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Increases in CPI: Good or bad? January 30, 2012

Posted by tomflesher in Macro, Teaching.
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One of the nice things about WordPress is that I get a nice summary of the search engine terms that led people to my page. Bobby Bonilla is popular, as always – it’s nice to know that people are curious about him – but another common way people end up on my blog is by looking for pros and cons of the Consumer Price Index. One searcher this week asked:

Is an increase in CPI good or bad?

As with all economics, the answer is, “It depends,” but let’s start by asking a refining question: Good or bad for whom?

  • The Government: Good.

An increase in the CPI represents an increasing cost of living, which is related to inflation. Inflation, as measured by an increase in the CPI, means that the government can sign contracts to pay employees or purchase materials in current dollars and then pay them back in inflated dollars; that is, if I sign a contract today, January 30, 2012, to pay you $100 on January 30, 2013, then the $100 I have now is worth more than the $100 I’ll pay you back with. (This is one reason for interest payments.) Of course, if everyone expects the inflation, they’ll take that into account when contracting with the government and demand higher payments. A government can, in fact, use large unexpected inflation to cut their costs this way – it’s called an inflation tax, and we’ll talk about it a little later on – but it’s not  a strategy that works well or often.

  • Businesses: Good.

Businesses can take a beating if they’re contracting with governments, but consider wage contracts – when I worked at a factory, pay rates were set by position in January, so my only hope of getting a raise was to move to a higher position. If CPI rose over the course of the year, which it almost always did, I took what was effectively a pay cut until the next round of  cost-of-living adjustments in January. That means that the business could negotiate contracts throughout the year for supplies and sales, but its real wage expenses actually fell.

  • Consumers: Bad (mostly).

And who takes the brunt of the drop in real wages? Households, or consumers. Since I lack the power to demand my wages rise throughout the course of the year, then my wages on January 1 are going to buy fewer goods than my wages on December 31, even though they’re nominally the same amount of money.

On the other hand, a small, predictable amount of inflation allows for a few things to happen. If it’s small, it means that prices more or less stay the same. (A large inflation rate would make it impossible for me to keep the same wage from January 1 to December 31 without built-in monthly or quarterly raises, for example.) If it’s predictable, we avoid a couple of ugly problems like the inflation tax or surprises when repaying loans. If it’s inflation, rather than deflation, people and businesses have a smaller incentive to hold on to their money to wait for prices to drop, so there’s an argument, weak though it is, to be made that inflation encourages spending.

All told, an increase in CPI means that a household has to spend more dollars to maintain the same standard of living; that’s mostly bad for the households, but it can be good for businesses and the government.

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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.

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

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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.

What’s so Gross about the Domestic Product? March 17, 2011

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The Gross Domestic Product (GDP) is one of the fundamental ideas of introductory macroeconomics. That’s because GDP is the core of one of the best ways to measure citizens’ well-being. We’ll get to that in a future post, though. For now, let’s talk about what GDP measures, and pretend that we’re not going to allow international trade. That makes this a closed economy model.

Let’s start with a simple premise: Everything that’s produced is purchased by someone. That makes sense in a couple of ways. A household can buy something, another business can buy something, the government can buy something, or… businesses can produce goods and store them for future use. For now, let’s treat this as the business buying its own goods to resell later.

GDP is defined as the market value of all final goods and services produced within a country in a given period of time. If we’re talking about the United States’ GDP for 2010, then it amounts to the prices of everything that was made in the US in 2010. The word ‘final’ means that if one company produces something that’s used as an input for another product, then only the last product counts. That means that some goods, like flour, might be final goods sometimes and intermediate goods other times. If I own a bakery, then I’ll buy a five-pound sack of flour to use in making bread, and so the flour is intermediate (since it’s used to produce another, final good). If I buy flour to make the same loaf of bread at home, then the flour might be used in home production, but since home-produced goods aren’t sold, then the flour is last sold to a consumer, and so it’s a final good. Since a consumer makes the purchase, it’s called Consumption.

Imagine that a box factory produces 600 boxes on December 31, 2010 and then sells them on January 1, 2011. Then, we have a sale of final goods, but the final goods weren’t produced in 2011, so they can’t count toward 2011 GDP. This requires the idea of inventory, which can be defined as goods that are produced but not sold. Inventory sales need to be subtracted from spending when calculating GDP.

