Econ 3102: Problem Set 4

Econ 3102: Problem Set 4

This problem set is due on Tuesday, April 25th at the beginning of class. The maximum score is 100 points. Be sure to review the Syllabus for details about homework assignments and their grading. Feel free to contact me online (through the website) if you have speci_c questions about the assignment. Be sure to provide the intuition for relevant mathematical results.

Present all _nal answers neatly on these provided pages, except for the presentations section. Anything written on the back of each page will not be graded. Show any relevant calculations neatly. Please do your scratch work somewhere else. Please remember to attribute help received and collaboration.

Name:

(By writing your name or providing any means of personal identi cation you agree to the instructions listed above in this page and you are aware that in case you fail to comply with these rules your will lose 10 points in this problem set)

Names of students you worked with:

Food and Housing (if it isn’t answered you lose 64pts, 8pts each)

Consider a static (one-period) model of consumption. The representative consumer receives an endowment of food y. He has preferences over two goods, food (f ) and housing (h). His preferences are represented by the following utility function

u(f ) + u(h)

The function u has the usual properties (increasing in food and housing, and strictly concave in each good). Assume that the price of housing relative to consumption is 1.

• What share of the income y will the consumer allocate to food and housing? Explain your answer intuitively. Hint: Do not assume any utility function

Next, extend the model to T > 0 periods. Now, assume that = 1 and r = 0. With these changes the new lifetime utility is

_∑T

((u(f_t) + u(h_t))

t=1

In this case, assume that the consumer receives an arbitrary stream of food endowments fy₁; y₂; : : : ; y_T g. That is, the stream of endowments do not need to be constant. It could vary over time. Recall that the consumer knows the stream since t = 1. Finally, suppose that the price of housing relative to consumption is 1. With this new information answer the next couple of questions:

• What share of the income y will the consumer allocate to food and housing in each period t? Explain your answer intuitively. Hint: Do not assume any utility function

• What share of wealth will the consumer allocate to food and housing? Hint: Wealth is de_ned here as the present value of the stream of endowments received over the lifetime (from t = 1 to t = T ). Explain your answer intuitively. Hint: Do not assume any utility function.

Assume now that the consumer has no access to capital markets. That is, he cannot save or borrow. Still the lifetime utility function is given by

_∑T

((u(f_t) + u(h_t))

t=1

With this new information, answer the next couple of questions:

• What share of the income y will the consumer allocate to food and housing in each period t? Explain your answer intuitively. Hint: Do not assume any utility function

• Will the amount allocated to food and housing in each period t be the same over time? Ex-plain your answer intuitively. Hint: Do not assume any utility function

Now, assume that the consumer discounts the future with . That is, the lifetime utility function is

∑^T

^{t  1}((u(f_t) + u(h_t))

t=1

To gain intuition on this equation notice that if T = 2 (the case discussed in class), we have that the lifetime utility function is

u(f₁) + u(h₁) +  (u(f₂) + u(h₂))

With this new information (consider a general T ), answer the next question:

• Will your answers in (d) and (e) change? Why? Explain

Next, assume = 1, r = 0, T = 2, y₁ = 1, and y₂ = 2. Consider the following two cases separately

• Full access to capital markets (consumer can save and borrow)
• No access to capital markets whatsoever

• Under cases (1) and (2), how much would this consumer allocate to food and housing?

• In which case will this consumer be better-off? Provide arguments to your answer. You could use a graph (with indifference curves and a budget constraint, as in class, to support your point)

Presentation (if it isn’t answered you lose 36pts, 6pts each)

Note: Everyone is asked to solve this problem

(This time do not prepare Power Point slides. The remaining groups will make the presentation using the board. Read the paper by Aguiar and Hurst (2008), pages 1-3 (skip the section\Empirical tests of canonical model”) and answer the following questions (notation follows that of Aguiar and Hurst (2008)). Consider the endowment economy without government in which the consumer takes the interest rate as given. Always assume that (1 + r) = 1 and do not assume any utility function in any case. Finally, assume that the consumer receives an endowment in each period of his lifetime. The initial period is t = 0. In each of the following cases _nd an algebraic expression for optimal consumption in period t = 0. You will _nd that the formula for optimal consumption obeys the following form: c₀ = W , where is a constant and W is the present value of endowments during his lifetime.

1. (-5pts) Assume that the consumer lives one period. His endowment is denoted by y₀. Find in this case. Call it ¹

2. (-5pts) Assume now that the consumer lives two periods (from t = 0 to t = 1). His endow-ments are denoted by y₀ and y₁. Find for this case. Call it ²

3. (-5pts) Assume now that the consumer lives three periods (from t = 0 to t = 2). Find for this case. Call it ³

4. (-5pts) Assume now that the consumer lives T + 1 periods (from t = 0 to t = T ). Find for this case. Call it ⁴

5. (-5pts) Take the limit as T ! 1 of the expression for optimal consumption you found in (4) and argue that there is a typo in one of the equations of the paper (pages 1-3). Hint: You

may want to use the following fact. Suppose d 2 (0; 1). Then it follows that ^∑1 d^t =  ¹

t=0          1  d

6. (-5pts) Find in (5). Call it  ⁵

7. Bonus (5pts) What is the relationship among ¹,  ²,  ³,  ⁴,  ⁵? Explain