# Macroeconomics

Macroeconomic Theory and Policy – Midterm Exam | © Sanjay K. Chugh 8

Problem 2: Low Wages, High Wages, and Taxes (35 points). There are two categories of people: those that receive high real wages and those that receive low real wages. Denote these two real wages as Hw (high wages) and Lw (low wages), respectively, with

H Lw w . The utility function for each person, regardless of the real wage he/she receives, is identical: l( n, l) nu c c ll = + , in which, exactly as in Chapter 2, c stands for consumption and l stands for leisure. Furthermore, after defining n as labor, keep in mind that n + l = 1 (which is also identical to the framework considered in Chapter 2). The labor income tax rate for individuals that earn high wages is H (the Greek lowercase letter “tau”) and the labor income tax rate for individuals that earn low wages is L ; it is always the case that H L . The budget constraints for the low-wage individuals and high-wage individuals are, respectively,

)(1 )(1L L L L L Lc lw w−+ − = and

)(1 )(1H H H H H Hc lw w−+ − = , in which the “L” and “H” subscripts indicate, respectively, low-wage individuals and high- wage individuals.

The Lagrange function for low-wage individuals is

1) ln ln ) ( )( , , ( 1L L L L L L L L L L L L LL c l w wc l c l+ − −−+ − , and the Lagrange function for high-wage individuals is

1) ln ln ) ( )( , , ( 1H H H H H H H H H H H H HL c l w wc l c l+ − −−+ − .

Macroeconomic Theory and Policy – Midterm Exam | © Sanjay K. Chugh 9

Problem 2 continued a. (3 points) Based on the Lagrange function above for low-wage individuals, provide

the first-order conditions (FOCs) for consumption and leisure. Display the two FOCs clearly by drawing a box around each.

b. (3 points) Based on the Lagrange function above for high-wage individuals, provide

the first-order conditions (FOCs) for consumption and leisure. Display the two FOCs clearly by drawing a box around each.

Macroeconomic Theory and Policy – Midterm Exam | © Sanjay K. Chugh 10

Problem 2 continued c. (5 points) Based on the results obtained above, construct the consumption-leisure

optimality condition for low-wage individuals, which should be written as

… )(1 L Lw= − , in which the term in ellipses (“…”) is for you to determine. Display the final expression clearly by drawing a box around it. Provide important algebraic steps.

d. (5 points) Based on the results obtained above, construct the consumption-leisure optimality condition for the high-wage individuals, which should be written as

… )(1 HH w= − , in which the term in ellipses (“…”) is for you to determine. Display the final expression clearly by drawing a box around it. Provide important algebraic steps.

Macroeconomic Theory and Policy – Midterm Exam | © Sanjay K. Chugh 11

Problem 2 continued For the remainder of the analysis, suppose that H L= . For the sake of simplicity, suppose that neither of the wages Hw and Lw is affected by the particular fiscal policy H L= . Use the results obtained above in addressing part e. e. (12 points) With the policy setting H L= , could it be the case that low-wage

individuals’ optimal choices for both c and l are identical to the high-wage individuals’ optimal choices for c and l? More precisely, is it possible that the optimal values are such that both * *H Lc c= and

* * H Ll l= ?

If so, clearly and carefully provide important algebraic steps explaining why. If not, clearly and carefully explain why not. If it is impossible to determine, clearly and carefully describe why it is impossible to determine. (Note: no numerical analysis is needed and should not be included in your solution; the result is entirely analytic. Points will be deducted if the result relies on a numeric example.) Be as precise as possible in your results.

