Unformatted text preview: and state all
assumptions.
(b) State  but do not solve – a “simplified” Utility Maximization Problem (UMP). That is, state the UMP by simplifying
and/or eliminating some constraints from the general UMP. Show all calculations and state all assumptions. Assume all
pecuniary parameters are
.
(3.9) Given a utility function of the form ( ) defined over consumption set { } (a) Solve the UMP and derive the demand functions for goods 1 and 2 and (any) Lagrange multipliers for a consumer.
Show all calculations and state all assumptions.
()
()
[Hint: 2
ECO 204 Chapter 3: Practice Problems & Solutions for Utility Maximization in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. (b) Calculate the price, crossprice and income elasticity of goods 1 and 2. Show all calculations and state all
assumptions.
(c) Calculate the marginal optimal utility of income by the envelope theorem and the “value function” approach, i.e.
calculate . Show all calculations and state all assumptions. (d) All else equal, what is the impact on the demand functions for goods 1 and 2 and the optimal utility due to a
successful advertising campaign for good 2? Show all calculations and state all assumptions.
(3.10) Consider a consumer who has monotone preferences and perceives goods 1 and 2 as “perfect substitutes”.
{
}
Assume all pecuniary parameters are strictly positive and that the consumption set is
(a) Without solving the UMP, state and graph the various possible optimal choice “cases”.
(b) In general, which of the following two excise tax scheme will have a larger impact on the consumer’s optimal utility: a
$0.01 quantity tax on good 1 versus a $0.01quantity tax on good 2? Show all calculations and state all assumptions.
[Hint: Don’t forget there can be various possible optimal choices].
) defined over the consumption set
(3.11) Consider a consumer with an arbitrary utility function (
{
}. Prove that if the consumer has monotone preferences then the marginal optimal utility of income is always
strictly positive, i.e. . Show all calculations and state all assumptions. [Hint: Set up the UMP and write down the first order conditions and (any) KuhnTucker cases]. Please remember that the consumer has an arbitrary utility function
(so that we have no idea what her indifference curves look like).
(3.12) A consumer has the following utility function over two commodities (“goods”): (a) For what range of and does the consumer have monotone preferences over goods 1 and 2? Show all
calculations and depict the direction of “more is better” preferences graphically.
(b) Based on the answer to part (a) sketch some indifference curves.
(c) Does this consumer have strictly convex preferences? What about convex preferences? Briefly explain your answer.
[Hint: Use the answers to parts (a) and (b)].
(d) Now suppose the consumer does not have an income constraint (i.e. the consumer has unlimited income) and
. Solve for...
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 Fall '09
 AJAZHUSSAIN
 Microeconomics, Utility, Department of Economics, S. Ajaz Hussain, Sayed Ajaz Hussain

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