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Bobby Bonilla’s Deferred Deal: A Case Study August 2, 2011

Posted by tomflesher in Finance, Macro, Teaching.
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Note: This is something of a cross-post from The World’s Worst Sports Blog.  The data used are the same, but I’m focusing on different ideas here.

In 1999, the Mets owed outfielder/first baseman Bobby Bonilla $5.9 million dollars. They wanted to be rid of Bonilla, who was a bit of a schmoe, but they couldn’t afford to both give him his $5.9 million in the form of a buyout and meet payroll for the next year. No problem! The Mets sat down with Bonilla and arranged a three-part deal:

  1. The Mets don’t have to pay Bonilla until 2011.
  2. Starting in 2011, Bonilla will receive the value of the $5.9 million as an annuity in 25 annual installments.
  3. The interest rate is 8%.

Let’s analyze the issues here.

First of all, the ten-year moratorium on payment means that the Mets have use of the money, but they can treat it as a deposit account. They know they need to pay Bonilla a certain amount in 2011, but they can do whatever they want until then. In their case, they used it to pay free agents and other payroll expenses. In exchange for not collecting money he was owed, Bonilla asked the Mets to pay him a premium on his money. At the time, the prime rate was 8.5%, so the Mets agreed to pay Bonilla 8% annual interest on his money. That means that Bonilla’s money earned compound interest annually. After one year, his $5.9 million was worth (5,900,000)*(1.08) = $6,372,000. Then, in the second year, he earned 8% interest on the whole $6,372,000, so after two years, his money was worth (6,372,000)*(1.08) = (5,900,000)*(1.08)*(1.08) = $6,881,760. Consequently, after ten years, Bonilla’s money was worth

5,900,000 \times (1.08)^{10} = 12,737,657.48

The prime rate didn’t stay at 8.5%, of course, as the Federal Reserve Economic Data (FRED) webpage maintained by the St. Louis Fed shows. This is a monthly time series of prime rates, which are graphed to the right. The prime rate  can be pretty volatile, so it was a bit of an odd choice to lock in a rate that would be effective for 35 years. As it turns out, the prime rate fluctuated quite a bit. Taking the annualized prime rate by dividing each monthly rate by 12 and adding them together, we can trace how much a bank account paying the full prime rate would have paid Bonilla:

\begin{tabular}{c||cc} Year& Annualized interest rate & Current Value \\ \hline 2000& 0.09233 & 6444766.67 \\ 2001& 0.06922 & 6890851.93 \\ 2002& 0.04675 & 7212999.26 \\ 2003& 0.04123 & 7510355.16 \\ 2004& 0.04342 & 7836429.74 \\ 2005& 0.06187 & 8321243.53 \\ 2006& 0.08133 & 8998038.00 \\ 2007& 0.08050 & 9722380.06 \\ 2008& 0.05088 & 10217006.15 \\ 2009& 0.03250 & 10549058.85 \\ 2010& 0.03250 & 10891903.26 \\ \end{tabular}

So since his agreement with the Mets paid him $12,737,657.48 and he could have invested that money by himself to earn around $10,891,903.26, Bonilla is already better off to the tune of (12737657.48 – 10891903.26) or about 1.85 million.

In addition, we can measure the purchasing power of Bonilla’s money. Using FRED’s Consumer Price Index data, the CPI on January 1, 2000, was 169.3. On January 1, 2011, it was 221.062. that means the change in CPI was

\frac{221.062 - 169.300}{169.300} \times 100 = 30.57 \%

So, the cost of living went up (for urban consumers) by 30.57% (or, equivalently, absolute inflation was 30.57% for an annualized rate of around 3%). That means that if the value of Bonilla’s money earned a real return of 30.57% or more, he’s better off than he would have been had he taken the money at the time.

Bonilla’s money increased by

\frac{12,737,657.48 - 5,900,000}{5,900,000} \times 100 = 115.89 \%

Bonilla’s money overshot inflation considerably – not surprising, since inflation generally runs around 3-4%. That means that if Real return = Nominal return – Inflation, Bonilla’s return was

115.89 - 30.57 = 85.32 \%

Because Bonilla’s interest rate was so high, he earned quite a bit of real return on his money – 85% or so over ten years.

Bonilla’s annuity agreement just means that the amount of money he holds now will be invested and he’ll receive a certain number of yearly payments – here, 25 – of an equal amount that will exhaust the value of the money. The same 8% interest rate, which is laughable now that the prime rate is 3.25%, is in play.

