MicroIII_Lecture4_经典的需求理论(1)Application2 Consumption Over Multiple Periods

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• We are left with only:
• Which simplifies to:
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• Now, look at periods j and j+1:
• What is the ratio?
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• Canceling terms we have simply: • Or:
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Utility Function
• But again we would like to assume separability. • The question is, what weight (or discount factor should we put on each factor) period?
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■ Today is Special
• People seem to show a high level of patience when planning for the future. – E.g. they are willing to put off having something in five years for something bigger in six. • However, they seem to really value having something today (very soon) compared to tomorrow (in the near future). • This is true at all points in the lives.
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Borrowing and Lending Revisited
• We assume that consumers can borrow and lend across each period at a gross interest rate of 1+r. • For now, assume that r is the same in every period. • Again the rate for borrowing is the same as the rate for lending. There are no credit constraints. • Borrowing and lending must balance out at the end of the last period.
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■ Hyperbolic Discounting
• Solution: a slight modification to exponential discounting:
• This is called hyperbolic discounting. • Tradeoffs at all future times are exponential, but today vs. the future gets an extra “kick” of < 1.
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■Spending Over the Life-Cycle
• What we have just done says that—assuming interest rates are more or less assuming constant—we should see a steady increase or decrease in consumption over the life we lifecycle. • Empirically this is totally false. • For most people consumption closely tracks current income, and falls in retirement. • Why do you think this is?
■Time Inconsistency
• Imagine you are planning out all your future consumption. • You consume a lot in period 1, because you want instant gratification. • But when planning future consumption, you are relatively patient, and not biased very much (if at all) towards consumption “today” vs. “tomorrow”.
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■Hyperbolic Discounting and FOCs • As you might expect, the FOCs with this utility function are:
• So what is “irrational” about this?
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• So you want to go back on your original plan and consume more than you had initially allotted for c2. • This is time inconsistency. Your preferences today are not consistent . with your preferences tomorrow. – Note that this was not the case with regular exponential discounting. • So what should you do about this?
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Exponential Discounting
• One nice thing about this formulation is that the relative difference between all periods is the same. • For example: – The discount from period 1 to period 4 is – The discount from period 5 to period 8 is • We will explore this more later.
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4.经典的需求理论 经典的需求理论
Application3- Consumption Over Multiple Periods 郑江淮
南京大学经济学院产业经济学系
Intermediate Microeconomics
Utility Function
• Now, instead of two periods, what if there are n periods? • How should we generalize our model from last time? • In general, we have
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■How to Model This?
• Exponential discounting does not capture this. • With constant interest rates, it says that for every two consecutive periods the tradeoff is the same. • We want a model that says: – In the long term I behave this way. – But in the short term I am impatient.
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■The Euler Equation: Extensions
• You should be able to show that between periods j and j + t the formula is:
• And if the interest rate is different every period (i.e. it is ri in period i) then the equation is:
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Exponential Discounting
• Usually we assume that the discount factor from one period to the next is proportionally constant:
• This is called exponential discounting discounting. • Why is this a natural assumption?
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• This is exactly the same as the two period case. • What is the intuition here? • Is consumption rising or falling with time? How do you know?
• Then our Lagrangian is:
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• This is easier than it looks. • Take the derivative with respect to cj:
• These are just sums, so all the terms with i≠j drop out!
• And the value of your period t consumption in period 1 dollars is:
• So we just convert everything to period 1 dollars.This is the general form of net present value.
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The Budget Constraint • Just as before, the loan market establishes an exchange rate between all periods. • If I borrow 1 dollar in period 1, how many dollars do I have to pay back in period 4?
• If I want 1 dollar in period 5, how many dollars do I have to invest in period 1?
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• So, the value of your income in period t in period 1 dollars is:
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■I Want It Now!
• There are a lot of explanations for this: – Credit constraints. – Changing needs over time. – Changing preferences over time. • We are going to focus on something more “irrational”: the desire for immediate gratification
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• In period 1 dollars:
• More succinctly:
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The Maximization Problem • We are solving:
• Subject to:
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Maximizing with Lagrangians
• Let Y be defined as the NPV of income:
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• But what happens when tomorrow comes around? • Suddenly your utility function looks like (note the change in the starting point:
• And your first FOC becomes: