Question

In: Economics

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

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?

Expert Solution

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

Rahul Sunny answered 2 years ago

Ella’s preferences can be represented by the utility function u(x,y) = min{5x, y}. If the price...

Ella’s preferences can be represented by the utility function u(x,y) = min{5x, y}. If the price of x is $10 and the price of y is $15, how much money would she need to be able to purchase a bundle that she likes as well as the bundle (10, 25). Select one: a. $475. b. $85. c. $425. d. $209. e. $440.

Dr Pepper’s preferences can be represented by the utility function u(x,y) = x + y where...

Dr Pepper’s preferences can be represented by the utility function u(x,y) = x + y where x is his consumption of Coca Cola (hereafter, referred to as Coke) and y is his consumption of orange juice (hereafter, referred to as OJ). Initially, both types of drinks are not taxed and with an income of $12 he faces prices ($1, $2). On the advice of nutritionists, the government decides to impose a specific tax of $2 on Coke which leads to...

A consumer's preferences for food (x) and clothes (y) are represented by the utility function u(x,y)=...

A consumer's preferences for food (x) and clothes (y) are represented by the utility function u(x,y)= x + 2 ln(y). The prices of food and clothes are px and py euros/unit, respectively, and the consumer's income is I euros. A. (10 points) Describe the consumer's problem, including her budget constraints, and calculate her ordinary demand functions, x(px,py,I) and y(px,py,I). B. (10 points) For prices and income (px,py,I)=(1,1,3), calculate the substitution and income effects over the demand of y of a...

Brianna’s preferences can be represented by the utility function u(x,y) = min{x,y}. Initially she faces prices...

Brianna’s preferences can be represented by the utility function u(x,y) = min{x,y}. Initially she faces prices ($2,$1) and her income is $12. If prices change to ($3,$1) then the compensating variation Select one: a. There is not enough information to determine which variation is greater. b. is $2 more than the equivalent variation. c. is $1 less than the equivalent variation. d. equals the equivalent variation. e. is $1 more than the equivalent variation.

"Suppose a consumer has preferences represented by the utility function U(X,Y) = X(^2)Y Suppose Py =...

"Suppose a consumer has preferences represented by the utility function U(X,Y) = X(^2)Y Suppose Py = 1, and the consumer has $360 to spend. Draw the Price-Consumption Curve for this consumer for income values Px =1, Px = 2, and Px = 5. Your graph should accurately draw the budget constraints for each income level and specifically label the bundles that the consumer chooses for each income level. Also for each bundle that the consumer chooses, draw the indifference curve...

A consumer has his preferences represented by the utility function U(x,y) = min {5x + 4y,...

A consumer has his preferences represented by the utility function U(x,y) = min {5x + 4y, 4x + 7y} if x is on the horizontal axis and y is on the vertical axis, what is the slope of his indifference curve at the point (10,10) a. -4/7 b. -5/4 c. -4/5 d. -7/4 e. -5/7

Suppose a consumer has preferences represented by the utility function U(X,Y) = X2Y Suppose PY =...

Suppose a consumer has preferences represented by the utility function U(X,Y) = X2Y Suppose PY = 1, and the consumer has $300 to spend. Draw the Price-Consumption Curve for this consumer for income values PX = 1, PX = 2, and PX = 5. Your graph should accurately draw the budget constraints for each income level and specifically label the bundles that the consumer chooses for each income level. Also, for each bundle that the consumer chooses, draw the indifference...

Consider a consumer with preferences represented by the utility function: u(x, y) = x1/4y1/2 Suppose the...

Consider a consumer with preferences represented by the utility function: u(x, y) = x1/4y1/2 Suppose the consumer has income M = 10 and the prices are px=1 and Py = 2. (a) Are goods x and y both desirable? (b) Are there implications for the utility maximization problem for the consumer from your finding in 1a? If so, explain in detail. (c) Derive the utility maximizing bundle.

Consider a consumer with preferences represented by the utility function: u(x; y) = x1/4y1/2 Suppose the...

Consider a consumer with preferences represented by the utility function: u(x; y) = x1/4y1/2 Suppose the consumer has income M = 10 and the prices are px = 1 and py = 2. (a) Are goods x and y both desirable? (b) Are there implications for the utility maximization problem for the consumer from your finding in a? If so, explain in detail.

Your utility function over x and y is U ( x , y ) = l...

Your utility function over x and y is U ( x , y ) = l n ( x ) + 0.25 y. Your income is $20. You don’t know the prices of x or y so leave them as variables (p x and p y). a) (8 points) Find x*, your demand function for x. Find y*, your demand function for y. b) (10 points) Find the cross-price elasticity of demand for x (E x ∗ , p y:...