In general, an annuity can be expressed using a couple of parameters. Assuming you don’t want any money left over, you can set the present value equal to the sum of the discounted series of payments. That equates to the formula:

PV = PMT \times [\frac{1 - \frac{1}{(1 + r)^t}}{r}] \\ 12737657.48 = PMT \times [\frac{1 - \frac{1}{(1.08)^25}}{.08}] \\ PMT = 1193248.20

where PV is the present value of the annuity, PMT is the yearly payment, r is the effective interest rate, and t is the number of payments. Bonilla will receive $1,193,248.20 every year for 25 years from the Mets. It’s a nice little income stream for him, and barring a major change in interest rates, it’s one he couldn’t have done by himself.

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Equilibrium in Macroeconomics April 22, 2011

Posted by tomflesher in Macro, Teaching.
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One of the things macroeconomists focus on quite a bit is calculating equilibrium conditions, or equilibria. Sometimes these account for random shocks or long-term growth – these have names like Dynamic stochastic general equilibrium and they’re outside the scope of this blog, which has so far focused on introductory-level material. We’re going to develop an idea of what an equilibrium is supposed to be and show how to figure out an equilibrium in a simple, open macroeconomy.

Equilibrium has a connotation of balance. The idea is that two (or more) things need to be balanced for some definition of balance that makes sense in the discussion. In economics, we generally think of equilibrium as representing a point where everything that’s produced is consumed. In a market for an individual good, that means we need to find a point where enough of those goods are produced so that everyone who wants to buy a good can do so.

That’s not very exact, though – it’s very rare that we see a goods market where everyone who wants the good can get it. There has to be some sort of incentive for the item to be produced, and generally that isn’t the satisfaction of seeing people using your trinket. (Sometimes it is – for example, the satisfaction of producing this blog and the fact that it forces me to think clearly are incentives for me to produce.) In general, that incentive is a price for the good – by producing, you get the opportunity to sell the good and get some money in exchange. That also provides a mechanism by which people self-select whether or not they participate in the market – at a certain price, people are willing to buy if they value the product at least that much. If the price is too high, they might still want the item, but they aren’t willing to pay for it so they’re no worse off.

How does that generalize to a macroeconomy, where we’re concerned about lots of goods and lots of prices? Well, it can be difficult to do so. That’s part of what makes grad macro so difficult. We, however, are going to make a couple of simplifying assumptions.

First of all, think back to the idea of the GDP Factory, where everyone works. Instead of producing individual goods, imagine that everyone just produces Stuff. The Stuff goes out on the market and is sold for money, the money is used to pay workers, and the workers go back to work and produce more Stuff. So, we can think of all goods as being part of the larger concept of production. So, everything we produce is supplied to the market. Remember that the supply equation is

Y^S = A \times f(K,H,N,L)

where Y^S is Stuff supplied, A is technological knowledge, K is capital, H is human capital, N is natural resources, and L is labor.

Then, remember that all the Stuff we produce has to be bought, stored in inventory, or exported, so our demand equation is

Y^D = C + I + G + NX

where Y^D is Stuff demanded, C is consumption, I is investment, G is government spending, and NX is Stuff exported less Stuff imported.

In order to find an equilibrium, we need to make another pair of assumptions:

  • The more you can sell for, the more you want to produce.
  • The more goods cost, the less you’ll want to buy.1

Since we’re simplifying away from individual goods, instead of a price, think about the price level of the economy as a whole (which we’ll abbreviate as PL). Also since we’re not thinking too much about individual goods, we don’t have to worry too much about changes in relative prices. (We can talk about those a little bit later.) So, basically, we’re looking at things statically – we don’t need to figure out what happens if coffee’s price goes up more than tea, for example.

The price level determines, on average, how much Stuff sells for. As the price level increases, we’ll produce more. As the price level decreases, we’ll buy less. There’s just one more condition we need to allow an equilibrium:

  • At 0 production, we need an incentive to produce more. So, at 0 production, demand is greater than 0 and supply is 0 by definition. At infinite production, demand is less than infinite.

So, these conditions say 2 things: At a low level of production, demand outstrips supply. As the price level increases, we produce more and demand less. These conditions guarantee that there’s a price level at which we’ll want to supply exactly as much as we want to demand – a little bit lower in price and more will be demanded than produced, and a little bit higher and more will be produced than demanded. So, at that price level,

Y^D = Y^S

Supply equals demand.

Equilibrium.

