The Substitution Bias in the Consumer Price Index
Student Instruction Sheet

Introduction
In this exercise, we will seek to understand why the traditional Laspeyres price index overestimates the true cost of living inflation rate.  This is an important topic because the Consumer Price Index is based on the Laspeyres methodology.  We will then use the Goal Seek tool of Excel to find an ideal price index that exactly measures the change in the cost of living.

Suppose an individual has Cobb-Douglas preferences represented by the utility function:
					U = x^ay^b						
where a and b are positive constants.  For an individual facing fixed prices (p_x and p_y) and a fixed income (I), the utility maximizing quantities x^* and y^* are given by the formulas:
			    and   			
Suppose initially a = b = 1 and that p_x = $4, p_y = $1, and I = $400.  Note that the relative price of good x to good y is 4.

We will be using this information to examine how a Laspeyres price index (L^t) measures a change in prices over time.  Recall that the formula for a Laspeyres price index is:
										
Where the superscript refers to the time period, which is either any given year (t), or the base year (base).

Instructions

1.	Set up a spreadsheet to contain the information above.  Use the first column (A) to list the parameter and variable names (a, b, p_x, p_y, I, x^*, y^*, U ), and the second column (B) to contain the associated values and formulas.  (If you don’t know how to do italics or subscripts, just use regular formatting.)  You may wish to start on row 2, and use row 1 as a label row.  (Label the values and formulas as “Original values” in cell B1.)  When typing in values, for a, b, p_x, p_y, and I, use the values above.  For x^*, y^*, and U, type in formulas based on the appropriate equations above.  Remember to start your formulas with an equal sign, use parenthesis to differentiate between the numerator and the denominator of the fractions (as well as different fractions), use an asterisk (*) for multiplication, and use a carrot (^) for exponents.  For example, your equation for x^* should be:  =(B2/(B2+B3))*B6/B4  If you have set up the problem correctly, you should get x^*= 50, y^*=200, and U=10,000.
2.	Assume that a year passes, and there has been inflation such that the price of good x rises to $4.20 and the price of good y rises to $1.40.  (Note that the relative price has fallen to 3, as the price of good y has risen faster than the price of good x.)  We want to show what this price change will do to the equilibrium consumption values, initially assuming that income stays the same.  Copy the parameter values you just computed (cells B2-B9) to column D (cells D2-D9) of the spreadsheet, so the rows line up as before.  Label this new block of cells “New values” in row 1 (cell D1).  In this new values block, change the prices to the new values.  Note what happens to the quantity demanded of each good, and then note how these new consumption levels affect utility.
3.	We now want to calculate the Laspeyres price index for the inflation.  A couple of rows below the “New values” block of cells (cell D12), type in the formula for the Laspeyres price index based on the L^t equation above.  Use the cell addresses for p_x, p_y, x^*, and y^* in your formula.  Again, be careful to put parentheses around the numerator and the denominator, and use the * for multiplication.  Use the “Original values” as the base year values, and the new values as the current (t) year.  If you typed in the formula correctly, the cell should read 1.225.  [If you need to adjust the number of decimal places, click on the “Increase Decimal” or “Decrease Decimal” buttons (next to the comma button) in the standard Excel toolbar.]  The 1.225 corresponds to a 22.5% average inflation rate.  So that you can keep track of the information that you are creating, type the phrase “Laspeyres price index” in cell D11 above this calculation.
4.	We are now ready to examine the impact of a cost of living raise.  Copy the original block of parameters and values (cells A1 through B9) to rows 15-23 (with row 15 as the label row) in columns A and B.  Change the label on this block of cells to “Cost of living raise”.  Change the prices to p_x=$4.20 and p_y=$1.40, so this block looks just like the new values.
5.	We want to give the consumer a raise that corresponds with the 22.5% inflation rate.  To do this, write a formula for the income value cell that does this, based on the original income level and the Laspeyres price index cell you calculated above (your formula should be =B6*D12).  If you have done this correctly, the consumer should have an income of $490.  Note what happened to total utility – it increased from the original value.  This occurred because the consumer chose (because of her demand functions) to purchase relatively more of good x and less of good y because the relative price of good x fell from 4 to 3.  That is, she substituted away from the good whose price rose in relative terms, and this allowed her to shelter herself from some of the inflation.  This is an example of the “substitution bias” that is a problem for the consumer price index.
6.	Now we want to find an “ideal” price index.  This price index would leave the consumer exactly as well off after the inflation (and the associated cost of living raise) as before.  To find an exact ideal price index for this consumer, we will use the “Goal Seek” tool.  Copy the “New values” cells to column D, just to the right of the “Cost of living raise” block of cells.  Make sure that you pick rows so the values match up with the parameter names (copy cells D1-D9 to cells D15-D23 given that the original parameter and variable names appear in cells A16-A23).  Label this block of cells “Holding U constant.”
7.	We want to find the level of income that will give the consumer the same utility as before the inflation (10,000 units).  To do this, click on the utility value cell (which should read 6,802.72), then click on the Tools menu, then choose “Goal Seek…”.  A window should appear with entries:  “Set cell”, “To value” and “By changing cell.”  This is fairly self-explanatory; next to “Set cell”, the utility function cell address should appear (because you clicked on it before starting Goal Seek).  Next to “To value”, type in 10,000.  Next to “By changing cell”, type in the address of the income cell (D20). Alternatively, you can click on the income cell; if the cell is blocked by the Goal Seek window, click on the name of the Goal Seek window and drag the window to a new location.  Click OK.  You should see the values of I, x^*, y^*, and U change to correspond to U=10,000.  If you have done this correctly, you should see I=$484.97, x^*=57.74, and y^*=173.21.  (Note:  Excel may round to more or fewer digits on your computer.  To change the number of digits, click on the cell, then on the “Increase Decimal” or “Decrease Decimal” buttons next to the comma button.)  These values correspond to the demand for x and y at an income level of $484.97, which leaves the consumer at a utility level of 10,000.
8.	We are ready to compute an ideal price index.  Recall that in step 5 we found the new income associated with the cost of living raise by multiplying the old income level by the price index.  We can rearrange a version of this formula to find our ideal price index, P_I:
					I^new/I^old = P_I ^new						
That is, the ideal price index is just the ratio of the new income (that leaves the consumer at the original utility level after the inflation) to the old income level.  Write a formula to do this calculation a few rows below the U value (cell D26). Be sure to write an “Ideal price index” label for this calculation (in cell D25).  If you have done the calculation correctly, you should get an answer of 1.212. Note that this price index is significantly lower than the Laspeyres index computed earlier; in this example the Laspeyres price index overstates the true increase in the cost of living by 1.3 percentage points.
9.	To further study this substitution bias, redo the exercise for the price changes in the table below.  You will not have to redo every step of the process; just change the prices in the “New values”, “Cost of living raise”, and “Holding U constant” parts of the spreadsheet, and redo the Goal Seek procedure.

New p_x | New p_y | New relative price ratio (p_x/p_y) | Value of the Laspeyres price index | # of percentage points the price index overstates the cost of living increase
$4.20 | $1.40 | 3 | 1.225 | 1.3
$3.20 | $1.60 | 2 | ? | ?
$5.00 | $1.25 | 4 | ? | ?