Miles's Teaching Tumblog

If you have a hard retirement saving problem with more than one thing contributing to the present value in or more than one thing contributing to the present value out, you need to go the the earlier post

http://profmileskimball.tumblr.com/post/33141391872/retirement-saving-calculations

But let me give you some helpful hints when you have one thing contributing to the present value in and one thing contributing to the present value out.  

1. If something is a lump (either initial assets A or intended bequest B), multiply the lump by the real interest rate r, and treat it as equivalent to a flow of rA or rB per year, starting now for initial assets A, and starting when you intended to give the bequest for B.  From here on, I will discuss only flows, since you could have done this conversion for a lump. (But if there are initial assets AND another flow in, then you need the earlier post, and if there is another flow out AND a bequest, then you need the earlier post.) 

2. Line up your information on the real interest rate r, the starting time of the flow in S_1, the duration (years from beginning to end) of the flow in D_1, starting time of the flow out S_2 and duration (years from when it begins to when it ends, NOT the ending time, but the ending time minus the beginning time). In section I had the students write  

r=

S_1 =

D_1 = 

S_2 = 

D_2 = 

Remember that S_1 and S_2 are measured by how many years in the future they are from “now.” Often, we have had “now” be 30 years old, but there could be problems when “now” is a later age. If you use a different age for “now” you will still get the right answer if you do it carefully, but you might make the calculations harder.  If you turned a lump into an equivalent flow in step 1, the duration of that equivalent flow is infinity. Remember that taking e to the power negative infinity always yields zero.   

3. Whenever you have e to a negative power, ask yourself “How many doublings?” using the rule of 70. Then since it is a negative power, if there are, say two doublings, that would be 1/2, if two doublings, that would be 1/4, if three doublings, that would be 1/8, etc.   

4. If the two durations D_1 and D_2 are the same, for example if you save for 35 years and then you withdraw money for 35 years, then you just need to know the number of doublings in the difference between the two start dates, since that is how much your money will grow. X will always be smaller than Z by that many doublings if the durations are equal.  

5. If the durations are different, then I would go with the formula for flows. First, on top, figure out how many doublings for the start date of withdrawals. Then multiply by 1 minus what you get from the number of doublings in the duration for withdrawals. On the bottom, write down what you get for the number of doublings for the start date of contributions. Often the start date will be “now” and so there will be no doublings and you will just write down the number 1. Then multiply on the bottom of the fraction by 1 minus what you get from the number of doublings for the duration of the flow of money in.(Note that if you converted lumps into a flow, the “duration” will be infinity.)  This fraction is then what you have to multiply Z by to get X.  (If you need to go from X to Z, just write it down as an equation and solve.) 

6. Remember that Z is the amount of money you take out of your retirement saving account each year during your withdrawal period (which should be your retirement). It will be subject to taxes. Then what you have left of Z after paying taxes gets added to your social security.  

Those who oppose increasing the allowed levels of legal immigration argue that Americans will be hurt by that additional immigrations. In particular, they argue that the wages of American will be driven down. But even if this is true, those allowed to immigrate benefit from immigration—something that should also be considered. This exercise looks at the tradeoff between a harm to Americans from immigration and a benefit to the immigrants themselves in a Utilitarian welfare analysis that counts the welfare of citizens more than the welfare of non-citizens. The welfare of non-citizens is multiplied by an “altruism to foreigners” parameter that is typically less than 1.

Before the adjustment of multiplying foreigner’s utility changes by the “altruism to foreigners” parameter, the utility of both Americans and foreigners is given by the equation

U = -1/C,

where C = Income/40000.  In the table, the columns, whose labels are abbreviated, are

Number of Americans Affected by Allowing the Extra Immigration  

Income of These Americans Before the Immigration

Income of These Americans After the Immigration

Number of Foreigners Affected by Allowing the Extra Immigration

Income of These Foreigners Before Immigration

Income of These Foreigners After Immigration

Altruism to Foreigners Parameter

Change in Total Citizen-Equivalent Utility, Measured in Big Utils

In the first exercise, please fill in the implied change in citizen-equivalent big utils. Positive numbers mean the policy raises overall citizen-equivalent welfare after giving foreigners some (though reduced) weight. Answers are lower down. 