1Mathematically,

\frac{\partial Y^D}{\partial PL} > 0
\frac{\partial Y^S}{\partial PL} < 0

Don’t Make A Production About GDP April 11, 2011

Posted by tomflesher in Macro, Teaching.
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I’ve often heard people complain that economics growth models assume that the economy can grow infinitely. Some of these arguments sound plausible – what happens when we run out of resources? – but others seem to fall into the fallacy that wealth exists and is spread around, rather than created. The title of this post – Where does growth come from? – might well be restated as, “How is wealth created?”

To answer that question, think back to how electricity is produced. There’s a small amount of input electricity that’s required to produce any extra electricity, along with natural resources like coal and water to produce steam. It’s possible to recapture some of the electricity produced and use it as input in the next procedure. Note that this doesn’t imply there’s some sort of perpetual motion machine or a finite amount of electricity being used to produce an infinite amount, just that sometimes you need to use a small amount of the final product to prepare more of it. In that case, we could say that the amount of electricity produced is a function of the inputs, which are coal, water, land to produce the plant, and electricity.

In the same way, businesses produce goods and services. There’s a fancy term for production. It’s … production. You can also think of production as real GDP, which is equivalent to the amount of stuff produced in a certain division. Generally GDP will be production within a country over the course of one year.

The input factors for production can be a couple of things. First of all, you need money, but what do you use the money for? Generally, companies use money to buy one of a couple of different things.

  • A factory, for example, requires machinery. Even offices need machinery, like copiers and printers. That’s called capital (more properly, physical capital) and (because Marx wrote Das Kapital in German) abbreviated K.
  • Any company needs well-trained people who have skills and can operate the machinery or perform services. That’s called human capital and abbreviated H. Human capital is, at the broadest level, synonymous with special skills or (especially) education. It refers to talent or skill, not to the people themselves.
  • Companies need to know how to produce whatever it is they produce. This is called technological knowledge and abbreviated with the letter A or occasionally z. Technological knowledge allows everyone to be more productive.
  • Companies need natural resources, abbreviated N, to operate their machinery or keep their employees comfortable.
  • Finally, companies need people to do the work. This is called labor and abbreviated L.

There are many ways to express this relationship. The broadest is to say that we can put a bunch of machines and raw materials in a room with some laborers, some people who have special talent, and the energy to run them, and the people will produce something. How much they produce is a function of how much of each of those factors of production is available, and then there will be a bonus for extra technological knowledge that represents what level of production our factory was already at. We can express that mathematically, using Y to represent output as usual, as

Y = A \times F(K, H, N, L)

This represents all of the production of our factory or country. We could, then, figure out how much we’re producing per worker. This has a special name: productivity. We can represent productivity mathematically simply by dividing each factor through by the number of workers (which is the exact plain-sense meaning of “production per worker”):

\frac{Y}{L} = A \times F(\frac{K}{L}, \frac{H}{L}, \frac{N}{L}, 1)

(This requires an assumption called constant returns to scale, which is in turn related to the assumption that when we produce, we produce at around an optimal point where we’re as efficient as we can be given our current level of technology.)

Basically, we produce goods, which are a form of capital. To do so, we use capital and natural resources, which can be used up, along with technological knowledge, labor, and the special skills and talents we call human capital, none of which are used up. Any economy can continue to grow as long as it continues to operate efficiently. That means, as we mentioned in the Comparative Advantage entry, that a country like the US, which has easy access to capital and human capital, should produce things that make the best use of those factors. China, on the other hand, is best off producing labor-intensive goods. India is spread thin with respect to natural resources but has a lot of labor and a lot of human capital.

The key here is that we may see changes in what a country produces over time, but growth can continue indefinitely, as long as we make good choices about production.

Comparative Advantage April 8, 2011

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So far, we’ve done a lot of discussion of macroeconomics where the economy is closed – that is, we assume all trade takes place in the country, or, in plain terms, there’s no importing and no exporting. Now, we can extend that idea into allowing international trade.

The first question, though, is why would we want to do any international trade at all? Why shouldn’t we – the United States – produce all the goods we need at home instead of sending money outside the country to buy things produced somewhere else?

The first thing to think about is called absolute advantage. In some cases, goods are just cheaper to produce in another country than here. An example might be labor-intensive goods (those are goods produced using more human input than machinery). A lot of clothes purchased in the US are produced in India and Bangladesh, for example, and that makes sense: there are many people, wages are relatively low, and so it’s cheaper to produce goods that can be made by people. On the other hand, the US is more adept at producing capital-intensive goods. An example might be circuitboards, which require a lot of machinery to produce right. It’s easier to substitute people for sewing machines than to substitute them for photoengraving equipment.

