# Prices

Price Adjustment, Price Indices, and Chain-Weighting

- Modified: Sep 6th, 2022

## ¶ Real GDP

To measure aggregates, our unit is dollars.

Problem: Dollars themselves change in value. Solution: We need to adjust for this by using a price index.

Conceptually:

$\text{Nominal GDP} = \text{The Actual Data} \approx \text{Quantities} \cdot \text{Prices}$ $\text{Real GDP} \approx \text{Quantities}$

We want to find some way to cancel out the prices so we are left with just the quantities.

### ¶ The Simple Method

The simplest way to do this:

1. Choose a base year. Only use prices from this specific year
2. Use those prices for every other year. (Current Quantity) x (Base Year Price) = “Real GDP”

This method is simple, and easy to understand. But it can lead to problems when making comparisons over long time periods.

Most “Real” aggregate data nowadays uses chain-weighting (as opposed to the simpler “fixed-weighting” described above).

### ¶ The Chain Weighting Method

Problem: Price ratios change, which means the choice of base year matters.
Solution: Use all the base years!

This is how Real GDP is calculated in most data nowadays.

The recipe for chain weighting:

1. For each pair of years, calculate gross real gdp growth using two different ‘base years’.
2. Average those growth rates to get our chain-weighted growth rate between the two years, called the “Fisher index”.
3. Choose some actual base year. Multiply or divide by our chain-weighted growth rates to get the real gdp for other years.

The Fisher index formula used for calculating chain-weighted GDP growth is

$Q_{t}^{F}=\color{#a00}\sqrt{ \textcolor{#080}{ \frac{\sum p_{t-1}q_{t}}{\sum p_{t-1}q_{t-1}} } \times \textcolor{#00d}{ \frac{\sum p_{t}q_{t}}{\sum p_{t}q_{t-1}}} }$

(Our textbook labels this $g_{c}$, while the BEA NIPA handbook labels this $Q_{t}^{F}$)

This formula says that to find the growth in real GDP between years $t-1$ and $t$, take the geometric mean of real GDP growth calculated with year $t-1$ prices and real GDP growth calculated with year $t$ prices.

One minor downside of chain-weighting: components don’t add up to the whole. So if you want to calculate, eg, consumption as a percentage of GDP, then you need to use nominal data or data deflated with some fixed base year.

## ¶ Implicit GDP Deflator

Once we have a time series for “Real” quantities, we can extract prices as well.

The basic idea looks like this:

$\text{Prices} \approx \frac{\text{Quantities} \cdot \text{Prices}}{\text{Quantity}} \approx \frac{\text{Nominal GDP}}{\text{Real GDP}}$

Multiply by 100 to put in % terms, and you get the price index called the implicit GDP deflator.

$\text{Deflator}=\frac{\text{Nominal}}{\text{Real}} \times 100$

Then to use a deflator to adjust nominal data into real data, we just rearrange the formula:

$\text{Real}=\frac{\text{Nominal}}{\text{Deflator}} \times 100$

This is not the only way to get aggregate prices.

## ¶ Price Indexes

In reality, there isn’t one single price in the economy. There are many thousands of different prices which are changing at different rates.

To talk about prices in the aggregate, we need some way of averaging out many different prices. We need a “Price Index”. Because there are many different ways to average prices, there are many different price indexes in use.

• The Implicit GDP Deflator or GDP Price Index is implicitly the price index we get by averaging prices over all goods produced in the country.
• The Personal Consumption Expenditures Price Index averages prices only over Household purchases.
• The Gross Domestic Purchases Price Index likewise averages prices over C, I, and G, but ignores the prices of exported goods.
• The Consumer Price Index is a survey-based price index compiled by the BLS. Goods are weighted based on a basket of goods that reflects the consumption patterns of a “typical” household.
• The Producer Price Index is another BLS price index, reflecting the prices that producers face when they sell things.
• “Core” price indexes (Core PCE, Core CPI) are calculated by excluding the prices of food and energy.
• This is done to try to get a grasp on longer term patterns in price changes because those two types of goods have relatively volatile prices (energy especially).

### ¶ The Price Level and Inflation

When a price index represents all or most of the goods in the economy, we might call it the “Price Level”.

The “Inflation Rate” is the percentage change in the price level. In other words, positive inflation represents an average increase in prices across the economy, or equivalently a decrease in the average amount of goods or services that a dollar can buy.

(Change in the CPI is the measure of inflation most commonly seen in headlines.)

### ¶ Other Price Indices

While the above price indices are useful for thinking about aggregate macroeconomic phenomena, there are also many smaller price indices that are in use. Price indices for specific industries, specific commodities, locations, etc.

At the extreme, every household has their own consumption patterns, and so experiences price changes in different ways. Measures of the overall price level in the economy won’t be a perfect reflection of any individual households experiences. But this is simply the nature of aggregate statistics.