Assume that a consumer's utility over two years is a function of consumption during the twe years. Let the consumer's utility function be U(C.C) = In C + 3 In C2. where Ci is consumption during the 1st year, and C2 is the consumption during the 2nd year The consumer's income is Y, during the Ist year and Y, during the 2nd year. Let R denote the annual interest rate (0 < R< 20%), at which the consumer can choose te borrow or lend across the two years. The consumer's budget constraint is

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Chapter1: Making Economics Decisions
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Assume that a consumer's utility over two years is a function of consumption during the two
years. Let the consumer's utility function be
U(C1, C2) = In C + 3 In C2,
where Ci is consumption during the 1st year, and C2 is the consumption during the 2nd year.
The consumer's income is Y, during the Ist year and Y, during the 2nd year.
Let R denote the annual interest rate (0 <R< 20%), at which the consumer can choose to
borrow or lend across the two years. The consumer's budget constraint is
Ci +
C2
1+R
= Y +
1+R
We aim to find out the values of C, and C2, at which the consumer's utility is maximised.
(1) Write down the Lagrangian function, which can be used to maximise the consumer's utility
function subject to the budget constraint.
(2) Use the Ist order condition to derive the values of C and C2, at which the consumer's utility
is maximised subject to the budget constraint.
(3) Derive the Lagrange multiplier when the utility function achieves its maximum under the
budget constraint.
Transcribed Image Text:Assume that a consumer's utility over two years is a function of consumption during the two years. Let the consumer's utility function be U(C1, C2) = In C + 3 In C2, where Ci is consumption during the 1st year, and C2 is the consumption during the 2nd year. The consumer's income is Y, during the Ist year and Y, during the 2nd year. Let R denote the annual interest rate (0 <R< 20%), at which the consumer can choose to borrow or lend across the two years. The consumer's budget constraint is Ci + C2 1+R = Y + 1+R We aim to find out the values of C, and C2, at which the consumer's utility is maximised. (1) Write down the Lagrangian function, which can be used to maximise the consumer's utility function subject to the budget constraint. (2) Use the Ist order condition to derive the values of C and C2, at which the consumer's utility is maximised subject to the budget constraint. (3) Derive the Lagrange multiplier when the utility function achieves its maximum under the budget constraint.
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