Spending by businesses is on two things: intermediate goods (to produce final goods) and capital production (that is, stuff that allows them to be more efficient). All together, we call this spending by businesses Investment, which has a special definition in macroeconomics. Make sure not to confuse ‘investment’ in macro with the idea of putting money in stocks and bonds and hoping it grows. When taking a macro class, ‘investment’ pretty much means ‘spending by businesses.’ Inventory gets subtracted from investment, because it represents using past-produced goods. Those goods would have been counted as GDP in a previous year, so they need to be subtracted now even though a consumer purchased them.

Consumption and business spending aren’t the only things that need to be counted, though. Sometimes a business will produce a good that isn’t bought by a consumer. (I, for example, have never purchased a space shuttle, even though clearly someone’s producing them.) This is why we need to count Government spending.

Everything that’s produced is purchased, as long as we define ‘purchased’ to include ‘stored in inventory,’ and then we can subtract inventory sales from future GDP. Even though someone consumes a good that might have been produced in the previous year, subtracting it as inventory spending allows us to maintain the definition of GDP as ‘everything produced in the US in 2010’ while at the same time having an easy way to calculate it: just add up everything we buy!

This leads directly to the expenditure method for calculating GDP: just add up all the spending by consumers, by businesses, and by the government. In math, the letter Y is often used to represent output, and GDP represents the production (i.e. output) in an economy. So, we can use the formula

Y = C + I + G

where C is consumption spending, I is business spending (including subtracting inventory) and G is government spending. (In an open economy we’d need to account for imports and exports. That will come later.)

These two definitions (“the final market value of all goods and services produced in an economy in a given period of time” and “C + I + G”) are equivalent. In a future post, we’ll talk about how to put that to use.

Shortcomings of CPI March 14, 2011

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In the previous post, we talked about the Consumer Price Index (CPI). Basically, the CPI is a number that indicates how much the price of a basket of goods purchased by the typical consumer has changed since a base year we choose when we calculate it. To review, in the base year, CPI is always 100, and in other years a number greater than 100 indicates that prices are higher than the base year while a number less than 100 indicates that the price level is lower than the base year.

One way to measure inflation is just by calculating the change in CPI. The percentage change in CPI – \% \Delta CPI – can be calculated by figuring the year you’re interested in as the base year and then just subtracting 100 from the current CPI. However, if you’re using other data where the base year is already calculated (such as from FRED), you can use the regular percentage change formula:

\%\Delta CPI = \frac{CPI_{current} - CPI_{base}}{CPI_{base}}

However, the nature of the CPI leads to a few problems. They stem from the use of the basket of goods, which has to stay constant from one year to another in order for the comparisons to be meaningful. That means that whatever we decide to use as the basket of goods in 2007 has to be what we calculate the value of in 2011. There are some immediate problems that come to mind.

First, think about the changes in relative prices of goods to each other. Let’s say the basket contains a pound of flank steak. A lot of people use that as the filling for tacos, so a substitute good might be roast pork. If those goods are around the same price, people are indifferent between them. What if the price of pork increases a little, but the price of flank doubles? In that case, a lot of people will stop buying flank steak and switch over to pork instead. The basket doesn’t reflect this, though, so the CPI will rise a lot more than the relative cost of living does, so the CPI doesn’t really accurately reflect the change in the cost of buying what a typical household buys. This is called substitution bias.

Second, I’m addicted to my iPad. When the current basket was created in 2007, it didn’t exist. Now, it’s practically a requirement for a grad student. The basket can’t account for the introduction of new goods like this, since in order for it to be a useful comparison, the basket has to stay the same from year to year.

Finally, I think it’s fair to say that some goods are getting more durable. An iPod Touch, for example, lasts longer than it did when it was introduced in 2007. It offers more features than it used to (such as voice control and a camera). Even if prices had stayed the same,  fourth-generation Touch is worth far more than a first-generation Touch. The same expenditure generates a lot more happiness, and so the quality of goods isn’t accounted for in the CPI.

CPI isn’t a perfect measure of inflation or the cost of living, but it’s a common and important one. Know how to calculate it, and know its shortcomings.

