Decompose the total effect of the price change into the income and substitution effect.

Universidad Carlos III
Master in Economics/Master in Industrial Economics and Markets
Microeconomics I
2015-2016
Problem set 11

Exercises in Goolsbee, Levitt, and Syverson, Microeconomics, 2013.
1. Chapter 4: 1, 7, 17.
2. Appendix to Chapter 4: 1, 2, 3, 4, 5.
3. Chapter 5: 4, 6, 7, 8, 9, 13, 18, 19, 20
4. Appendix to Chapter 5: 1, 2

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Marshallian (uncompensated) demand functions

Let prices and income be denoted by (pX , pY , m). Consider consumers the following utility functions
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1

U (x, y) = x 3 y 3

(1)

U (x, y) = ln x + ln y

(2)

U (x, y) = min {3x, y}

(3)

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1

2

U (x, y) = x 2 + y 2
1

(4)

U (x, y) = 2x 2 + y,

(5)

U (x, y) = x + ln y

(6)

U (x, y) = 2x + y

(7)

1. For each one of them:
(a) Derive marginal rates of substitution.
(b) Derive marshallian demand functions. For cases (5)-(7), pay attention to the cases in which solutions
are not interior.
(c) Compute marshallian demands and utility when (pX , pY , m) = (5, 10, 100).
(d) Compute marshallian demands and utility when (pX , pY , m) = (5, 5, 100).
2. Assume that pX = 5 and that there are are 30 consumers with utility function (1), 20 consumers with
utility function (6) and 10 consumers with utility function (7) and that all of them have income m = 100.
Derive the market demand function for good Y .

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Hicksian (compensated) demand functions

Let prices and income be denoted by (pX , pY ). Consider utility functions (1)-(6) above. For each one of them:
1. Derive hicskian demand functions. For cases (5) and (6) you can limit yourself to the interior solutions.
2. Compute hicksian demands when (pX , pY ) = (5, 10) and when utility is the one that can be attained
under (pX , pY , m) = (5, 10, 100).
3. Consider the case in which (pX , pY , m) = (5, 10, 100). Suppose that the government subsidizes the consumption of good Y with a 50% subsidy on its price, so that the price the that the consumer has to pay
is only 5.
(a) Decompose the total effect of the price change into the income and substitution effect.
(b) What would be the cost of the subsidy to the government?
(c) What income tax would exactly compensate the price subsidy?
(d) What income subsidy would be equivalent to the price subsidy for the consumer?
(e) Which would be better for the government, the price subsidy or the income subsidy?

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