Macroeconomic Theory and Policy – Midterm Exam | © Sanjay K. Chugh 12

Part 2e continued (more space if needed)

Macroeconomic Theory and Policy – Midterm Exam | © Sanjay K. Chugh 13

Problem 2 continued (NOTE: IF THE RESULT(S) IN PART E ARE INCORRECT, IT MAY BE DIFFICULT, IF NOT IMPOSSIBLE, TO AWARD ANY POINTS FOR THE ENSUING PART F.) f. (7 points) In the ONE SINGLE consumption-leisure diagram below, clearly sketch

the result(s) obtained in part e. (Note: The qualitative diagram may be awarded ZERO points if it is not clearly and carefully labeled.)

consumption

leisure

Macroeconomic Theory and Policy – Midterm Exam | © Sanjay K. Chugh 17

Problem 4: Ricardian Equivalence in a Period of Low Interest Rates (25 points). You are a junior economist at the U.S. Treasury Department. In your first meeting with Treasury Secretary Steven Mnuchin, he informs you that lump sum taxation is pointless because it is institutionally difficult to collect lump sum taxes from a broad swath of people. After further discussion, you both agree that the “operational” or “working” definition of Ricardian Equivalence (or “quasi-Ricardian Equivalence”) is:

“Holding constant the flow of government spending across time periods, Ricardian Equivalence occurs if period-one private consumption c1 does not change despite a change in the timing of proportional asset taxation between period 1 and period 2.”

Secretary Mnuchin asks you to attempt to design a tax system that satisfies this modified version of the Ricardian Equivalence Theorem. Neither lump-sum taxes nor consumption sales taxes can be used in the tax system you attempt to design. However, asset taxes (aka, wealth taxes) imposed on the private sector can be used. More precisely, the new policy tools available for the design of the tax system are:

,1w : period-1 tax rate on end-of-period-1 (aka, beginning of period-2) private-sector assets 1a .

,2w : period-2 tax rate on end-of-period-1 (aka, beginning of period-2)

private-sector assets 1a . The period-1 and period-2 budget constraints of the representative consumer are:

1 ,1 1 1 0

2 2 2 ,2 1 1

(1 () 1 )

(1 ) w

w

c a y r a c a y a ra + − = + +

+ = + − +

The period-1 and period-2 budget constraints of the government are:

1 1 ,1 1 0

2 2 ,2 1 1

(1 )

(1 ) w

w

g b r b g b r b

a a

+ = + +

+ = + +

(OVER)

Macroeconomic Theory and Policy – Midterm Exam | © Sanjay K. Chugh 18

Problem 4 continued Because interest rates have been low the last several years, suppose that the real interest rate is r = 0. Both the representative agent and the government begin period 1 with zero net wealth (i.e.,

0 0a = and 0 0b = ), and, as usual in a two-period analysis, both 2 0a = and 2 0b = . Both the private sector and the public sector are free to borrow or lend as much as needed during the course of period 1. All of your tax-policy analysis is conducted at the very beginning of period 1. a. (5 points) At the very end of period 1 (aka, the very beginning of period 2), what is

the sum 1 1a b+ ? (Hint: There is no international trade or international finance in this framework.) Describe in no more than three brief sentences/phrases/equations.

Macroeconomic Theory and Policy – Midterm Exam | © Sanjay K. Chugh 19

Problem 4 continued b. (20 points — DIFFICULT) Based on the limitations of the tax system (i.e., no lump-

sum taxes and no proportional consumption sales taxes) that Secretary Mnuchin has informed you about, is it possible to design a tax system that is consistent with the modified version of Ricardian Equivalence above (i.e., a change from an initial setting of tax rates ,1w and ,2w to a different setting of tax rates ,1w and ,2w )?

If it is impossible to design such a tax system, carefully, clearly, and briefly describe to Secretary Mnuchin why it is impossible. Your informative brief “memo” must use some combination (which is left up to you) of both mathematics and economic interpretation. If it is possible to design such a tax system, carefully, clearly, and briefly describe to Secretary Mnuchin how such a tax system can indeed be designed. Your informative brief “memo” must use some combination (which is left up to you) of both mathematics and economic interpretation. (Note: If your step-by-step analysis is unclear and/or confusing, your solution may be awarded ZERO points.)

Macroeconomic Theory and Policy – Midterm Exam | © Sanjay K. Chugh 20

Problem 4e continued (more space if needed)

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