What Good is the Consumer Price Index, Anyway? March 14, 2011

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One of the first things people learn in their Intro Macro course is that prices change. Before we get into why that happens, let’s think about one of the other things we’ve already talked about: purchasing power and the associated concept, purchasing power parity. One way to measure purchasing power is to look at how much a dollar buys, but in my Beerflation post I referred to how the relative price of beer may have increased since the 70s but the relative price of laptops has decreased. So, let’s consider how we can solve that problem.

One common way to do so is to consider not the price of one good, like a beer at a bar or a can of black beans or a copy of Introduction to Modern Economic Growth, but the overall price of many goods. Think about, for example, the hypothetical cost of being a new college student. I’ll ignore tuition for now (since that’s variable based on where you go to school and how you did in high school) and only consider things a typical freshman would need to buy over the course of the year.

First, the student needs a place to sleep and food to eat. At my current employer, the University at Buffalo, a double room costs $5928. A meal plan with 14 meals per week plus $300 in flexible spending costs $1950 per semester or $3900 for the year. Then, the student needs clothes to wear. A typical wardrobe might be five pairs of jeans, five t-shirts, five polo shirts, ten pairs of underwear, ten pairs of socks, and a winter jacket. If purchased at Target, Wal-Mart and Old Navy, this would run a total of about $250.

As for actual academic expenses, the student will need books for about eight to ten classes (let’s settle on 9 for now). Assuming the price of books is around $80 each (allowing for a class or two that has a novel as the assigned text and two math classes that might use the same book, like Calculus I-II), that puts book expenditure around $720. The Calculus class probably allows the use of a graphing calculator, which currently runs about $100 at Amazon.

Finally, the biggie will be the student’s computer. I tend to go cheap and pick up laptops around $450 from Best Buy, so let’s use that number.

If we add up all of these numbers, we get $11,348, which is a fair number to use when we discuss “the out-of-pocket cost of being a college student.”1 Then, we can compare the cost to last year’s rates and next year’s rates and look at how the out-of-pocket cost of being a college student changes from year to year.

That’s basically what the CPI does. It looks at what’s called in technical terms a “basket of goods.” That is, it assumes that people buy more or less the same goods every year and then looks at the changes in the total price of those goods.2 In order to make things easier to deal with, they perform a mathematical trick called normalizing. That allows us to use an index, or a single number, to show changes. It’s easier remembering that CPI in the base year is defined as 100 than trying to remember what a particular year’s total cost was. They choose a base year and divide that total cost for the basket of goods by the base year’s total cost. So, the formula for CPI in a given year is

CPI \equiv \frac{Cost_{current}}{Cost_{base}} \times 100

So, there are two things to note here. The first is that when the current cost equals the base cost, the CPI will equal 100 (so, CPI is  always 100 in the base year). The second is that we can find the approximate percentage change in prices – that is, the approximate level of inflation – using either of the following formulas:

\%\Delta CPI = \frac{CPI_{current} - CPI_{base}}{CPI_{base}} \times 100 = (\frac{CPI_{current}}{CPI_{base}} - 1) \times 100

So, summing up, the Consumer Price Index (CPI) is a number that represents how the price of a certain basket of goods has changed since a chosen base year. The basket of goods is meant to represent typical products purchased by a typical household in a given year. It’s one way to measure inflation, since the goods are the same from year to year so the real value should stay the same even if the prices change.

Notes.
1 Note that this is explicitly the out-of-pocket cost. The opportunity cost is much different and would have to take into account the wages that the student gave up to go to college, but wouldn’t necessarily take into account a computer or clothing since those would need to be bought anyway. Again, this cost is an estimate and many of the numbers are estimated or rounded. If anyone cites this as “the cost of going to college,” I’ll be very sad.

2 If we have a model economy where we know that we have a well-defined basket of goods x1, x2, …, xn with associated prices p1, p2, …, pn, then the price of the basket would be

\displaystyle\sum\limits_1^n x_n \cdot p_n

Purchasing Power Parity and Real Exchange Rates March 13, 2011

Posted by tomflesher in Macro, Teaching.
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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

Posted by tomflesher in Macro, Teaching.
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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.