However, that ignores some possibilities. If we only took absolute advantage into account, we’d come to the conclusion that a few very smart, very productive nations should do just about everything. Breaking this down to individuals, imagine an economy where there are only two people: a writer and her teenage neighbor. The writer can produce 80 pages of quality material in eight hours and do her dishes in an hour. The teenager can produce 8 pages in eight hours and they aren’t very good, and it takes him two hours to do the dishes. (Not a very productive kid.) If the writer wants a novel, she should do it, and if the writer’s dishes need to be washed, then under the theory of absolute advantage, she should do them, since her absolute cost to do so is lower.

Still, that leaves her with two fewer hours to write, kicking her down to only seven hours and 70 pages. The kid has six pages written and one load of dishes. She’s had to give up 10 pages of production – that’s her opportunity cost, or the best thing she gave up to go mow the lawn. It’d be fair to say that doing the dishes cost her 10 pages of writing. The tally: 76 pages plus two set of dishes (70 + 1 from the writer, and 6 + 1 from the kid).

Suppose instead that the writer negotiates with the kid – she’ll do all his writing, and he’ll do all her dishes. She writes 80 pages. He does two loads of dishes. The total: 80 pages plus two loads of dishes, PLUS the kid has five hours free to put together another five pages of material. We have 85 pages and two loads of dishes. That’s an extra 9 pages. Everyone’s better off.

This is called comparative advantage. The kid isn’t faster than the writer at anything, but his opportunity cost to do a load of dishes – two hours of time – could only produce two pages of writing. The writer’s opportunity cost for a load of dishes is 10 pages. So, since his opportunity cost is lower, the teenager’s comparative advantage is in doing dishes. On the other hand, the opportunity cost to the writer of writing 10 pages is one load of dishes. The opportunity cost to the teenager of writing 10 pages is five loads of dishes. The writer’s opportunity cost is lower, so her comparative advantage is in writing.

You can extend that same idea to two different countries. In some, there are lower opportunity costs to produce goods. It’s correct in a quick and dirty way to say that the opportunity cost of producing labor-intensive goods in the US is higher than in India, and vice versa for capital-intensive goods. Basically, the theory of comparative advantage tells us that even if we have the capability to produce something good, we should allow another country to produce it and then import it if we can produce something better.

Purchasing Power Parity and Real Exchange Rates March 13, 2011

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In an earlier post, I talked about purchasing power through the lens of Beerflation. (Again, hat tip to James at the Supine Bovine.) There, purchasing power was used to compare the relative values of the minimum wage in the past to the minimum wage now. Another way we can use purchasing power is to compare whether current prices in one country are the same as prices in the local currency of another country.

That requires some understanding of foreign exchange rates. The convention is to list the exchange rate as a fraction with the foreign country’s value in the numerator and the local currency in the denominator. For example, as of today, one dollar would buy about .72 euro, so the exchange rate in the US would be listed as €0.72/$1. One dollar would buy about 81.89 yen, so the exchange rate there is ¥81.89/$1.1 If we wanted to list the exchange rate of yen for euro, we could work it out using the US exchange rate to be ¥81.89/€0.72, or, dividing out, about ¥113.74/€1. Since we’re comparing currency instead of goods, we call this a nominal exchange rate.

If we stop to think about prices, this should mean that$1 worth of some random good that’s of uniform quality worldwide should be the same as €0.72 or ¥81.89 worth of that same good. Think, for example, about white sugar. Currently, Domino Sugar is listed on Amazon.com at $8.99 for a ten-pound bag. If my purchasing power is the same worldwide, then I should be able to get that same ten-pound bag for around (8.99*0.72) = €6.47 or for around (8.99*81.89) = ¥736.19. If I can, then my money is worth the same around the world – prices are all the same after I change my money. This is called purchasing power parity.

