Suppose the demand for good X can be represented by the following equation: QX = 50 - 1.25P. Furthermore, suppose that the demand for good Y can be represented by QY = 20 - 0.5P. a. Find the elasticity of demand for both good X and good Y when the price of X is $10 and the price of Y is $15. b.If the community’s goal is to raise tax revenue as efficiently as possible, what should be the ratio of the tax on X to the tax on Y? (Hint: You can use the regular elasticity formula to calculate the elasticity. Elasticity at a given price can also be found using the formula: elasticity = - (1/s)(P/QX), where s is the slope of the inverse demand curve, QX is the quantity demanded, and P is the price. )
(a) The elsticity of X would be as
or
or
or
or
, and for the price $10 and quantity
, we have the elasticity as
or
.
The elsticity of X would be as
or
or
or
or
, and for the price $15 and quantity
, we have the elasticity as
or
.
(b) The after tax demand would be
and
, and the tax revenues would be
and
. The tax revenue maximization problem would be to maximize
with respect to tx and ty.
The FOCs would be as below.
or
or
or
.
or
or
or
.
Solving for the FOCs, we have
, which is the optimal ratio of tax on X and tax on Y that
maximizes the total revnue T.
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