Suppose you have the following preferences u(x,y) = xa - yb. Calculate the optimal demand functions....

Suppose you have the following preferences u(x,y) = x^a - y^b. Calculate the optimal demand functions. Is good y a normal or inferior good? Please show work.

Solutions

Expert Solution

We have the following information

U(X,Y) = aX – bY

The marginal rate of substitution between X and Y is –(a/b), which is a constant, independent of the quantities consumed of the goods. The indifference curves between the two goods are straight lines.

Marginal utility of X = ∂U/∂X = a

Marginal utility of Y = ∂U/∂Y = – b

Using lagrangian multiplier

µ = aX – bY + λ(M – P_XX – P_YY)

P_X = Price of X

P_Y = Price of Y

M = Money Income

Mathematically, the first-order conditions

∂µ/∂X = a – λP_X

a = λP_X

∂µ/∂Y = – b – λP_Y

– b = λP_Y

could both hold only if –(a/b) = P_X/P_Y, which would happen by coincidence. Usually, the consumer will choose to be at a corner solution, spending all her money on the good for which a_i/P_i is highest (a_i is coefficient of the ith good in utility function and Pi is the price of ith good).

Demand for Good X

X = M/P_X

Demand for Good Y

Y = – (M/P_Y)

The Good Y is an inferior good. The negative sign in the demand equation for Good Y shows that as income increases the demand for Good Y declines.

Rahul Sunny answered 1 year ago

Next > < Previous

Related Solutions

For a demand function u (x, y) = xy, show the demand functions for good x...

For a demand function u (x, y) = xy, show the demand functions for good x and good y. (Remember that MRS = (du/dx) / (du/dy) = px / py in the point of interest, the tangency point of budget line and indifference curve. The budget condition is given by pxx + pyy = m) for u (x, y) = x1/3 y2/3

Suppose a consumer's preferences are given by U(X,Y) = X2*Y. Thus, the marginal utility of X,...

Suppose a consumer's preferences are given by U(X,Y) = X2*Y. Thus, the marginal utility of X, MUx = 2XY and the marginal utility of Y, MUy = X2. Suppose the consumer has $180 to spend and the price of good Y is $1. Sketch the price-consumption curve for the prices of Px = $1 Px = $2 Px = $3 To do this, carefully draw the budget constraints associated with each of the prices for good X, and indicate the...

"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...

Write the demand functions for the following utility function: U = ln(x) + ln(y)

Consider the following utility functions: (i) u(x,y) = x2y (ii) u(x,y) = max{x,y} (iii) u(x,y) =...

Consider the following utility functions: (i) u(x,y) = x2y (ii) u(x,y) = max{x,y} (iii) u(x,y) = √x + y (a) For each example, with prices px = 2 and py = 4 find the expenditure minimising bundle to achieve utility level of 10. (b) Verify, in each case, that if you use the expenditure minimizing amount as income and face the same prices, then the expenditure minimizing bundle will maximize your utility.

Suppose Mike’s utility function is u(x,y)=2lnx +lny. 1. Derive the demand functions. 2. Is y a...

Suppose Mike’s utility function is u(x,y)=2lnx +lny. 1. Derive the demand functions. 2. Is y a Normal good? 3. Is x an ordinary good? 4. Assume the price of x is initially 1 dollar, and the price of y is also 1 dollar. Given that income is 9, if the price of x doubles to 2 dollars, decompose the change in consumption of x into substitution effect and income effect. Illustrate your answer with a graph. 5. Now, given the...

1. Suppose Mike’s utility function is u(x,y)=2lnx +lny. a. Derive the demand functions. b. Is y...

1. Suppose Mike’s utility function is u(x,y)=2lnx +lny. a. Derive the demand functions. b. Is y a Normal good? c. Is x an ordinary good? d. Assume the price of x is initially 1 dollar, and the price of y is also 1 dollar. Given that income is 9, if the price of x doubles to 2 dollars, decompose the change in consumption of x into substitution effect and income effect. Illustrate your answer with a graph. e. Now, given...

1A) For a demand function u (x, y) = xy, show the demand functions for good...

1A) For a demand function u (x, y) = xy, show the demand functions for good x and good y. (Remember that MRS = (du/dx) / (du/dy) = px / py in the point of interest, the tangency point of budget line and indifference curve. The budget condition is given by pxx + pyy = m). 1B) Calculate the own-price elasticities as well as the income elasticities of demand for goods x and y based on your results in A)...

Suppose a consumer has preferences given by U(X,Y) = MIN[2X,Y]. Suppose PX = 1 and PY...

Suppose a consumer has preferences given by U(X,Y) = MIN[2X,Y]. Suppose PX = 1 and PY = 2. Draw the Income Consumption Curve for this consumer for income values • M = 100 • M = 200 • M = 300 To do this, carefully draw the budget constraints associated with each of the prices for good X, and indicate the bundle that the consumer chooses in each case. Also, be sure to label your graph accurately.

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...

Subjects