Question

In: Economics

Randy consumes two goods: X and Y. Randy’s preferences over consumption bundles (X,Y) are summarized by the...

Randy consumes two goods: X and Y. Randy’s preferences over consumption bundles (X,Y) are summarized by the utility function:

u (X,Y) = X³Y.

Write algebraic expressions for Randy’s demand functions for goods X and Y to be

If P_X= 1, P_Y= 1, and m = 100, what would be Randy’s optimal consumption of goods X and Y?

Suppose now that the price of X rises to 3, while the price of Y and income remain unchanged. What is Randy’s new optimal consumption bundle?

Calculate the equivalent variation.

Expert Solution

1. The utility is given to be , and the budget line would be . The MUs would be or or and or or . The optimal combination of goods would be where or or . Putting it in the constraint, we have or or , and since , we have . These are the demand for X and Y.

2. For the given prices and income, Randy's optimal consumption of goods would be or and or .

3. For the new price of X, Randy's optimal consumption of Y would not change and remain 25, but optimal consumption of X would be or .

4. Randy's old utility level was and new utility level is . For the Marshallian demand of X be and Y be c, the indirect utility level would be as or or .

Now, for the indirect utility function be , the inverse of this function would be the expenditure function. Hence, from the indirect utility function, we have or or or . This is the expenditure function. This is the expenditure required for U* utils with prices Px and Py.

This means that for the old prices and new utility, the income required is or or or . Hence, the equivalent variation would be 100-43.86 or $56.14. This means that to reach the new utility a old price, the income must be reduced by $56.14. The graph is as below.

Rahul Sunny answered 1 year ago

1. Consider the preferences of an individual over two goods, x and y, with prices px...

1. Consider the preferences of an individual over two goods, x and y, with prices px and py and income I. (a) If the individual's preferences can be represented by the utility function u(x,y) =2x1/2 + y, what is the marginal rate of substitution? What does this MRS imply about how this consumer would trade y for x? Are the underlying preferences homothetic (explain)? Graphically illustrate a typical indifference curve and explain how you know the shape. (b) If the...

Consider a consumer with preferences over two goods (good x and good y) given by u...

Consider a consumer with preferences over two goods (good x and good y) given by u ( x , y ) = x ⋅ y. Given income of I and price of good yas $ P yper pound and price of good xgiven as $ P xper pound, the consumer chooses the optimal consumption bundle given as x ∗ = I 2 P x and y ∗ = I 2 P y . Given P x = $ 1per pound...

A consumes two goods, x and y. A ’s utility function is given by u(x, y)...

A consumes two goods, x and y. A ’s utility function is given by u(x, y) = x 1/2y 1/2 The price of x is p and the price of y is 1. A has an income of M. (a) Derive A ’s demand functions for x and y. (b) Suppose M = 72 and p falls from 9 to 4. Calculate the income and substitution effects of the price change. (c) Calculate the compensating variation of the price change....

For two goods x and y, the individual’s preferences are measured by the utility function ?(?,?)=...

For two goods x and y, the individual’s preferences are measured by the utility function ?(?,?)= ?0.5?0.5, the price of good y is $10, income equals $100, and the price of good x increase from $5 to $10. Draw the price consumption curve for x and y and compute the slop of the price consumption curve. Show your calculation step by step.

Consumer Theory. A consumer has preferences over goods x and y that can be represented by...

Consumer Theory. A consumer has preferences over goods x and y that can be represented by the utility function ?(?,?) = ?+??(?) where ln is the (natural) log function. The consumer has income I (all to be spent on x and y) and the price of x and y are px and py respectively. (You may assume the “at least as good as x” set B(x) is a convex set, so the solution to the consumer’s problem will be a...

13. Pip consumes two goods, x and y. Pip’s utility function is given by u(x, y)...

13. Pip consumes two goods, x and y. Pip’s utility function is given by u(x, y) = x 1/2y 1/2 The price of x is p and the price of y is 1. Pip has an income of M. (a) Derive Pip’s demand functions for x and y. [5 marks] (b) Suppose M = 72 and p falls from 9 to 4. Calculate the income and substitution effects of the price change. [5 marks] (c) Calculate the compensating variation of...

Leon consumes two goods, X and Y. His utility function is given by U(X,Y) = 56XY....

Leon consumes two goods, X and Y. His utility function is given by U(X,Y) = 56XY. The price of X is $14 a unit; the price of Y is $8 a unit; and Leon has $672 to spend on X and Y. a. Provide the equation for Leon’s budget line. (Your answer for the budget line should be in the form Y = a – bX, with specific numerical values given for a and b.) b. Provide the numerical value...

Ron consumes two goods, X and Y. His utility function is given by U(X,Y) = 33XY....

Ron consumes two goods, X and Y. His utility function is given by U(X,Y) = 33XY. The price of X is $11 a unit; the price of Y is $4 a unit; and Ron has $352 to spend on X and Y. a. Provide the equation for Ron’s budget line. (Your answer for the budget line should be in the form Y = a – bX, with specific numerical values given for a and b.) b. Provide the numerical value...

Consider a student who purchases education (x) and other goods (y). The student has preferences over...

Consider a student who purchases education (x) and other goods (y). The student has preferences over these goods given by u(x, y) = ln(x) + 3ln(y). The prices of education and other goods are, respectively, px = 10 and py = 5, and the student’s income is I = 20. 1. Using the first-order conditions you derived, find the student’s optimal consumption of education x ∗ and other goods y ∗ and derive an algebraic expression for the student’s income...

An Individual consumes commodity bundles including two goods: P and Q. The satisfaction/value/utility obtained for the...

An Individual consumes commodity bundles including two goods: P and Q. The satisfaction/value/utility obtained for the individual is provided by the following equation – 3*(units of P)^2 + 2*(units of Q). Now answer the following questions a. Compute the utility got the following bundles – (12, 5), (2,7) and (8,5) b. Considering a utility level of 75 units; identify one consumption bundle that provided that level of utility c. Now draw an indifference curve for the individual with an utility...