In the second exercise, please find the “break-even” level of the altruism to foreigners parameter that makes the change in citizen-equivalent big utils zero. A larger level of altruism to foreigners than the break-even level would make the immigration look like a good policy, while a lower level of altruism to foreigners would make the immigration look like a bad policy. So the way to judge the policy after this calculation of a break-even level of altruism to foreigners is to decide whether that number is too low for how much we should count foreigners or too high.  

For the equations linking real and nominal exchange rates, it is hard to keep track of which end is up. This exercise is meant to help with that. 

First, in Mankiw’s textbook, the nominal exchange rate is expressed as the number of units of foreign currency that can be purchased with one unit of domestic currency. The real exchange rate E is then the effective price of domestic goods compared to foreign goods, and the effective price of domestic goods for foreigners is eP, where P is the domestic price and e is the nominal exchange rate. So

E = eP/P*

where P* is the foreign price.  This equation can also be solved for the nominal exchange rate, yielding

e = EP*/P.

Here the intuition is that the higher the price in the foreign country, the more units of its currency we should get, for things to be fair in the sense of having a reasonable real exchange rate.

Normally different countries use different systems for their price indices, that are not comparable. In the exercise below, the price indices listed have been carefully constructed to be comparable between the two countries. For example, looking at the price of Big Macs is a simple example of a price index that is constructed to be comparable across countries. In this case, imagine that there is a basket of things that you could buy in either country, and the price index tells the number of units of the relevant currency needed to buy that basket.

“Purchasing Power Parity” means the same thing the real exchange rate being equal to 1: E=1. But the real-world principle associated with Purchasing Power Parity is that the real exchange rate seldom wanders too far from 1. All of the examples in the tables below have a real exchange rate between .5 and 2 inclusive.  

Please fill in the missing number in each row using the equations above. (There is another exercise based on annual percent changes below these questions and answers.) 

Here are the answers:

The equation E = eP/P* can be transformed into percent change form as 

%ΔE = %Δe + %ΔP - %ΔP*

Dividing by the number of elapsed years yields the equation

(%ΔE/yr) = (%Δe/yr) + (%ΔP/yr) - (%ΔP*/yr)

The percent change in a price per year is the same thing as inflation. Let’s call the domestic inflation rate π and the foreign inflation rate π*.  Then

(%ΔE/yr) = (%Δe/yr) + π - π* 

To keep things straight, think of the percent change in the real exchange rate, which is how expensive things look in the U.S. to foreigners, as coming partly from the percent change in the nominal exchange rate, and partly from our prices going up faster than theirs.   

Rearranging this equation to solve for the annual rate of change in the nominal exchange rate yields a closely related equation:

(%Δe/yr)  =  (%ΔE/yr) + π* - π

Here think of the intuition as saying that if the foreign prices are going up faster than the domestic prices, we should get more units of the foreign currency to be fair in the sense of having a reasonable real exchange rate not too different from where it is now.  

The exercise below is based on these equations about percent changes per year in real and nominal exchange rates and rates of inflation in the two countries.  Since the real exchange rate E seldom gets too far from 1, over long time periods, its rate of change is likely to be quite small. The exercise below reflects that.  

Here are the answers:

In the table below, for various government policy changes in the international trade arena, I have listed the changes in consumer surplus (CS), producer surplus (PS), government revenue (GovRev) and foreigner surplus (ForeignS). With C=1 meaning $40,000 per year, the consumption levels relevant for each of these contributions to welfare are listed. Assume that the degree of inequality aversion is equal to 2, so that marginal utility is    1/[C squared]. For foreigner surplus, you need to multiply not only by marginal utility but also by the parameter theta that tells how much policy should pay attention to foreigner welfare as compared to the welfare of citizens.  After adjusting each $ changes by marginal utility and how much the foreigners count, just add, since your job is to fill in the change in utility, measured in small utils.  

