information be used to locate (x1*,x2*) for given p1, p2 and m?

Two ways to do this

1. Use Lagrange multiplier method

2. Find MRS and substitute into the Budget Constraint

COMPUTING DEMAND Lagrange Multiplier Method

Suppose that the consumer has Cobb-Douglas preferences

and a budget constraint given by

mxpxp 2211

aaxxxxU 12121 ),(

Set up the Lagrangian

mxpxpxx aa 2211

121 tosubject max

xpxpmxxL aa

Differentiate

(2) 0)1(

xpxpmL

From (1) and (2)

Then re-arranging

Rearrange

Remember

mxpxp 2211

Substitute

mxpxp 2211

Solve x1* and x2*

COMPUTING DEMAND Method 2

Suppose that the consumer has Cobb-Douglas preferences.

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

Suppose that the consumer has Cobb-Douglas preferences.

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

ax xa b1

bx xa b2

So the MRS is

MRSdxdx

U xU x

So the MRS is

At (x1*,x2*), MRS = -p1/p2 , i.e. the slope of the budget constraint.

MRSdxdx

U xU x

So the MRS is

At (x1*,x2*), MRS = -p1/p2 so

MRSdxdx

U xU x

** *. (A)

(x1*,x2*) also exhausts the budget so

p x p x m1 1 2 2* * . (B)

So now we know that

* * (A)

p x p x m1 1 2 2* * . (B)

So now we know that

* * (A)

p x p x m1 1 2 2* * . (B)

Substitute

So now we know that

* * (A)

p x p x m1 1 2 2* * . (B)

p x pbpap

x m1 1 21

Substitute

and get

This simplifies to ….

a b p11

*2 )( pba

Substituting for x1* in p x p x m1 1 2 2

then gives

*1 )( pba

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 )(21

),(pba

COMPUTING DEMAND Method 2: Cobb-Douglas

a b p11

bma b p

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

Rational Constrained Choice

But what if x1* = 0 or x2* = 0?

If either x1* = 0 or x2* = 0 then the ordinary demand (x1*,x2*) is at a corner solution to the problem of maximizing utility subject to a budget constraint.

Examples of Corner Solutions: Perfect Substitutes

MRS = -1

Slope = -p1/p2 with p1 > p2.

MRS = -1

MRS = -1 (This is the indifference curve)

MRS = -1

Slope = -p1/p2 with p1 < p2.

ANOTHER EXAMPLE

So when U(x1,x2) = x1 + x2, the mostpreferred affordable bundle is (x1*,x2*)where

*1 ,0),(

if p1 < p2

if p1 > p2.

MRS = -1

Slope = -p1/p2 with p1 = p2.

The budget constraint and the utility curve lie on each other

All the bundles in the constraint are equally the most preferred affordable when p1 = p2.

Examples of ‘Kinky’ Solutions: Perfect Complements

X1 (tonic)

X2 (gin) U(x1,x2) = min(ax1,x2)

x2 = ax1 (a = .5)

MRS = 0

U(x1,x2) = min(ax1,x2)

x2 = ax1

MRS = -

MRS = 0

x2 = ax1

MRS = -

MRS = 0

MRS is undefined

x2 = ax1

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

x2 = ax1

Which is the mostpreferred affordable bundle?

x2 = ax1

The most preferredaffordable bundle

x2 = ax1

and p1x1* + p2x2* = m

x2 = ax1

(a) p1x1* + p2x2* = m(b) x2* = ax1*

(a) p1x1* + p2x2* = m; (b) x2* = ax1*

(a) p1x1* + p2x2* = m; (b) x2* = ax1*.

Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = m

(a) p1x1* + p2x2* = m; (b) x2* = ax1*.

Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = mwhich gives

*1 app

(a) p1x1* + p2x2* = m; (b) x2* = ax1*.

x2 = ax1

p ap11 2

amp ap

Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility...

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