## (solution) Public Economics I need help with this problem set. I can pay 40

Public Economics
I need help with this problem set. I can pay 40 tutor credits if you answer all. Full answers
I completed question 2 and 4 but I am not sure if there are fine, so I prefer you to do it.

Problem Set 1

ECON 4821 - Public Economics

Fall 2016 This problem set is due on Sept. 27, 2016 at the beginning of class. Please hand in a hard copy.

Department policy requires that all homework assignments be typed except those portions which are

mainly computational/math. Problem 1 (Optimization)

Suppose that a consumer wants to maximize his happiness as expressed by utility function u(c1 , c2 ). Her income is M

and the prices of goods 1 and 2 are p1 and p2 , respectively. The consumer faces non-negativity constraints.

1. Write down the problem of the consumer. (5 pts.)

2. Write down the first order necessary conditions for c?1 , c?2 and ?? . Use the tools in the Math Review seen in Class

(Hint: You may solve this problem using either the Lagrange Multiplier Method or Kuhn-Tucker Theorem. If

using the Lagrange Multiplier Method, be sure to defend why it is appropriate, i.e. why the problem can be

characterized by equality constraints only.). (5 pts.)

3. Suppose that u(c1 , c2 ) = log c1 + A log c2 . Use the first-order conditions found above to find optimal c?1 , c?2 as

functions of p1 , p2 and M . (10 pts.)

( 1 ) ( A )

4. Show that if the utility function is changed to the Cobb-Douglas case u(c1 , c2 ) = c1 1+A c2 1+A , then the optimal

c1 and c2 are the same as in #3. (Hint: No need to re-derive the demand functions here. A few sentences

explanation should suffice.) (5 pts.)

5. Now suppose that instead of being equipped with income, the consumer purchases goods using the sale of her

endowment (e1 , e2 ). Show that in this case, her demand is unaffected if prices all are multiplied by a common

number. (5 pts.)

6. Now suppose that u(c1 , c2 ) = log c1 + c2 . Also, suppose that the consumer faces prices and income (p1 , p2 , m) =

(1, 5, 3). The consumer is faced with a budget constraint and two non-negativity constraints, one for each good.

Solve for c?1 and c?2 . Hint: The utility function guarantees that the demand for one of the goods is strictly positive.

What does this imply about its multiplier? (10 pts.) Problem 2 (Gruber 5th Edition, Chap. 2 #8)

Consider an income guarantee program with an income guarantee of \$5,000 and a benefit reduction rate of 40%. A

person can work up to 2,000 hours per year at \$10 per hour.

1. Draw the person?s budget constraint with the income guarantee. Be sure to place appropriate labels similar to

those found in the slides/lecture. (5 pts.)

2. Suppose that the income guarantee rises to \$7,500 but with a 60% reduction rate. Draw the new budget

constraint. Make sure it is visible. (10 pts.)

3. Which of these two income guarantee programs is more likely to discourage work? Explain briefly. (5 pts.)

1 Problem 3 (Data Work)

Go to the webpage of Economic Report of the President http://www.gpo.gov/fdsys/browse/collection.action?collectionCode=ERP

and click on the link for year 2015. Obtain Federal government receipts, outlays, surplus/deficit, debt and GDP from

Table B-19.

Construct the following time series:

1. federal receipts as % of GDP (3 pts.)

2. outlays as % of GDP (3 pts.)

3. deficit as % of GDP (3 pts.)

4. debt as % of GDP (3 pts.)

Then create the following charts.

1. A line chart with receipts and with outlays as % of GDP (both time series plotted together) (6 pts.)

2. A line chart with deficit as % of GDP (6 pts.)

3. A line chart with debt as % of GDP (6 pts.)

For each chart, write a paragraph that explains the trend of the line chart using your economic intuition. In other

words, think of economic events that may affect the evolution of the time series, such as an economic recession. Problem 4 (Social Welfare)

This problem comes in two parts. The second part is asked in Problem Set 2.

Consider an island with Tom Hanks and Wilson, and one good - coconuts. There is NO endowment of coconuts,

and to have something to eat Tom Hanks and Wilson have to work - climb the palm trees and gather coconuts. In one

hour, Tom Hanks can gather wT H and Wilson wW coconuts. Both Tom Hanks and Wilson have T hours at disposal.

Suppose that the utility functions for person i with i ? {T H, W } is

Ui (ci , `i ) = B i log ci + log `i

where ci is consumption of coconuts, `i is the time spent lying on beach and surfing (leisure), and B T H and B W are

positive constants.

1. Write down the problem of person i ? {T H, W }, if Tom Hanks and Wilson are on their own (in autarky), so if

no trade is possible. (5 pts.)

2. Write the constraint for person i from part (a) in the form where `i is on the same side as ci , keeping T on the

other side of the constraint. Interpret this constraint: what is the ?price? that person i has to ?pay? for one

hour of leisure? (5 pts.)

3. Find the expressions for optimal choices ci? and `i? . How do they depend on wi and B i , explain the intuition

behind this. (10 pts.)

4. Suppose that Tom Hanks and Wilson could trade if they wanted. Would they trade? Hint: Think about what

the person who would like to buy a coconut could offer in return for the coconut bought. Given your answer

what can we say about the competitive equilibrium allocation (allocation with trade), how is it different from

the allocation cT H? , `T H? , cW ? , `W ? from part (3)? (5 pts.)

5. Suppose now that Tom Hanks and Wilson decide to pool the coconuts they gather. Write down the aggregate

resource constraint for coconuts on this island - an equation that shows how total consumption of coconuts by Tom

Hanks and Wilson cT H +cW is limited by the amount of coconuts available that depends on `T H , `W , wT H , wW , T .

(5 pts.) 2

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