# 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.

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### 1 Answer

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.