1. (15 marks) Suppose that a decision maker has $100. They need to choose how many kilos of chocolate (c) and vegetables (v) they will purchase. Chocolate costs $2 per kilo and vegetables cost $1 per kilo. They are choosing c and v to maximise the utility function U (c, v) where a is a constant and 0 < a < 1. = 5 α 1 ca + -va α a) (5 marks) Formally write down the decision maker's optimisation problem and the asso- 1 2 ciated Lagrangian. b) (5 marks) Write down the first order conditions and find the values of c and v that satisfy those conditions. Your answer should be a function of a. c) (5 marks) Check the second order conditions for the solution(s) you found in part (b). Do these values of c and v solve the decision maker's problem?
1. (15 marks) Suppose that a decision maker has $100. They need to choose how many kilos of chocolate (c) and vegetables (v) they will purchase. Chocolate costs $2 per kilo and vegetables cost $1 per kilo. They are choosing c and v to maximise the utility function U (c, v) where a is a constant and 0 < a < 1. = 5 α 1 ca + -va α a) (5 marks) Formally write down the decision maker's optimisation problem and the asso- 1 2 ciated Lagrangian. b) (5 marks) Write down the first order conditions and find the values of c and v that satisfy those conditions. Your answer should be a function of a. c) (5 marks) Check the second order conditions for the solution(s) you found in part (b). Do these values of c and v solve the decision maker's problem?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:1. (15 marks) Suppose that a decision maker has $100. They need to choose how many kilos of
chocolate (c) and vegetables (v) they will purchase. Chocolate costs $2 per kilo and vegetables
cost $1 per kilo. They are choosing c and v to maximise the utility function
U (c, v)
where a is a constant and 0 < a < 1.
=
5
α
1
ca
+
-va
α
a) (5 marks) Formally write down the decision maker's optimisation problem and the asso-
1
2
ciated Lagrangian.
b) (5 marks) Write down the first order conditions and find the values of c and v that satisfy
those conditions. Your answer should be a function of a.
c) (5 marks) Check the second order conditions for the solution(s) you found in part (b). Do
these values of c and v solve the decision maker's problem?
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