# Assume that a person's utility over two goods is given by  U(x,y) = (x-5)^(1/3) x (y-10)^(2/3) The price of the good x is equal to px and the price of good y is py. The total income of the...

Assume that a person's utility over two goods is given by

U(x,y) = (x-5)^(1/3) x (y-10)^(2/3)

The price of the good x is equal to px and the price of good y is py. The total income of the individual is given by I.

a) Determine the demand functions for x and y.

b) Do x and y appear to be complementary or substitute goods? Explain.

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

a) You should first solve for the optimality condition, hence, you need to set equal the marginal rate of substitution and the negative of the price ratio, such that:

`-(p_x)/(p_y) = ((1/3(x-5)^(1/3 - 1))(y-10)^(2/3))/(2/3(y - 10)^(2/3-1)(x - 5)^(1/3))`

`-(p_x)/(p_y) = -(1/2)(y - 10)/(x - 5)`

`1 = (1/2)(y - 10)/(x - 5)(p_y)/(p_x) `

`2xp_x - 10p_x = yp_y - 10p_y`

`2x = y(p_y)/(p_x) - 10(p_y)/(p_x) + 10`

`x = (y/2)(p_y)/(p_x) - 5(p_y)/(p_x) + 5`

You need to use the budget equation for the demand function, such that:

`p_x*x + p_y*y = I`

You need to plug the equation `x = (y/2)(p_y)/(p_x) - 5(p_y)/(p_x) + 5` into budget equation, such that:

`p_x*[(y/2)(p_y)/(p_x) - 5(p_y)/(p_x) + 5] + p_y*y = I`

The demand functions for the products x and y are the followings, such that:

`x(p_x,p_y,I) = (y/2)*I/((y/2 - 4)p_x)`

`y(p_x,p_y,I) = I/((y/2 - 4)p_y)`

b) You need to determine if the goods are complementary or substitute, hence, you need to use the partial derivative of the demand functions, such that:

If `(del x)/(del p_y) >0` , then the product x is a substitute of product y, while if

`(del x)/(del p_y) <0` , then the product x is a compliment of product y.