c.7.f. The Compensating and Equivalent Variations Applied to Demand Curves
Figure 7.f.1 shows a consumer with the utility function U=xy, with an income of $80, and initially facing prices of Px=$2, Py=$1. The consumer’s marginal rate of substitution is MRS=MUx/MUy. Marginal utility of x, MUx, is the partial derivative of U with respect to x, which is MUx=y. Similarly, MUy=x, so:
MRS=y/x.
The two conditions that must hold at the consumer’s optimum point A are:
MRS= Px/Py
Pxx+Pyy=ISubstituting the given values, the above two equations become
y/x=2
2x+y=80
which yields the optimal consumption of x and y as x=20, y=40. Note that this yields a utility of U=20x40=800.
If the price of x falls from $2 to $1, then the consumer’s optimum moves to point B, and the above two equations become:
y/x=1
x+y=80
which yields the optimal consumption of x and y as x=40, y=40 at the new optimum, point B.
To find the compensating variation, we imagine taking away enough money to return the consumer to his old indifference curve at point C. We find the x, y coordinates of point C by noting two conditions that must hold at point C:
U=800 (i.e., the consumer is on the old indifference curve, where utility=800.)
MRS=1 (i.e., the slope of the indifference curve at point C is 1.)
Substituting the known functions for U and MRS:
xy=800
y/x=1
which yields x2=800, or x=28.28, and y=28.28 at the point C. A line (with a slope of -1) that passes through point C would have a vertical intercept of 56.56 (=28.28+28.28). Thus the compensating variation is 80-56.56=23.44. This means that the consumer would be willing to pay $23.44 in order to have the price of x reduced from $2 to $1.
The compensating variation is also equal to the area to the left of the compensated demand curve, between the prices of $2 and $1, or areas FG in panel (b). To understand the compensated demand curve, start by looking at points A and B in panel (a). The movement from A to B shows the substitution effect of reducing the price of x from $2 to $1. In other words, points A and B show what would have happened to consumption of x if the price had fallen from $2 to $1, but the consumer had been held on his old indifference curve. In panel (b), points A’ and B’ are just a re-plotting of the information in points A and B. Panel (b) shows that as the price of x falls from $2 to $1, and the consumer is held on his old indifference curve (utility=800) consumption of x rises from 20 to 28.28. Thus the compensated demand curve passes through points A’ and B’.
The equation of the compensated demand curve is found by noting that two conditions must hold along the compensated demand curve:
(1) The consumer’s utility is held at 800
(2) the slope of the indifference curve (MRS) is equal to the slope of the budget constraint (Px/Py), or MRS=Px, since Py is assumed to be $1.
These two conditions become:
xy=800
y/x=Px
We want to use these two equations to express Px as a function of x, so use the second equation to get y= Pxx. Substituting this into the first equation yields xPxx=800, or
Px=800/x2This is the equation of the compensated demand curve.
If you were to integrate the area to the left of the compensated demand curve so as to calculate the dollar value of areas FG, you would get an area of $23.44, which is the compensating variation.
To find the equivalent variation, we imagine starting the consumer at point A, and giving him enough money to reach the new indifference curve at point E. We find the (x,y) coordinates of point E by noting two conditions that hold at point E:
1) U=1600
2) MRS= Px=2
Substituting the given values, these equations become:
xy=1600
y/x= 2
which gives the coordinates of point E as x=28.28, y=56.56. The dashed line passing through point E has a slope of 2, so its vertical intercept is 113.12 (=56.56+56.56). Thus the equivalent variation is 33.12 (=113.12-80). This means that the consumer would be indifferent between (a) a gift of $33.12, or (b) having the price of x fall from $2 to $1.
To find the equivalent demand curve we observe the two conditions that must hold as the consumer moves between points B and E:
1) U=1600
2) MRS= Px
Substituting the given values, these equations become:
xy=1600
y/x= Px
Solving so as to express Px as a function of x yields
xPxx=1600 or
Px=1600/x2
This is the equation of the equivalent demand curve. Note that the equivalent demand curve passes through points E’ and B’, since those points are just a re-plotting of points E and B in the upper panel. If we were to integrate the area to the left of the equivalent demand curve so as to calculate areas FGHJ, we would get a value for the equivalent variation of $33.12—the same value we got in panel (a).
The last demand curve we need to find is the ordinary demand curve, sometimes called the Marshallian demand curve, after Alfred Marshall, the father of supply/demand analysis. The ordinary demand curve is derived from points A and B in panel (a), and it passes through points A’ and B’ in panel (b). At point A, the price of x (i.e., the slope of the budget constraint) is $2, and the consumer buys 20 units of x. Point A’ in the lower diagram also shows the consumer buying 20 units when Px=$2. At point B, the price of x is $1 (i.e., the slope of the budget constraint is $1) and the consumer buys 40 units of x. Point B in the lower diagram shows the same thing: when Px=$1, the consumer buys 40 units.
To find the equation of the ordinary demand curve, we must satisfy two conditions:
Pxx+Pyy=I (This is the equation of the budget constraint)
MRS=Px/Py(The slope of the indifference curve equals the slope of the budget constraint.)
We are given that Py=1 and I=$80, but we must treat Px as a variable, since Px varies along the demand curve. Substituting our known values, these two equations become:
Pxx+y=80
y/x=Px
Solving to express Px as a function of x, this yields Pxx+Pxx=80, or
Px=40/xThis is the equation of the ordinary demand curve.
If we integrated the area to the left of this demand curve, so as to obtain the area FGH, we would get a value of $27.72, which is the change in consumer surplus that results when the price of x falls from $2 to $1. Note that $27.72 is in between the compensating variation ($23.44) and the equivalent variation ($33.12). Thus consumer surplus provides a compromise estimate of the effect of price changes on consumers’ wealth.
Study tip: The compensating variation, and the compensated demand curve, are found using the consumer’s OLD indifference curve. The equivalent variation, and the equivalent demand curve, are found using the consumer’s NEW demand curve. An easy way to remember which is which is to note that there is an “O” in “old” and an “O” in “compensated”.
Study questions:
1. Redraw figure 7.f.1 for the case where x is an inferior good, and show that in this case:
a. the compensated demand curve and the equivalent demand curve are both flatter than the ordinary demand curve.
b. the equivalent variation is smaller than the change in consumer surplus, while the compensating variation is larger.
2. Redraw figure 7.f.1 for the case where there is no income effect (i.e., the point B lies directly above the point C, and point E lies directly above point A). Show that in this case all three demand curves are identical, and that the compensating variation and equivalent variation are both equal to the change in consumer surplus.