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

Posted by tomflesher in Macro, Teaching.
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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 RatesMarch 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. Tags: , , , , , , , 2 comments 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.