If purchasing power parity holds, then the price of a good in US dollars is the same, after conversion, as the price of a good in euro. That means the exchange rate of those currencies should be the same as the ratio of those prices. In symbols, we could define e as the nominal exchange rate, P as the local price, and P* as the foreign price. (This is the convention that Greg Mankiw uses in Brief Principles of Macroeconomics.) Then,

e = \frac{P\star}{P}

A real exchange rate, like all real variables, relates the prices of goods to each other. The classic example of a real price is to compare the price of a good you wish to buy with the number of hours you’d need to work to get it. If I make $22 per hour, and the price of Daron Acemoglu’s Introduction to Modern Economic Growth is $61.38 (currently), then regardless of the inflation rate or where our prices are pegged, the price of the book for me is 2.79 hours of work. In order to avoid having individual prices, I could peg the price not to hourly wages (which differ from person to person) but to some staple good. Given a $1.95 can of black beans, Daron’s book is worth 31.48 cans of beans, and I make around 11.28 cans of beans per hour. If we divide those out, we’ll get 31.48/11.28 = 2.79 hours per copy of Introduction to Modern Economic Growth.

Neat, huh?

So, if a real price relates the price of goods to each other, a real exchange rate relates local prices to each other. In general, this is done using what’s called a basket of goods that’s supposed to represent what a typical consumer buys in a year. Here, for simplicity, we’ll stick with one good.

If purchasing power parity holds, then the ratio of those prices should be 1/1 after we correct for the foreign exchange rate. So, if we define RER as the real exchange rate between two countries, then

RER = e \times P \times \frac{1}{P\star}

Or, in other words, prices are the same after you exchange your money. That is, with purchasing power parity, the real exchange rate is 1. More simply, if purchasing power parity holds, then prices should be the same whether you change your money into a foreign currency or not.

1 I’m rounding all prices to the nearest hundredth for simplicity.

Money Neutrality (or, the Quantity Equation) March 4, 2011

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The Macro class I’m TAing has just gotten to money growth and inflation, chapter 12 in Mankiw’s Brief Principles of Macroeconomics. As usual, the quantity equation, MxV = PxY, confuses some of the students a little bit, so I thought I’d see what I can do to clarify it a little.

First, let’s define some terms. M is the size of the (nominal) money supply. V is the velocity of money, or the number of times a given dollar is spent in a year. (It represents how fast people spend money, so that if the money supply is only $100 but GDP, the total of expenditures on all final goods and services in the US in a year, is $500, the velocity of money is 5/year.) P is the nominal price level – that is, just the average price of stuff in the economy. Y is real GDP, so it represents the production level or the amount of stuff produced in the economy.

For reasons of a mathematical nature1 explained at the end of the page, you can think of PxY as all of the expenditures in the economy, and because of that, you can think of it as the product of the average price and the quantity produced. So, the right-hand side of the quantity equation is nominal GDP.

V is determined by many different things. For example, when people feel less confident about the economy, V might drop because people would spend less money. When people feel more confident, they spend more easily, and so V might rise. A lot of things that affect V are difficult to talk about if we only have Principles-level tools, though, so the conventional wisdom is to leave V constant for now.

Then, we basically have the equation:

M = P \times Y

What this means is that a change in the money supply has to be matched by the change in price level and the change in production.2 If all we know is that the money supply changed, then it could be due to a change in the price level, a change in real production, or some combination of the two. If on the other hand the price level changes and production doesn’t, then it must be due to a change in the money supply (like if the government started printing too much money).

When there’s too much money in the economy, price levels rise; when businesses produce more, then the real GDP level (Y) will increase, so if the money supply doesn’t increase then price levels would be expected to fall3; when price levels are changed for some outside reason, then either GDP has to change, the money supply has  to change, or both.

Note:
1 Under the expenditure method, the Gross Domestic Product or GDP is the total market value of all final goods and services produced in the economy in a given period of time. For each good or service, it will either be recorded as a sale if someone buys it or as inventory by the business that produced it if it isn’t sold. So, for every good i in the economy with a given nominal price i, the contribution to GDP of that good is

p_{i} \times q_{i}

So, in an economy with n goods, you can add up all of the expenditures on those goods and generate nominal GDP. Thus,

GDP_{Nominal} =\displaystyle\sum\limits_{i=1}^n p_{i} \times q_{i}

This is equivalent to multiplying the average price by the total number of goods in the economy.

2 This can be shown using the natural logarithm transformation. Since the interpretation of a change in the natural logarithm is the percentage change in the untransformed variable,

ln(M \times V) = ln (P \times Y) \implies ln(M) + ln(V) = ln(P) + ln(Y)

\Delta V = 0 \implies \Delta ln(V) = 0 \implies \Delta ln(M) = \Delta ln(P) + \Delta ln(Y)

So, the percentage change in M can be decomposed into two pieces: the percentage change in P and the percentage change in Y.

3Thanks to Roman Hocke for catching an error in an earlier version of this post.