# The Effects of Shifts in the NCO Curve on the Medium-Run International Finance Diagram are Not Ambiguous After All

• Assume that the Supply of Loanable Funds curve is upward-sloping
• Assume that the I-curve is downward sloping.
• Both of these assumptions are crucial to what follows.
• Now, Consider a case when the NCO-curve shifts out. (After you read the rest of this post, make sure you work through the case when the NCO curve shifts back for yourself. Actually, in general you should always work through the opposite directions of things as part of your studying. The simplest way to make an exam question is to do the opposite direction from what was covered in class.)
• Since the Demand for Loanable Funds is the horizontal sum of the I-curve and the NCO-curve, the outward shift in the NCO-curve also shifts the Demand-for-Loanable-Funds out.
• This raises the interest rate r.
• The increase in r moves things up and left along the NCO-curve.
• It might seem that the outward shift in the NCO curve combined with a shift up and to the left along the NCO-curve might make the direction the quantity of NCO goes ambiguous. But not so!
• Since the I-curve does not shift, the increase in r lowers the quantity of investment I.
• Since the Supply-of-Loanable-Funds does not shift, the increase in r raises the quantity of saving S.
• S = I + NCO, since physical investment and sending funds abroad with NCO are the two possible uses of overall national saving. (Remember that national saving is household saving + corporate saving + government saving.)
• Since quantity of S increases, while the quantity of I decreases, the only way that the equation S = I + NCO can be satisfied is for the quantity of NCO to increase.
• Thus, the quantity of NCO moves in the same direction as the NCO-curve shifts.
• The quantity of NCO determines the location of the vertical Supply of Dollars. With the quantity of NCO increasing, the Supply of Dollars curve shifts to the right.
• This is actually a nice example of a negative feedback loop that is too weak to change the direction of something.
• NCO –>+  Demand for Loanable Funds –>+  r –>-  NCO

# Guidelines for Blog Posts

1. Examples: Most of your question about what you can do in the blog posts will be answered by reading the posts that I liked best from last year
2. Topics: You can write about anything related to economics. Try to do some of the posts more specifically about macroeconomics or finance, but I don’t want to stifle your creativity. Just start out writing about any economics-related topic that you have an angle on (=something to say about) and we will let you know if your topics are running too far afield. But that has never been a problem before.
3. References: Each blog post needs to link to at least two other sources that you integrate into the post. At least one of the sources needs to be dated 2015. For roughly half of the blog posts, one of the sources should be in the Wall Street Journal. A good reference for a blog post is a link that works, plus enough information to google the source if the link stops working for whatever reason.
4. Length: 400 to 700 words. At least 2/3 of the words must be your own words, and you must clearly signal which words are not your own.
5. Revised Posts: Spread your 5 revised blog posts out over the semester. Don’t bunch them all up at the end or all at the beginning. On revised posts (not on regular posts), send an email to both of us with a link to the revised post. Put Econ 411 at the start of the subject line.

# Online Resources for Reading, Writing and Data Analysis

1. Monetary Policy Releases, 2014 (on the Fed’s official website)
2. Links I am Thinking About—this is a companion blog of mine, entirely made up of links to articles I thought looked interesting. I post new links frequently.
3. FRED: Federal Reserve Economic Data—a graph-making website with a huge amount of economic data to make the graphs with.
4. Economistsview (top economics aggregation blog)
5. John Cochrane’s blog The Grumpy EconomistJohn Cochrane is one of the top financial economists in the world. He announces his blog posts on Twitter @JohnHCochrane but doesn’t do much else on Twitter.
6. Grammar Girl blog—an excellent source of grammar tips.
7. thesaurus.com—in case you need help finding just the right word. I just googled “thesaurus.” You can find many online dictionaries, as well.

