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 ...

1. 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−+ − =

2. 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

3. 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

4. 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.

5. 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

6. 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)

7. 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

8. 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:

9. ,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:

10. 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,

11. 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

12. 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.)

13. Macroeconomic Theory and Policy – Midterm Exam | © Sanjay
K. Chugh 20
Problem 4e continued (more space if needed)

14. Tasks to be completed.
Document #1 requires a detailed outline (MS Word) of the topic
Document #2 requires a report discussing the advantages and
Document #3 requires a presentation (PPT) presenting detailed
advantages and disadvantages of the topic selected (200 points
max) ( 5 ppt slides)
Appropriate Topics: Select one of the following research areas
to investigate: i) Biometrics ii) Drones iii) Robotics iv) Virtual
Reality (VR) v) Intelligence Apps (I-Apps) vi) Cognitive Cloud
Computing vii) Web Sports Technology viii) Search Engine
Optimization ix) Disaster Recovery Methodologies x) 3D
Printing