ECM190 Seminar 1 - outline answers PDF

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ECM190 – MICROECONOMICS II ECM101 – MICROECONOMIC POLICY (Part 2) AUTUMN 2016 Outline Answers

SEMINAR 1: Further Topics in Consumer Theory 1. Marshallian and Hicksian demand a) Using indifference curve analysis, derive the Hicksian demand curve. Explain why the Hicksian demand curve is called the compensated demand curve. See diagram in separate file. Given the initial price of x1, p1, the budget line is BL1 and utility is maximised at point A, where x1* units of good x1 are consumed. If the price rises to p1’, the budget line rotates inwards to BL2, and the new utility maximising choice is at B. The compensated budget line shows the consumer’s opportunity set if they were given sufficient monetary compensation to make them as well off as they were before the price change; in this case, utility is maximised at C and x1H units of good x1 are consumed – this represents the consumer’s Hicksian or compensated demand when the price rises to p1’. These two points are illustrated in the lower panel, and used to derive the consumer’s Hicksian demand curve. The Hicksian demand curve is known as the compensated demand curve because it represents the level of demand as the price changes if the consumer were given sufficient monetary compensation to make them as well off (i.e. allow them to achieve the same level of utility) as they were before the price change.

b) The Hicksian demand curve will always be steeper than the Marshallian demand curve if the good is a normal good. Is this statement true or false? Justify your answer. This is true, since the Hicksian demand curve captures the pure substitution effect. If the good is a normal good, the income effect will go in the opposite direction of the price change, so it reinforces the substitution effect, which the impact of a change in price on demand will be greater.

2. If two goods are perfect complements, the consumer’s utility function is given by: u ( x 1 , x 2 ) =min { x 1 , x 2 } . If the goods are perfect substitutes, the utility function is: u ( x 1 , x 2 ) =a x 1+ b x 2 . Discuss the nature of the expenditure function and the Hicksian demand function in each of these two cases. Perfect complements:

1

The expenditure function represents the minimum expenditure required to achieve a given level of utility, say u*: E=min p1 x 1+ p2 x 2 subject ¿ min { x1 , x2 } ≥ u¿ If the goods are perfect 1:1 complements, at the utility maximising choice: ¿

¿

¿

x 1 =x 2=u

So for any target level of utility u, the expenditure function is given by: E ( p1 , p2 , u¿) = p1 x❑1 + p2 x 2❑=( p1+ p 2) u (This could also be shown/explained using a diagram. One of the key things to remember here is that the expenditure function is a function of prices and utility, and not directly of quantities of the goods) With perfect complements, there is no substitution effect, so after compensation you end up with exactly the same utility maximising bundle as initially. Therefore if two goods are perfect complements, the Hicksian demand curve will be a vertical line.

Perfect substitutes: The expenditure function solves the minimisation problem given above, but in the case of perfect substitutes, the consumer will spend all their income on the cheapest good. To keep things simple, we will assume a=b=1. There are two main cases to consider: 1. p1< p 2 : Then the utility max choice is to spend all money/expenditure on good 1, so x 1=

E , x =0 p1 2

which implies utility

u=

E p1

or alternatively

E=u . p1

2. p1> p 2 : Then the utility max choice is to spend all money/expenditure on good 2, so x 1=0 , x2 =

E p2

which implies utility

u=

E p2

or alternatively

E=u . p2

So we could write the general expenditure function as: E ( p1 , p2 , u¿) =u . min { p1 , p2 }

2

(Again, this could be shown/explained using a diagram instead).

3. In the intertemporal choice model discussed in the lecture, it was assumed individuals could save or borrow at the same rate. In practice, capital markets are imperfect and individuals face different interest rates for saving and borrowing. a) In a diagram, illustrate the consumer’s budget constraint and choice set if the interest rate on borrowing is greater than the interest rate on savings. See diagram in separate file. The key thing is to make clear that the lower line segment (where the individual is a borrower) is steeper b) Illustrate and explain under what conditions a consumer would save and under what conditions they would borrow. When would it be rational for a consumer to neither save nor borrow? The consumer will save if: MRTP=− ( 1+r ) The consumer will borrow if:

MRTP=− ( 1+r ')

A complete answer should illustrate and explain these two conditions (using the intertemporal choice model covered in the lecture). It would be rational to neither save or borrow if utility is maximised at E (in diagram in separate file)

4. How can we explain why individuals buy both insurance against losses and lottery tickets? Is this behaviour inconsistent with expected utility theory? (You should read the article by Friedman and Savage on the reading list for this topic to help you in answering this question). To answer this question, you should illustrate and explain the Friedman-Savage utility function, and explain how it reconciles both risk averse and risk seeking behaviour.

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