Choosing Taxes with a Constant Elasticity of Substitution (CES) Utility Function
Move the slider labeled "tax rate" to the right to see the effect of a per unit tax on good . The per unit tax makes the budget line steeper and is equivalent to an increase in the price of . Compare the per unit tax to a lump sum tax by dragging the slider labeled "income tax" until the budget line passes through the new equilibrium point. Due to the convexity of the indifference curves (i.e., diminishing marginal rate of substitution), the lump sum tax is superior to the per unit tax, in the sense that the lump sum tax can raise the same amount of tax revenue as the per unit tax, but will leave the consumer better off (i.e., on a higher indifference curve).
Snapshot 1: a lump sum tax raises the same revenue as the per unit tax but leaves the consumer better off
Snapshot 2: a lump sum tax leaves the consumer just as well off as a per unit tax but generates more revenue
Snapshot 3: a lump sum subsidy makes the consumer better off than a per unit subsidy of the same cost
This Demonstration shows the indifference curve budget line for a consumer in a two-good world, with preferences represented by the constant elasticity of substitution (CES) utility function: ; , . You can vary the parameters and with the two sliders at the bottom of the controls.
A positive tax rate of makes the budget line steeper, and the consumer moves to a new equilibrium on a lower indifference curve. The per unit tax generates revenue of , where is the equilibrium choice of good one after the tax has been imposed. Moving the slider labeled "income tax" to the left shows the effect of a lump sum tax that does not change relative prices. Find the amount of a lump sum tax that raises the same revenue as the per unit tax by moving the slider to the left until this parallel budget line passes through . Due to the convexity of the indifference curves (i.e., diminishing marginal rate of substitution), the consumer will prefer the lump sum tax to a per unit tax that raises the same revenue. See , pp. 91–92 for additional details. Note that by slightly increasing the lump sum tax until the budget line is tangent to the indifference curve passing through , there is an alternative statement expressing the superiority of lump sum taxes to per unit taxes: a lump sum tax can raise more revenue than a per unit tax that leaves the consumer just as well off as she would be with the per unit tax. This is an example of Hicksian income compensation (, p. 155).
Additional uses of this Demonstration:
1. Partitioning the effects of price changes into substitution and income effects, using either Hicksian or Slutsky income compensation (, Chapter 8).
2. Comparing per unit and lump sum subsidies (i.e., by using a negative tax rate).
3. If , the change in the intercept can be used to find the compensating and equivalent variations of price changes (, Chapter 14).
4. A simple modification of the code could be used to make analogous comparisons of lump sum and ad valorem taxes or subsidies.
 H. R. Varian, Intermediate Microeconomics with Calculus: A Modern Approach, New York: W. W. Norton & Company, 2014.