Question

Judy's preferences over x and y can be represented by the utility function: U=xy³

The price of x is P_x = 8, the price of y is P_y = 4, and her income is I = 480.

At Judy's optimal consumption bundle, What is her demand for x? What is her demand for y?

Answer 1

Given:

Utility function: U=xy³

x and y represent the good x and good y.

Price of good x: P_x = 8

Price of good y: P_y = 4

Income (I) = 480

The budget constraint is:

Where X and Y are the quantities of good x and good y, I is the income.

Therefore, putting the values of prices and income, we get:

Therefore, the consumption maximization problem is that:

Judy maximizes U=xy³

such that, it satisfies the budget constraint.

The optimal bundle for Judy is determined where the marginal rate of substitution is equal to the ratio of prices.

That is the ratio of marginal utilities is equal to the ratio of prices of both the goods.

Therefore finding the marginal utilities,

The marginal utility of good x is the partial derivative of utility with respect to x.

Now the marginal utility of good y:

Therefore,

The marginal rate of substitution equals,

Putting the values of marginal utility,

Setting it equal to the ratio of prices, that is optimum bundle is where,

Therefore,

(equation 1)

Therefore putting this value of y into the budget constraints, we get the quantities of x and y.

Now putting this value of x into the equation 1.

i.e.

Therefore, the optimal bundles for Judy are 15 units of x and 90 units of y. Judy will demand the quantity that gives her maximizes her utility and also satisfies her budget. Therefore,

Therefore Judy's demand for good x = 15

Her demand for good y = 90