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

Posted by tomflesher in Macro, Teaching.
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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

Posted by tomflesher in Macro, Teaching.
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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

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 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.

Beerflation February 26, 2011

Posted by tomflesher in Macro, Teaching.
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One of the ways we measure well-being is through purchasing power. There are a lot of unusual measures of purchasing power; possibly the most famous is The Economist’s Big Mac Index, which measures the price of a Big Mac in different countries as a way of checking to see if reported exchange rates are correct.

This depends on idea of purchasing power parity. That is, if we’re going to compare prices in different currencies, the goods we’re comparing have to be precisely the same. The Big Mac is thus an ideal good to use for an exchange rate calculation since the point of branding McDonalds burgers is that they provide a consistent experience.

Another use of purchasing power is to estimate well-being across time periods. If your wage rose at a faster rate than the price level did, you’re better off than you used to be. James over at the Supine Bovine uses the rising price of beer relative to the minimum wage to illustrate that the average entry-level worker (personified as “twenty-somethings who are stuck living with mom and dad”) is worse off in the current economy than he used to be:

So I ask my dad what minimum wage was back in the day and what the price of beer (at a bar) was; he told me it was (about) $4/hour and $0.25, respectively.  This means that on one hour of minimum wage work he could get shitfaced – 16 beers is no small amount.  Anyway, today the minimum wage is $8/hour and a beer at a bar costs no less than $3.50 (but you’re looking at double that at most places).  So my peers can purchase at most two beer per hour worked at minimum wage.  So… beer costs about 8 – 16 times what it used to.

Now, I have a few quibbles with this (mainly to illustrate the problems with using a single good to compute inflation). For one, if That ’70s Show is any indication, bar beer hasn’t always been served in twelve-ounce containers. Adjusting down to an eight-ounce glass, that leaves us with approximately 12 beers. Second, are we discussing the same beer? Even moderate changes in alcohol content can adjust this calculation quite a bit. If, for example, James is using Labatt Blue at 5% alcohol by volume, then substituting the price of Sierra Nevada Pale Ale at 5.6% means that one Sierra Nevada is about 1.12 Blues. That kicks the old figure down to about 10.7 beers, or, equivalently, an old price of .09 hours worked for a beer. (Note that this doesn’t substantially change James’ point.) The current ratio, accepting James’ $3.50 figure as correct, is about .43 hours worked for a beer, meaning the real price of beer has risen approximately 386%. (When I say ‘real price,’ I am indicating that the price is expressed in terms of other goods, rather than in dollars.) It’s impossible to compute a rate of inflation because we don’t know what year James’ dad was talking about.

On the other hand, a lot of twenty-somethings these days have laptops. Just spitballin’.

Why is the Money Multiplier 1 over the Reserve Ratio? February 25, 2011

Posted by tomflesher in Macro, Teaching.
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Or, How does the banking system create money?

The Introduction to Macroeconomics class I’m TAing just got to the Monetary System. This was a difficult chapter for the students last semester, so I want to place a clear explanation online and give a mathematical proof for some of the more motivated students who look grumpy when I handwave away the explanation. For those who want to play with this themselves, I’ve created a spreadsheet that shows the process of money creation. You can choose an initial deposit and a reserve ratio (which I set at $1000 and 1/10), then trace out the changes as people borrow the money.

It boils down to what banks are for. The biggest mistake you, in your capacity as a consumer, can make is to think that the purpose of a bank is to give you a place to save money. Really, the purpose of a bank is to serve as a financial intermediary, which allows people who want to lend money (savers) to find people who want to borrow money (borrowers). When you put your money in a savings account, you’re putting money into the pool of loanable funds, and the bank will pay you a small amount of interest. Why don’t they let you do that yourself? Two reasons:

  1. They have more information about who wants to borrow money (specifically, businesses and consumers ask them for loan), and
  2. They offer a small return with certainty, whereas if you found a borrower yourself, you’d have to collect your return yourself.

In order to provide this certainty, the banks have to hold a certain amount of each deposit in reserve, in order to make sure that people who want to take their money out can do so. This is the reserve ratio, and it’s set by the Board of Governors of the Federal Reserve. Adjusting the reserve ratio upward means that a bank needs to hold a larger portion of each deposit, so they have less to lend out. That means that all things being equal, money will be harder to get than with a smaller reserve ratio, and so interest rates rise and less money is created. Conversely, lowering the reserve ratio means that banks have more money they can lend out and more money will be created.

When you deposit $1000 in the bank, if the reserve ratio is 1/10 or 10%, the bank has to hold $100 (or .1*$1000, or 1000/10) as reserves, but they can lend out $900. That means they’ve effectively created $900 out of thin air. They can loan out an additional 9/10 or 90% of the initial deposit. They’ll loan that money to a company which then uses it to, for example, pay an employee.

The employee will deposit $900 in the bank. The bank has to hold $90 ($900/10) as reserves, but they can lend out $810. Money is created, but the creation is smaller. As you can see, this cycle will repeat, with 10% being taken out every time, and if you iterate this enough times, you can see that the process approaches a final value because the new loanable funds that turn into new deposits are smaller every time.

In the case of a reserve ratio of .1 or 1/10, using my spreadsheet above, you can see that $1000 jumps to $6500 in deposits by the tenth person to be depositing the money, and the amount continues growing but at a slower rate. By the fiftieth person, the total deposits have reached about $9950, and by the one-hundredth depositor, it’s at $9999.73. The extra loanable funds here are only about 3 cents, so there’s going to be an almost imperceptible change. This is called convergence, and by playing with different reserve ratios, you can see that each reserve ratio converges. The total new deposits represent new money created. So, the change in the money supply is equal to the initial deposit divided by the reserve ratio, or, mathematically,

\Delta MS = (1/rr)*\Delta D

Since 1/rr is a little inconvenient to write, we made up a new term: the Money Multiplier.

\Delta MS = MM*\Delta D

Mathematically, this process is called a geometric series. Initially, the deposit is $1000 and the new loanable funds are (9/10)*1000. The second deposit is (1/10)*1000 and the new loanable funds are (1/10)*(1/10)*1000. This process continues, and can be represented using the following notation:

\displaystyle\sum\limits_{i=0}^{\infty} r^i * 1000

Rearranging slightly,

1000*\displaystyle\sum\limits_{i=0}^{\infty} r^i

A geometric series with a ratio of less than one – which our ratio, 9/10, is – has the solution

1/(1-r) = 1/(1 - 9/10) = 1/(1/10) = 10

Thus, the sum of all of these new deposits will be

10*1000

or, more generally,

MM*Deposit

That’s exactly what we wanted to show. So, if you’re mathematically inclined, remember that the Money Multiplier is the reciprocal of the reserve ratio because the process of borrowing and depositing money is a geometric series.