In the last three problems, $ to foreigners are treated as something negative. Here I have in mind the effect of oil tariffs on Iran. Fewer dollars in the hands of Iran means less money to make nuclear weapons with. Since this is different from the usual case, I just put in directly a negative value of theta to show how much we *don’t* want them to have $, and put in C=1 for simplicity, since in this case it isn’t really about whether the Iranians are poor or rich. But the math is just the same, as long as you respect the negative sign.

Sorry to make these numbers so small. I couldn’t fit everything in without doing that.   

Update: I think the numbers in the exercise are correct (please let me know if they aren’t) but I made a huge error in the equations near the top of the post. In the present value formulas, flows should be divided by the real interest rate r, as well as discounted and truncated in the ways I had it. In the revised equations, the first thing I do is multiply through by the real interest rate r so that the fractions are cleared for the flows X and Z, but the lumps A (initial assets) and B (intended bequest) are then multiplied by r.  

The key principle for retirement saving calculations is that present value in must equal present value out. So if you have initial assets of A at time t=0 and want to make a bequest of B and time t=S_2+D_2, then, 

where 

S_1 = how many years in the future you start putting money in your 401k

D_1 = how many years you put money in your 401k from start to finish

S_2 = how many years in the future you start pulling money out of your 401k

D_2 = how many years you pull money out of your 401k from start to finish

r = real interest rate

X = annual real contributions to your 401k (before the tax break you get)

Z = annual withdrawals from your 401k each year (that will then be taxed before you add them to Social Security)

A = assets at time 0

B = bequest to your kids or to a charity at time S_2 + D_2

Two pages worth of problems are immediately below and two pages worth of answers are further down. As before, fill in the missing number. All dollar amounts are in 1000’s of dollars, so that X = 50 means $50,000 per year, and A = 1000 means you start out with one million dollars.  On lengths of time, 1000 years may as well be infinity as far as the calculations are concerned, so treat 1000 years as if it were infinity and you will be fine.  

ANSWERS:

The price elasticity of supply is defined as %ΔQ/%ΔP when moving along a supply curve. It is typically a positive number. (That is, supply curves slope up.)

The price elasticity of demand is defined as %ΔQ/%ΔP when moving along a demand curve. It is typically a negative number. Economists often talk about price elasticities of demand in terms of their absolute value, so that you have to supply the negative sign yourself from your general knowledge that increases in price reduce the quantity demanded. (That is, demand curves slope down.)

In the exercise below, the last four problems are more difficult, but for the rest, here is the procedure:

1. Figure out whether the elasticity should have a negative sign because it is a demand elasticity or a positive sign because it is a supply elasticity. Together with the absolute value of the elasticity in the second column, that will give you the elasticity with a + or a - in the third column. 
2. Multiply the (Platonic) percent change in the price by the elasticity to find the percent change in the quantity. 
3. Add the percent change in the price and the percent change in the quantity to get the percent change in revenue PQ.  

Of the harder questions, some are missing %ΔP but have %ΔQ. All you need to do there is to divide %ΔQ by the signed elasticity to get %ΔP. Then you can add %ΔP to %ΔQ to get the percent change in PQ.  The last two questions are the hardest. There you need to do a bit of algebra:

%Δ (PQ) = (1 + elasticity) %ΔP

You can solve for %ΔP, and then multiply by the elasticity to get %ΔQ.

The answers are below as a filled-in table. 

This equation is called either “the Quantity Equation” or “the Equation of Exchange.” It is a very important equation for understanding the effects of money on the economy, but in this exercise, you can just treat the percent change version of the equation as a mathematical fact. Fill in the blank in each row and then check your answers. 

Note that if you divide each thing in the percent change version of the equation by time elapsed, you will get an equation in growth rates. The growth rate of the aggregate price index P is called inflation. Note also that if Divisia indices are used for money M (which is an aggregate of several different components), prices and quantities, then the percent change version of the equation is primary (most basic) and the MV=PY version would have to be deduced from the percent change version.  