# Finding the Optimal Level of Consumption with a Finite Horizon

All of these calculations are based on the assumption that, trough a Roth IRA you can avoid all taxes on money that you sock away for the future, though you have to pay taxes on your labor earnings before you have money that can be put into the Roth IRA. If you have a regular IRA or retirement saving account instead, the biggest adjustment you need to make is to take the amount in your account statement and cut it in half or by some other factor to allow for all the taxes that will have to be paid on it when you withdraw the money. In other words, what looks like $1 million is probably only about$500,000 for the calculations below. Assuming you can save all you want in a tax-sheltered account (which may not be true) everything else (other than realizing that the money in your account needs to be cut in half or so because of taxes) should work the same in the ultimate calculation with a regular retirement saving account instead of a Roth one, but explaining why is complex.

To find the optimal level of consumption, first, compute the present discounted value of your lifetime resources as best possible. This present discounted value of lifetime resources is called Modigliani full wealth. Key things to include are

1. Put in your current financial wealth as is; since it is already there in the present, its present value is whatever your account statements say. That is, unless you will owe taxes on withdrawals. In that case, its after-tax present value is something like ½ what your account statement says.
2. Find the present discounted value of your labor income (salary and wages). If your future labor income is uncertain, doing things exactly right is very hard. As a rule of thumb, my best guess right now is that you should put in the 10th percentile outcome for your labor income to allow for risk. That is, put in a level of labor income where there is a 90% chance that you will do better than that. Make sure to allow for labor income taxes (including the income tax, the social security tax and the medicare tax), and only calculate the value of your after-tax labor income.
3. Find the present value of your future social security benefits.
4. Subtract the present value of future bequests and gifts to your kids that you want to be sure to make. (Remember to discount: given reasonable interest rates, $1 million 70 years from now is a lot easier to come by than$1 million now.)

Once you have calculated your Modigliani full wealth, multiply it by the optimal finite horizon propensity to consume B_T, where T is the number of years you have left to live (or estimate that you have left to live).

The Formula for the Optimal Finite-Horizon Propensity to Consume:

B_T = B_infinity/( 1 - e^{-B_infinity T} )

To calculate the optimal finite-horizon propensity to consume, first find the optimal infinite-horizon propensity to consume. Figure out how many doublings you have at rate B_infinity for T years, or whatever else you need to do to calculate e^{-B_infinity T}.  Divide B_infinity by (1 - e to the B_infinity times T) to get B_infinity/( 1 - e^{-B_infinity T} ) .

In the problems below, you are asked to find the optimal finite horizon propensity to consume using the rule of 70.  That is, treat ln(2) as if it were equal to exactly .7.

If B_infinity is negative, you will find that the formula for B_T still works fine.  If B_infinity is zero, L'Hopital’s rule gives the answer that B_T = 1/T

Exception to Formula Above: If the formula above would give 0/0, then the answer is 1/T.

Practice Problems: Find B_T Given T and B_infinity

# Finding the Optimal Infinite-Horizon Propensity to Consume

In some of the slides I called this R, but on the chalkboard I called it a better name: B_infinity.

Notation:

B_infinity = optimal infinite-horizon propensity to consume in the Merton model

S = elasticity of intertemporal substitution

rho = impatience

r = safe real interest rate

mu^2 / (2 * A * sigma^2)  = investor’s surplus

(See the earlier post about finding investor’s surplus)

Terminological Note.  What I call the “elasticity of intertemporal substitution” or “EIS” is often called the “intertemporal elasticity of substitution” or “IES.” My view is that intertemporal substitution is a phenomenon, of which the elasticity is a measure. So I hate to separate the words “intertemporal substitution.” Other economists think that “elasticity of substitution” is a well-known measure, and this is applying it in an intertemporal context. So my terminology emphasizes measuring a particular economic phenomenon, while the other way of talking about it emphasis applying a particular measure in a new context.

The Formula:

Using some very cool, but quite advanced math that extends Robert Merton’s model to carefully distinguish risk tolerance 1/A from the elasticity of intertemporal substitution S, the formula for the optimal infinite horizon propensity to consume is

B_infinity = S  rho + (1-S) [ r + investor’s surplus]

This formula is actually about average values of r and investor’s surplus over the rest of your life, using appropriate weights for the average. Thus B_infinity is affected very little by business cycle fluctuations.

