diploma micro 2007 week 2 lecture 2

7/31/2019 Diploma Micro 2007 Week 2 Lecture 2

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1. The Algebra of the Tangency Condition

Tangency

Slope of indiff. curve = Slope of budget constraint

(signs?)

marginal rate of substitution =2

1

P

P


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a) Utility

U = U(X1, X2 )

1 2 1 2( , ) ( , )?A A B B

U X X or or U X X > = <

1 2 1 2( , ) ( , )?A A C CU X X or or U X X > = <

Units of utility dont matter (well see why on next page)At any point on indifference curve:

U(X1, X2 ) = constant


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b) Marginal Utility

MU1 = increase in utility from an extra unit ofX1

Slope of utility mountainin calculus: 1 21

1

( , )U X XMU

X

=

For small changes,

1 1 2 2U MU X MU X = +

But as we move along an indifference curve 0U = !

1 1 2 2

1 2

2 1

0MU X MU X

X MURS

X MU

+ =

= =

Look carefully at signs and suffixesWhich assumption on preferences ensures that ICs

have negative slope?


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c) The Tangency Condition

Slope of indiff. curve = Slope of budget constraint

-MRS = -P2 / P1

cancel signs and substitute for MRS

MU2 / MU1 = P2 / P1

(or, using calculus:

2 2

1 1

/)

/

U X P

U X P

=

NB Only relative MU matters, so units of utilitydont matter

Q: Does tangency condition always hold?

A: No, may only hold as an inequality


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2. Interior vs Corner Solutions


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Corner solution inequalities vs tangency conditions

Take two combinations ofX1andX2 that both satisfy

the budget constraint:

2 21 12 2

1 1 1 1

21 1 1 2

1

;A BA B

B A

P PY YX X X X

P P P P

PX X X X

P

= =

= =

Now look at the associated increase in utility

21 1 2 2 1 2 2 2

1

22 1 2

1

PU MU X MU X MU X MU X P

PMU MU X

P

= + = +

=

Which gives a clear rule: if term in brackets is

positive, increaseX2; if it is negative, decrease it.

At a tangency condition:

2 2 22 1

1 1 1

0MU P P

MU MUMU P P

= =

So no change inX2can raise utility further.

At the corner solution in diagram

2 2 22 1

1 1 1

0MU P P

MU MUMU P P

<


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3. Income Effects

Slope of budget constraint doesnt change: whynot?

Is consumer better off?What happens if income andboth prices increaseby same %?

ie, ifkis some constant

( )

( )

( )

1 1

2 2

1

1

1 ?

B A

B A

B A

Y Y k

P P k

P P k

= +

= +

= +


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b) Income change with superior vs inferior

goods

Which good is inferior?With only two goods can both goods be inferior?


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4. Price changes

Which price has changed?Has it risen or fallen?(Hint: look at intercepts)

Is consumer better off?MustX2 increase?(Hence mustdemand curves slope downwards?)MustX1 decrease?


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Ambiguities when price changes

Is it possible that X2 < 0?


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Counter examples:

a) An inferior good

A to C = ?C to B = ?Does demand curve slope downwards?For a given substitution effect will it be more or

less elastic than for a normal good?

X2


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b) A Giffen Good

In general:

In all cases the substitution effect is ofopposite sign to price change

Income effects are of same sign for normalgoods but ofopposite sign for inferior goods

Caveat: income effects of price changes aretypically only significant for goods that have

large budget shares (as in our diagrams!)


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5. Limiting Cases of Price Effects

a) Initial equilibrium with.

Perfect Complements

Perfect Substitutes

Neither implies initial tangency condition


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b)Impact of price changes in limiting cases

Perfect complements

Substitution effect = ?Price changes do matter, but only via income

effects


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Perfect Substitutes

At point A,MU2 / MU1 < or > P2 / P1 ?At point B,MU2 / MU1 < or > P2 / P1 ?Effectively, only price matters


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6. Deriving Demand Curves

a) An individuals demand curve


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b) Market demand curves

Market demand curves are horizontal sums ofindividual demand curves

Does each individual demand curve need to beidentical?

NB, for mathematicians, economists appeartodraw diagrams the wrong way around!

2 2

1

( ) ( ) ( ) .... ( ) ( )N

N i

i

Q P Q P Q P Q P Q P=

= + + + =


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More to come from the 2 Good Model

Impact of taxes and subsidiesGoods vs leisureLink with firms cost functionsGoods today vs goods tomorrow(Eventually) general equilibrium


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Reading for Week 3

For Monday Lecture

P&R (5th Edition)7.1 (Measuring cost) [7.2 (Cost in the short run)]

8.1 (Perfectly competitive markets)

up to

8.4 (Choosing output in the short run)

Or

Varian (6th

edition)

16 (Equilibrium): 16.1 to 16.4

19 (Profit maximization): 19.1, [19.2]

22 (Firm supply) 22.1 to 22.5

For Tuesday Lecture

Perloff Handout

(Also available on website)

P&R 3.4 (Revealed Preference) [Ch 4 appendix

p144 (Duality in Consumer Theory)]

Varian Chapter 7 (Revealed Preference): 7.1 to 7.3


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Class Exercise for Week 3

1. As in the last exercise, a consumer has income in current prices of

Y, which she can spend on two goods, quantities of which are

measured byA andB, and prices of which are PA and PB. Her

preferences are well-behaved.

a) Give an algebraic statement of the tangency condition.

b) Show geometrically a utility-maximising equilibrium in which

the consumer does notconsume goodA (hint: P&R also give a

similar example).

c) From an initial tangency condition, in a diagram withA on they-

axis andB on thex-axis, show the optimal response, using budget

constraints and indifference curves, to a rise in Y,

(i) if both goods are normal;

(ii) if good A is inferior.

d) Can bothA andB be inferior goods?

e) Show in a diagram the optimal response to a fall in the price ofB

(with the price ofA unchanged), assuming that bothA andB are

normal goods, and show how the total change can be broken down

into substitution (impact of price changes holding utility constant)and income effects.

f) Show the impact of a price change if the goods are either perfect

substitutes or perfect complements.

g) What is the impact on the consumers choice ifPA, PB,and Yall

increase by the same percentage? Interpret your answer.


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h) Show how to derive the consumers demand curve for good B

using an indifference curve diagram as above, with another diagram

below it showing consumption ofB on thex-axis, and the price of

goodB on they-axis.

2. Are the following statements true or false?

(i)A rise in income will always cause consumption of bothA and B

to rise

(ii)For normal goods, substitution effects and income effects of

price changes always reinforce each other

(iii) For inferior goods the income effect of a price change more

than offsets the substitution effect

(iii)Demand curves can never slope upwards.

(iv)For normal goods, a fall in the price of goodB can never lead to

an increase in consumption of goodA.

(v)If the price elasticity of demand (P/Q.Q/P) is minus 1, total

spending (price times quantity) on the good will not change when its

price changes.

(vii)If the price elasticity of demand is less than -1 (P/Q.

Q/

P < -1) total spending (price times quantity) on the good will rise when

price rises.

(viii)Normal goods have income elasticities greater than one

(ix)Inferior goods have income elasticities less than zero

diploma micro 2007 week 2 lecture 2

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