I am used to using the following notation:

• i = nominal interest rate
• pi = inflation (usually “pi” would be the Greek letter pi, but I want to match the Excel file excerpt here)
• r = real interest rate

The real interest rate is defined by the equation 

r = i - pi

Notice that each of r, i and pi is a growth rate of some sort, going back to some sort of Platonic percent change divided by elapsed time. The real interest rate is how fast the real value of what you have in the bank grows. The nominal interest rate is how fast the nominal value of what you have in the bank grows. Inflation is how fast the price level grows. Inflation erodes the real value of what you have in the bank because it means you can’t buy as much with it if inflation has been positive.  

The exercise is simple. One piece of the equation above is missing in each row of the table. Fill in the blanks and then look at the answers down further below. Remember that in applications, the equation is often rearranged. For example, it often is written i=r+pi. In the tables, mentally shift the labels at the top over to the right to get them to be on top of what they refer to.   

Divisia indices are generally recognized by economists as superior to other kinds of price and quantity indices. However, they require a familiarity with percent changes and with the approximation for the percent change of a sum. They are actually quite easy to use. The only difference between the formulas for a price change index  and a quantity change index is whether the price changes of the individual goods are used or the quantity changes of the individual goods are used. Here are the steps for the case when there are two goods, X and Y.  

1. Find the shares of X and Y both before and after. In the tables below, before is labeled t and after is labeled t+1, but the formulas for the percent changes would be the same whatever the length of time between before and after. Of course, to go from percent changes to growth rates, one must divide by the time elapsed between before and after. But the focus in this post is on the percent change of the index. 
2. Average the share of X before s_X,t with the share of X after s_X,t+1; average the share of Y before s_Y,t with the share of Y after s_Y,t+1. Then, if the averaged shares are in % form, convert them to regular fractions.  
3. Multiply the averaged share of X in fraction form times the percent change in X (either the price of X or the quantity of X). Then multiply the averaged share of Y in fraction form times the percent change of Y (either the price of Y or the quantity of Y). Add. 
4. The result is the percent change in the Divisia index. For Divisia indices, the percent change is the central thing that is defined. If you every wanted the change in the index itself, you would deduce the change in the index itself from the percent change in the index. For example, if the percent change in the index was 70%, you could deduce according to the rule of 70 that the index doubled, so if you arbitrarily started the index off at 100, it would end up at 200.   

Let me make one important comment:

Notice how I am emphasizing the way Divisia indices use the average of the share of X at the beginning and ending of the elapsed time and the average of the share of Y at the beginning and ending of the elapsed time. The reason I am doing this is because not using weights averaged between the beginning and end of the elapsed time is the biggest problem with the Consumer Price Index and the GDP Deflator, as well as the current official measure of real GDP. The Consumer Price Index uses quantity weights (the “basket”) from the base year, which at best is at the beginning of the elapsed time. The GDP Deflator effectively uses the most recent quantity weights, which at best are at the end of the elapsed time. The usual measure of real GDP uses price weights from a reference year in the past, which at best is at the beginning of the elapsed time. For technical reasons, using price or quantity weights from before tends to make the change in an index too high. Using price or quantity weights from after tends to make the change in an index too low. 

For Divisia indices, In addition to the benefits of using weights (the shares) that are averaged between the beginning and the end of the elapsed time, shares tend to change much more slowly over time than price or quantity weights. So the average between beginning and end values of a share will tend to be very close to where the share is at as it evolves the whole intervening time. Once you get used to them, shares are also more intuitive and the real-world values for various shares are easy to learn.

There are two different exercises below.

• The first exercise gives you enough information to compute the shares of X and Y before and after, and the percent changes in X and Y (either prices or quantities of X and Y—the math doesn’t care). You need to fill in the blanks. You can fill in the blanks for the missing shares by using the principle that all the shares put together always need to add up to 100%. After you have filled in the missing shares, compute the percent change in the Divisia index.  The answers follow in a table with everything filled in. 
• The second exercise, down at the bottom asks you to compute nominal GDP for goods 1 and 2 combined as P_1 Q_1 + P_2 Q_2 and then to find the share of GDP accounted for by good 1, which is [P_1 Q_1/(P_1 Q_1 + P_2 Q_2)] and the share of GDP accounted for by good 2, which is [P_2 Q_2/(P_1 Q_1 + P_2 Q_2)]. For this exercise, I have made no attempt to make the shares come out even. You will want to use a simple calculator.