Hicksian Decomposition and Measuring Changes in Consumer Welfare
Student Instruction Sheet

Introduction
In this exercise, we will seek to understand the concept of Hicks’ method of decomposing a demand curve’s slope into its component income and substitution effects.  The Hicksian substitution effect is the change in quantity demanded that occurs when the price of the good changes, but the consumer is given enough additional income so that her total utility remains the same.  The income effect is the change in the quantity demanded of a good due to the change in real income caused by the price change.  The income effect is measured as the difference between the quantity demanded after the price change and the quantity demanded after the price change when income is adjusted to return the consumer to the original level of utility.  This exercise uses Excel’s Goal Seek tool.  We will also use the Goal Seek tool to compute two measures of the degree to which a consumer’s well-being is affected by a price change.  These two measures are called the compensating variation (CV) and equivalent variation (EV).  

Consider an individual with a Cobb-Douglas utility function over the consumption of two goods.
					U = x^ay^b					
where U is the level of utility, x is the quantity of one good consumed, and y is the quantity of the other good consumed.  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   			
We will use these functions to study the concept of income and substitution effects as well as the two measures of welfare change.  Suppose initially a = 2, b = 1, p_x = $2.00, p_y = $1.00, and I = $300.

Instructions

	Hicksian decomposition	

1.	Set up a spreadsheet to contain the information above.  It is suggested that you 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 format italics or subscripts, just use regular formatting.)  Start on row 2, and use row 1 as a label row.  Label the values and formulas as “Original equilibrium” in cell B1.  When typing in values for a, b, p_x, p_y, and I, use the initial values given at the end of the Introduction.  For x^*, y^*, and U, type formulas based on the appropriate equations in the Introduction.  Remember to start your formulas with an equal sign, use parentheses 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^*= 100, y^*=100, and U=1,000,000.
2.	Suppose that the price of good x rises to $4.00.  Copy column B to column D, and change the price of good x in column D to $4.00.  Change the label on this new column to “New equilibrium”.  Note that the individual’s utility has fallen from the doubling of the price of good x.
3.	According to the Hicks method, the substitution effect of a price change is the change in the quantity demanded of a good, given that the consumer receives enough extra income so that her total utility remains the same.  Thus, to find the substitution effect, we must first find the amount of additional income that will make the consumer as well off after the price change as before the price change.  Copy the “New equilibrium” column D to column E.  Label this column “Adjusted equilibrium”.  In this new column, click on the utility value cell (which should read 250,000), 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 1,000,000, since you want utility to return to its original value.  Next to “By changing cell”, type in the address of the income cell (F6).  Alternatively, you can click on the income cell; if the cell is blocked by the Goal Seek window, click on the name of the 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=1,000,000.  If you have done this correctly, you should see I=$476.22, x^*=79.37, and y^*=158.74.  (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 near the top of the screen.)  These values correspond to the demand for x and y at an income level of $476.22, which leaves the consumer at a utility level of 1,000,000.
4.	We are now in a position to compute the Hicksian substitution and income effects for good x in reaction to the price change.  Label cell F11 “Substitution effect”, and enter a formula in the adjacent cell (G11) that makes the correct computation.  Your answer should be –20.63.  (Since the substitution effect is the change in demand when utility is held constant, the answer is the difference between the “adjusted equilibrium” and the “original equilibrium” quantity demanded of good x.  The formula should be F7-B7.)
5.	Label cell F12 “Income effect.”  Given that the income effect is the change in quantity demanded of a good due to the change in real income caused by the change in prices, but not the direct effect of the change in prices themselves, write a formula in cell G12 to compute the income effect.  Your answer should be –29.37.  (Since the only difference between the “new equilibrium” and “adjusted equilibrium” is that the adjusted equilibrium has restored the lost real income caused by the price change, the income effect is the difference in quantity demanded between the new equilibrium and adjusted equilibrium.  The formula should be D7-F7.)
6.	For reference, label cell F13 “Total change in x^*”, and compute in cell G13 the total change in the quantity demanded of good x.  This total change should be the sum of the income and substitution effects and should check with the initial difference in x^* values (D7-B7).

Compensating and Equivalent Variation

7.	A natural question to ask is:  “How do we measure the degree to which the consumer is hurt by a price change?”  We cannot use the utility measure because utility is an ordinal (ranking) measure.  To study this issue, we will use the same set-up as above, but different parameter values.  First, copy columns A through D (the “Original” and “New” equilibrium models) to a new worksheet.  Now, change the value of a to 1 in the original and new equilibria.  In the “New” equilibrium only, change p_x to $2.50.  If your sheet is set up correctly, you should find that x^*=75, y^*=150, and U=11,250 in the original equilibrium, and x^*=60, y^*=150, and U=9,000 in the new equilibrium.
8.	Copy the new equilibrium to column F, as in step 3, above.  Change the label on this block of cells to “Hold utility at original level”.  Also as in step 3, use Goal Seek to find the level of income that returns the consumer to her original level of utility (11,250).  The difference between this income ($335.41) and her original level of income ($300) is called the compensating variation (CV) in income.  Compute the CV in cell H6 (=F6-B6) and put the CV label in cell H5.  The CV in income is the amount of extra income (given the price increase) that is necessary to make the consumer indifferent to the price change.  This is one way to measure the degree to which the consumer has been hurt by the price increase.
9.	Copy the entirety of rows 1 through 9 to rows 13-21.  In column F, change the price of good x back to $2.00, so it is identical to the original equilibrium.  Change the label on this block of cells to “Reduce utility to new level”.  Using Goal Seek, find the level of income before the price change that brings the consumer down to her new level of utility (9,000) after the price change.  The difference between the original level of income ($300) and the newly computed level of income ($268.33) is called the equivalent variation (EV) in income.  Compute the EV in cell H18 (=B18-F18) and put the EV label in cell H17.  The EV in income is the amount of income a person would be willing to give up to avoid being subjected to the price increase.  This too makes intuitive sense as a measure of the degree to which the price increase hurts the consumer.  But, you may notice that it gives a different answer than CV!
10.	(Optional) Using calculus compute the reduction in consumer’s surplus (CS) associated with the increase in the price of x.  Note that the reduction in CS is $33.47 and so “splits the difference” between the CV and EV values.
11.	(Optional) Copy the CV and EV scenarios to a new worksheet.  On this new sheet, change the utility function to a “quasilinear” utility function U = a ln x + y, and the corresponding demands to:
					    and    
Recall that to take the natural logarithm of a number (or cell address) in Excel, use =LN( ), with the number or cell address in parentheses.  Also, for this example, change the value of a to 60.
12.	(Optional) Recompute the CV and EV, and compare these numbers to the reduction in utility and the reduction in CS associated with the price increase.  All four values should be the same.  This is a special feature of the demand functions derived from a quasilinear utility function.