Question

2) Imagine a person who has coronavirus and is using cough syrup (x) and Tylenol (y) to treat the symptoms as modeled by the following utility function:

U (x,y) = x0.7y0.3

f. What is the cross-price elasticity of demand for Tylenol? What does this mean economically? Are Tylenol and cough syrup complements, substitutes, or unrelated?

g. What is the direct effect between cough syrup and income? What does this mean economically?

h. What is the income elasticity of demand for cough syrup? What does this mean economically? Is cough syrup a normal or inferior good?

i. What is the direct effect between cough syrup and its own price? What does this mean economically?

j. What is the price elasticity of demand for cough syrup? What does this mean economically? What category is this?

Answer 1

The utility function inclusing cough syrup (x) and Tylenon (y) is given as:

U= x^0.7y^0.3

Let the budget constrait of the consumer be

xPx+ yPy= M

Px= per unit price of x, Py= per unit price of y, M= income of the consumer.

The condition of optimality:

MUx/MUy= Px/Py

From the utility function,

U= x^0.7y^0.3

MUx= 0.7(y/x)^0.3

MUy= 0.3(x/y)^0.7

The optimality consition becomes,

0.7(y/x)^0.3/0.3(x/y)^0.7 =Px/Py

7y/3x= Px/Py

y= 3xPx/ 7Py

Putting this value of y in the budget constraint,

xPx+ yPy= M

xPx+ [3xPx/ 7Py] Py =M

10xPx= 7M

x= 7M/10Px

This is the demand function for x,

Similarly, the demad function for y,

y= 3M/ 10Py

f) The cross price elasticity for x is the percentage change in the qunatity of y, for a unit percentage change in the price of x.

From the demand function of y,

y= 3M/ 10Py

It is visible, that the demand for y does not depend on the price of x, so the cross price elsticity o y would be 0.

g) From the demand functtion of x,

x= 7M/10Px

It is cleas that the quantity demanded of x, bears a direct relationship with the income of the consumer M.

This ,means that when the income of the consumer increases the demand of x would also increase.

h) The demand function of x,

x= 7M/10Px

The income elasticity of demand for x is the percentage change in the quantity demanded of x, for a unit percentage change in the pric eof x.

The income elasticity is given as:

Differetiating the demand function with respect to M,

Thus,

We know that,

x= 7M/10Px

Thus, M= 10xPx/ 7

Puttinf this value os M, the income elasticity of x becomes,

Thus the income elasticity of demand for x is 1. This implies that for a unit pecentage chage in the price of x, the change in the demand for x would be 1 percent.

i) The demand function of x,

x= 7M/10Px

From the equation it is clear that the quantity demand for x bears a inverse relationsip with its price. As the price of x increases its quantity demanded falls.