U6301 Macroeconomics for International and Public Affairs Martsella Davitaya

Columbia University | SIPA

Problem Set #6
Suggested Solutions
_____________________________________________________________________________
Deadline. 11:59 pm Feb 20, 2024
Instructions. The electronic version (either scanned or typed up) of solutions must be uploaded to
CourseWorks by the indicated deadline. Solutions submitted after the deadline will receive 0 points.
Solutions submitted via email will receive 0 points. Solutions can be submitted individually or by a
group of up to 4 students. You can work in a group with students from sections 1, 2, 3, and 4 only. If
you work in a group, please indicate all names and UNI-s on top of the first page.

Problem 1.
Derive and illustrate the Laffer curve in a labor market where the labor demand is supply is perfectly
inelastic at 𝑁 = 25 and the equation of the labor demand is given by 𝑁𝐷 = 50 − 𝑤. What is the
proportional tax rate that maximizes tax revenue?

The equilibrium employment in this economy is equal to 𝑁 ∗ = 25 because the labor supply is perfectly inelastic.
With a proportional tax, the labor demand firms are facing is given by 𝑁𝐷 = 50 − 𝑤 𝑓𝑖𝑟𝑚 , and the wages that
workers are receiving is given by 𝑤 𝑤𝑜𝑟𝑘𝑒𝑟 = 𝑤 𝑓𝑖𝑟𝑚 (1 − 𝑡). Therefore, the amount that government collects on
each hour of employment is given by 𝑤 𝑓𝑖𝑟𝑚 − 𝑤 𝑤𝑜𝑟𝑘𝑒𝑟 = 𝑤 𝑓𝑖𝑟𝑚 𝑡. Since 𝑤 𝑓𝑖𝑟𝑚 = 25, the total tax revenue is
given by 𝑇 = 25𝑡 × 𝑁 ∗ = 625𝑡.
𝑇 is the increasing function of 𝑡, the proportional tax rate that maximizes 𝑇 is 100%.

Problem 2. Fiscal Policy

Read the following excerpt from President Obama’s “Remarks by the President in the State of the Union
Address” from February 2013:

“Over the last few years, both parties have worked together to reduce the deficit by more than
$2.5 trillion – mostly through spending cuts, but also by raising tax rates on the wealthiest 1
percent of Americans. As a result, we are more than halfway towards the goal of $4 trillion in
deficit reduction that economists say we need to stabilize our finances.”

a) Which deficit is President talking about? Could a tax rate increase cause the deficit to increase,
rather than decrease?

Government deficit. An increase in the tax rate could create so many distortions that would reduce the
incentive to work, save and invest so much that the overall tax revenue falls. In other words, if the economy
is to the right of the peak in the Laffer curve, any increase in the tax rate would cause the total tax revenue to
fall.

“Yes, the biggest driver of our long-term debt is the rising cost of health care for an aging
population. And those of us who care deeply about programs like Medicare must embrace the
 need for modest reforms – otherwise, our retirement programs will crowd out the investments
we need for our children and jeopardize the promise of a secure retirement for future
generations.”

b) Why an increase in government spending on retirement programs may crowd out the investment
for future generations?

An increase in TR to the old will decrease government budget, i.e., increase government deficit:

↓ 𝑆𝐺 = 𝑇 − 𝑇𝑅 ↑ −𝐼𝑁𝑇 − 𝐺

The increase in TR affects private saving in the same way as a cut in taxes.

• 𝑜𝑙𝑑
𝑆𝑝𝑣𝑡 = 𝑌 + 𝑁𝐹𝑃 − 𝑇 + 𝑇𝑅 ↑ +𝐼𝑁𝑇 − 𝐶 ↑ will not change because old have no incentive to save, and
increase in transfers is equal to increase in consumption.
𝑦𝑜𝑢𝑛𝑔
• 𝑆𝑝𝑣𝑡 ↑= 𝑌 + 𝑁𝐹𝑃 − 𝑇 + 𝑇𝑅 + 𝐼𝑁𝑇 − 𝐶 ↓ if young are Ricardian because they foresee increase in
taxes in the future. Note that private saving in this case increases by less than the increase in TR
because MPC<1.
𝑦𝑜𝑢𝑛𝑔
• 𝑆𝑝𝑣𝑡 = 𝑌 + 𝑁𝐹𝑃 − 𝑇 + 𝑇𝑅 + 𝐼𝑁𝑇 − 𝐶 will not change if young are myopic.

Overall, national saving would fall because decrease in government saving is larger than increase in private
saving.

In a closed economy, the drop in S would cause a disequilibrium in capital markets that would necessitate an
increase in the real interest rate that would cause investment to contract until it is equal to the new level of
saving. The drop in I causes a reduction in future capital and future GDP. The reduction in future GDP,
relative to what it would have been without the increase in government transfer, reduces the benefit that our
children will receive once they will retire.

A similar mechanism would apply if one would treat the U.S. as a large open economy. The contraction in
U.S. national saving causes a greater demand for international funds and increases the real interest rate in
the international funds market.

c) How would increase in the cost of Medicare affect national saving?

The increase in the cost of Medicare causes an additional transfer and a corresponding increase in income that
causes an additional reduction in the nation’s saving.

Problem 3. Money Supply

The Federal Reserve stopped its Large-Scale Asset Purchases (LSAP) in October 2014. However, the
size of the Fed’s Balance Sheet (its total assets) remained constant up to October 2017. Between October
2014 and October 2017, the Federal Reserve acquired new assets using the resources from the securities
that had reached maturity.
Explain the mechanism of this reinvestment and illustrate your answer on the Fed’s balance sheet.
The Fed wanted to keep the monetary base stable. If it stopped buying securities altogether, the monetary base
would decline as securities expired and borrowers paid back their borrowings to the Fed. This paying back would
decrease the monetary base. The effect of securities reaching maturity:

Assets Liabilities
• Gold • Monetary Base↓
• Foreign Assets o Currency
o Foreign currency o Commercial Banks’ Reserves
o Foreign securities • Deposits of Treasury
• Domestic Assets
o Loans to commercial banks Net Worth
o Treasuries
o Other domestic assets↓

The effect of reinvesting the proceeds of securities reaching maturity:

Assets Liabilities
• Gold • Monetary Base ↑
• Foreign Assets o Currency
o Foreign currency o Commercial Banks’ Reserves
o Foreign securities • Deposits of Treasury
• Domestic Assets
o Loans to commercial banks Net Worth
o Treasuries
o Other domestic assets ↑

Problem 4. Temporary Shocks and Asset Markets

Discuss the long-run effects on output prices of a temporary negative productivity shock.
Illustrate your answer on the asset markets graph.
What should the Central Bank do to prevent prices from changing in the long run?

In open economies:
• Temporary negative productivity shock causes Y to decrease.
• Since real money demand is a positive function of Y, real money demand decreases, and the L curve shifts
to the left.
• In open economies, r does not change. Nominal money supply (MS) remains constant. Therefore, P must
increase so that MS/P decreases and shifts to the left.
• Intuitively, P increases because there are less goods and the same amount of nominal money.
 r A: Old Equilibrium
r B: New
S(Ynew) S(Yold)
L(Yold)
L(Ynew)
rworld

rworld
B A
I
𝑀 𝑀
S*new S*old I* S,I 𝑃𝑛𝑒𝑤 𝑃𝑜𝑙𝑑