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Unformatted text preview: the optimal choice of goods 1 and 2, and the optimal utility. Show all calculations.
(e) For the answer in part (d) what is the impact on the optimal utility due to a small increase in
calculations. and ? Show all (f) Now suppose the consumer has an income of
. Assume the consumer always spends
her entire income. Solve for the optimal choice of goods 1 and 2, and the optimal utility. Show all calculations.
ECO 204 Chapter 3: Practice Problems & Solutions for Utility Maximization in ECO 204 (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. (g) For the solution to part (f), will a $1 increase in income (i.e. ) make the consumer “happier”? (h) For this part, suppose the consumer has an income of
(these are different prices from
earlier parts). Solve for the optimal choice of goods 1 and 2, and the optimal utility for the case when expenditure ≤
income. Show all calculations.
(i) For the solution to part (h), will a $1 increase in income (i.e. ) make the consumer “happier”? (3.13) Suppose a consumer is a price taker and faces uniform prices. True or false: the budget set
is a convex set and therefore the consumer has convex preferences? Briefly explain your answer.
(3.14) Consumers preferences are given by the utility function () () (a) Graph some indifference curves. Does this utility function have increasing, constant, or decreasing MRS? Show all
(b) Does the consumer have convex preferences? Explain.
(3.15) A consumer has the following utility function over the expenditure on good 1 ( The consumer’s income is and the price of good 2 is (a) Solve for the optimal and ) and good 2 ( ): . . Show all calculations and state all assumptions. (b) Derive the demand function for good 1. Show all calculations.
(c) Will the optimal solutions in parts (a) and (b) be identical to the solutions from the UMP with the utility function:
() () () ? Explain. (3.16) A consumer has the utility function: (a) What type of utility function is this?
(b) Trueo over the intuition once and for all):
The following graphs are for the problem: In each case, look at change in due to an infinitesimally small increase in 7
ECO 204 Chapter 3: Practice Problems & Solutions for Utility Maximization in ECO 204 (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. (3.2) Write down ● the Lagrangian objective function ● the first order conditions ● (any) Kuhn-Tucker conditions ● and
indicate the possible “signs” of the Lagrange multipliers (i.e. positive, zero, negative) for the following revenue
maximization problem (do NOT solve the problem):
The problem is:
Setup the Lagrangian objective function:
The FOC and KT conditions are:
() [ According to the way inequality optimization problems are done in ECO 204 Lagrange multipliers on inequality
constraints can be either positive or zero (but NOT negative). Review your lecture slides for why this is the case...
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