However, if people’s view of the long-run future changed, then B_infinity would change. For example, I think if China lets regular citizens get a decent return for their savings, this would change the Chinese people’s view of the long-run, lifetime average values of r and investor’s surplus and would cause B_infinity (and therefore consumption) to go up dramatically. (A preview of a future post on my main blog!)

Reasonable Values:

The Elasticity of Intertemporal Substitution S. What I just said about China only works if the elasticity of intertemporal substitution S is less than 1. And I truly believe that S is less than 1.  To me, any value between .1 and .6 or so is a thinkable value. But I need to tell you that some people truly believe that the elasticity of intertemporal substitution S is as high as 2.

You can see, however, that a value of 1 for S is extremely convenient. And many, many, economic models use a value S=1 for that reason.

Impatience. Many people are probably quite impatient–maybe up to 8% per year. Others are not impatient at all, which would be zero. It is even quite possible for someone to have rho be negative, which simply means that even at a zero interest rate and zero investor’s surplus, they would prefer to consume more in the future than now. However though there is no problem with negative impatience in a finite-horizon where death looms up ahead, negative impatience does very weird things in infinite horizon models, technically. These technical problems make many economists think that negative impatience is unreasonable, but it isn’t. Most people don't really expect to live forever.

The Real Interest Rate. I think of 3% per year as a reasonable value of the real interest rate to use for retirement planning. One part of the reasoning behind this is that the government has borrowed a lot of money. That tends to push real interest rates up when the economy is at the natural level of output (which it isn’t right now).

Investor’s Surplus. Let me tweak things a bit to get a round number. Suppose the risk premium is 3% per year, the standard deviation of the risky index mutual fund (or ETF) is 15% over a year (I am assuming the return of the Great Moderation over most of your life because of getting monetary policy better figured out, one way or another), and a risk aversion of 2. Then on an annual basis, investor’s surplus is .0009/(2 * 2 * .0225) = .01 = 1%. So that is 1% per year.

Interpretation Note:

Below, please calculate B_infinity according to the formula, even if the value turns out 0 or negative. If B_infinity is ever 0 or negative, it means that in an actual infinite horizon model, very weird things would happen. But a negative B_infinity can still be plugged into the finite-horizon formula in the next post. If B_infinity is zero, as the next post will argue, it means that the finite horizon propensity to consume is just 1/time left to live. So if you have 40 years left to live, consume 2.5% of what you have this year.

Problems:

# Finding Investor's Surplus

Investor’s surplus is equal to the risk premium squared divided by (2 times risk aversion times the standard deviation squared), where the risk premium   and standard deviation refer to the return of the risky mutual fund.  The risk premium is also sometimes called the mean excess return.

Notation:

sigma = standard deviation

A = risk aversion

Investor’s Surplus = mu^2 / (2 A sigma^2)

Tricky Units. The trickiest aspect is handling the percentages and the time units right (though if everything is on an annual basis, you won’t make a mistake with the time units).

Besides indicating a context, % is the same thing as multiplying by .01.  So for example, 10% is the same thing as .1, so 10% squared is .1 squared which is .01 which is 1%. Similarly, 20% squared is 4%.

Since the formula for investor’s surplus has mu squared on top and sigma squared on the bottom, if both mu and sigma are expressed as percentages, the % squared on top and the % squared on the bottom cancel. BUT, then you will want to express investor’s surplus as a percentage, so you need to multiply by 100%.  The 100 you are multiplying by is going to make the arithmetic look easier if you are doing it by hand. Anyway, you need to practice to get this right. A good benchmark to help get this right is that if the risk premium is 1% per year, the standard deviation is 10% over a year, and risk aversion is 1, then Investor’s surplus is .5% per year.

