Inflation

Price indices are useful for calculating inflation over time.  The consumer price index (CPI) measures changes in prices for the overall economy.  Researchers can also use price indices to understand the evolution of the price of health care over time.  For instance, the Bureau of Labor Statistics also calculates a CPI for Medical Care and Medical Care Services.

The question of how to calculate a price index is far from trivial however.  To calculate the change in the price of any good between years 1 and T, one could simply use the following formula:

• P_simple=p_iT/p_i1

However, a price index indicates the change in prices for a basket of goods.  If you are considering the change in price of 10 medical services, how much weight to you give to each one?

Economists have generally come up with the solution: the goods that make up a large share of total expenditures should be weighed more than those that make up a small share.  For instance, let us imagine a simple example where you have two expenses: food and medical care.  The price of food goes up by 10% and the price of medical care goes up by 20%.  Let us assume that food makes up a larger share of your budget than medical expenses and that the initial value of the price index is 1.0 (i.e., T=1).  Thus, if 80% of your income goes to food and 20% of your income goes to medical expenses, than the value of the price index one year from now would be would be 80%*1.1+20%*1.2=1.12.

Sounds easy right?  Not so fast.

I said that 80% of the person’s budget was made up by food, but does that figure refer to your budget expenditures in the first time period or the second time period?  Let us assume the following:

• P_food,1=$1; Q_food,1=800; E_food,1=$800;
• P_food,2=$1.1; Q_food,2=800; E_food,2=$880;
• P_med,1=$100; Q_med,1=2; E_med,1=$200;
• P_med,2=$120; Q_med,2=3; E_med,2=$360;

Above, P, Q and E refers to price, quantity and expenditures respectively; the first subscript in the formulas above refers to the good (food or medicine) and the second subscript refers to the time period (1 or 2).  In the example, 80% of the person’s budget in period 1 is for food and 20% is for medical supplies.  If we use the budget shares in the first period to weight the price changes, then we could calculate the price index as:

• (800*$1.1+2*$120)/(800*$1+2*$100)=1.120

This method is known as the Laspeyres price index.  The general formula is: [Σ p_itq_i0]/[Σ p_i0q_i0].

An alternative measure is the Paasche  price index.  In this case, we weight the price changes depending on the bundle of goods in the last time period under consideration.  In the example, our price index would be:

• (800*$1.1+3*$120)/(800*$1+3*$100)=1.127

The price index is higher now.  Why?  In the last period, the quantity of medical care we purchased increase (for 2 to 3) compared to the quantity of food purchased (stayed the same at 800).  This means that the Paasche price index will put relatively more weight on the price changes for medicine.  Since the price of medicine increased faster than the price of food, the overall price index level be higher in this example than in the case of the Laspeyres price index.  The general formula for the Paasche price index is: [Σ p_itq_iT]/[Σ p_i0q_iT].

However, both the Laspeyres and Paasche indices do not take into account substitution effects between goods. Goods are weighed statically based on the quantity purchased in either the first period (Laspeyres) or last period (Paasche). To solve this problem, one can use the Fisher price index. This index does account for individuals substituting across different types of goods. To calculate the Fischer index, one simply takes the geometric mean of the Laspeyres and Paasche indices. According to the example above, this means the price index would be:

• P_f=(P_p*P_l)^0.5=(1.120*1.127)^0.5=1.123

One can also chain the Fisher index calculations from each year in order to produce a chain-weighted Fisher price index, but I’ll save that explanation for another day.