Explain why his intertemporal budget set B can be de-scribed as follows:B =(c0; c1; c2) (0; 0; 0) : c0 + 12 c1 + 14 c2 2Initially suppose that Brianís choices over consumption streams are gov-erned by an exponential discounted utility family of preference relations,characterized by an ìinstantaneousîutility function u (c) = 2c c2=2 and adiscount factor = 1=2
Question
Explain why his intertemporal budget set B can be de-scribed as follows:B =(c0; c1; c2) (0; 0; 0) : c0 + 12 c1 + 14 c2 2Initially suppose that Brianís choices over consumption streams are gov-erned by an exponential discounted utility family of preference relations,characterized by an ìinstantaneousîutility function u (c) = 2c � c2=2 and adiscount factor = 1=2
Solution
The intertemporal budget set B represents all possible combinations of consumption in different periods that Brian can afford given his wealth and the interest rate.
Brian's wealth is $2 million, and he plans to consume this wealth over three periods (c0, c1, c2). The interest rate is 100% per period, which means that any money he does not spend in a period will double in the next period.
The equation c0 + 1/2 c1 + 1/4 c2 ≤ 2 represents Brian's budget constraint. Here's how it works:
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c0 is the amount Brian consumes in the first period. This comes directly out of his wealth, so it's just c0.
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c1 is the amount Brian consumes in the second period. However, because of the 100% interest rate, every dollar Brian doesn't spend in the first period becomes two dollars in the second period. So, the amount Brian can afford to consume in the second period is actually twice what he saves from the first period. This is why the equation has 1/2 c1 instead of just c1.
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Similarly, c2 is the amount Brian consumes in the third period. Every dollar Brian doesn't spend in the first period becomes two dollars in the second period and then four dollars in the third period, thanks to the 100% interest rate. So, the amount Brian can afford to consume in the third period is four times what he saves from the first period. This is why the equation has 1/4 c2 instead of just c2.
The inequality ≤ 2 means that the total amount Brian consumes over the three periods cannot exceed his initial wealth of $2 million.
Brian's preferences over consumption streams are governed by an exponential discounted utility family of preference relations, characterized by an instantaneous utility function u(c) = 2c - c^2/2 and a discount factor δ = 1/2. This means that Brian derives utility from consumption (the more he consumes, the more utility he gets), but his utility decreases the more he consumes (because of the -c^2/2 term). The discount factor δ = 1/2 means that Brian values consumption in the present more than consumption in the future.
Similar Questions
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