Why do I say the standard deviation is something like 10% or 20% “over a year” instead of “per year”? This is a very interesting technical thing. With uncorrelated shocks, it is the variance that adds nicely over time. 10% standard deviation over a year is a 1% variance per year, a 10% variance per decade, a .25% variance per quarter, etc.  Strangely enough, if you want to express it with “per,” the standard deviation in this example is 10% per square root of year, since that is what squares to yield the variance of 1% per year. This is one of the fascinating technical things I learned from Greg Mankiw.

What are reasonable values?

The Risk Premium. Historically, the risk premium for a broad stock index has been about 6% per year in the postwar era. Since it has been a very good 80 years, many people argue it is too much to expect 6% again in the coming years. But I think a 3% risk premium is pretty reasonable to count on, unless people start acting a lot less risk averse in the future than they do now (which would drive up the price and drive down the risk premium).

The Standard Deviation. I think of 20% over a year as a normal level of the standard deviation of a broad stock index. During the Great Moderation, it looked like it was down to 15% over a year.

As I said in class, we get a much better read on the standard deviation and how it evolves over time than the risk premium. Why? The standard deviation of returns is so big that it creates a huge amount of noise in trying to figure out what the risk premium is, without decades and decades of data. By contrast, every day that passes gives a good read on the standard deviation for that day. (And in the formal theory, even much smaller chunks of time give a good read on the standard deviation for whatever length of time it is.)

Risk Aversion. I argued in class (and hope to back it up in a blog post) that by the Raj Chetty argument, the appropriate risk aversion for most people is between 1 and 2.  This is much less risk averse than most people act. I was the instigator for a risk aversion measure in the Health and Retirement Survey that found that the average risk tolerance is something like 1/6, which would be a risk aversion of 6 for the representative investor. But many many people make choices on the survey that indicate a risk aversion considerably higher than 6.

Problems. Calculate the investor’s surplus in percent per year for the following situations. Express it in percent per year.

This is an easy exercise, but doing it will make the risk premium clear to you. The risk premium for the risky mutual fund is simply the expected return on the risky mutual fund minus the safe interest rate. (Think of the safe interest rate as the Treasury bill rate.) Because one rate is subtracted from the other, the risk premium calculation will come out the same whether you use real or nominal interest rates, but to stay in the spirit of everything else we have done in the class, it is best to think of these as real interest rates.

Find the risk premium in the following cases. The answers are down below, but don’t look until you have figured them out yourself.

# Winter 2014 Midterm Exam

• Winter 2012 Midterm ExamRemember that what was covered before that midterm exam was different than this year.
• Answers to Winter 2012 Midterm Exam
• Shell for Winter 2014 Midterm Exam
• Winter 2014 Midterm Exam
• Answers to Winter 2014 Midterm Exam (Note 1: Question 6 has 2 correct answers. Note 2: Raw scores on my exams have sometimes been low. I don’t think that will be true on this one, but it might be. Don’t worry: my curve is your friend. Ryoko and I will try to get information on the distribution to you soon.)
• Distribution of scores: 33 33 32 31 31 31 30 30 30 30 29 29 29 28 27 26 26 26 25 25 25 24 24 24 24 24 23 23 23 23 23 22 22 22 22 21 19 19 16

• Distribution of final course grades from Winter 2012:  A+ A+ A A A A A A A- A- A- B+ B+ B+ B B B B B B- B- B- C
• Distribution of final course grades from Winter 2011:  A+ A A A A A A A A- A- A- A- A- A- A- B+ B+ B+ B+ B+ B+ B+ B B B B B- C

BLOGGING HIATUS: BREAK GIVES YOU 4 BLOGGING ASSIGNMENTS OFF, BOTH SATURDAYS, A MONDAY AND A WEDNESDAY DURING BREAK. But, break is an ideal time to get ahead on your blogging assignments; remember that things will get busy toward the end of the semester, and it would be great at that point to be ahead. I will be reading blog posts over break from those who do send me the links.

Miscellaneous Note: I just realized that it is very helpful if your emails to me of links to your revised posts say this is my [first, second, third, fourth, fifth] revised post. That will help me avoid any errors in my recording of things.