Question

In: Economics

In a research paper an economist assumes that the typical consumer has a utility U(X, Y)...

In a research paper an economist assumes that the typical consumer has a utility U(X, Y) = X0.25Y0.75 and a budget of $1,000.

The price of X is $10 while the price of Y is $25. Suppose the price of X drops to $5 while the price of Y does not change.

e) In your diagram, illustrate how the price drop affects the consumer’s budget constraint.

f) Find the consumer’s new optimal bundle.

g) Adding an indifference curve, illustrate the consumers’ new optimal bundle in your diagram.

h) Does the consumer consider X and Y gross complements, gross substitutes or unrelated? Justify your answer.

i) Suggest a couple of goods

Solutions

Expert Solution

U = X^0.25Y^0.75

(e)

Initial Budget line: 1,000 = 10X + 25Y, or 200 = 2X + 5Y [Dividing by 5]

When X = 0, Y = 200/5 = 40 (Vertical intercept) & when Y = 0, X = 200/2 = 100 (Horizontal intercept)

After Px = $5, New budget line: 1,000 = 5X + 25Y, or 200 = X + 5Y [Dividing by 5]

When X = 0, Y = 200/5 = 40 (Vertical intercept) & when Y = 0, X = 200 (Horizontal intercept)

In following graph, AB and AC are initial and new budget lines with intercepts derived above.

(f) Utility is maximized when MUx / MUy = Px / Py = 5/25 = 1/5

MUx = U / X = 0.25 x (Y / X)^0.75

MUy = U / Y = 0.75 x (X / Y)^0.25

MUx / MUy = (1/3) x (Y / X) = Y / 3X = 1/5

3X = 5Y

Substituting in new budget line,

200 = X + 3X = 4X

X = 50

Y = 3X / 5 = (3 x 50) / 5 = 30

(g) In above graph, indifference curve I1 is angent to AC at point V with optimal bundle being (X1, Y1) = (50, 30).

(h) To answer this we need to find the original (X, Y) bundle.

MUx / MUy = Y / 3X = 10/25 = 2/5

6X = 5Y

Substituting in original budget line,

200 = 2X + 6X = 8X

X = 25

Y = 6X / 5 = (6 x 25) / 5 = 30

Since a fall in price of X does not change the demand for Y (which remains unchanged at 30), X and Y are unrelated.

(i) An example of such unrelated goods will be: Books and Shoes, or Laptop and Oranges.

Rahul Sunny answered 1 year ago

Next > < Previous

Related Solutions

Consider a consumer with the utility function U(X, Y) = X^2 Y^2 . This consumer has...

Consider a consumer with the utility function U(X, Y) = X^2 Y^2 . This consumer has an income denoted by I which is devoted to goods X and Y. The prices of goods X and Y are denoted PX and PY. a. Find the consumer’s marginal utility of X (MUX) and marginal utility of Y (MUY). b. Find the consumer’s marginal rate of substitution (MRS). c. Derive the consumer's demand equations for both goods as functions of the variables PX,...

Suppose a consumer has a utility function given by u(x, y) = x + y, so...

Suppose a consumer has a utility function given by u(x, y) = x + y, so that the two goods are perfect substitutes. Use the Lagrangian method to fully characterize the solution to max(x,y) u(x, y) s.t. x + py ≤ m, x ≥ 0, y ≥ 0, where m > 0 and p < 1. Evaluate and interpret each of the multipliers in this case. What happens to your solution when p > 1? What about when p =...

Suppose a consumer has a utility function u(x, y) = 2x + 3y. The consumer has...

Suppose a consumer has a utility function u(x, y) = 2x + 3y. The consumer has an income $40 and the price of x is $1 and the price of y is $2. Which bundle will the consumer choose to consume? Determine the demand functions for x and for y. Repeat the exercise if, instead, the consumer’s utility function is u(x, y) = min{x, 2y}.

Suppose that the utility function of a consumer is U(x,y) = x ¼y ¾, where x...

Suppose that the utility function of a consumer is U(x,y) = x ¼y ¾, where x and y are the quantities of the good X and good Y consumed, respectively. The consumer's income is 400. (a) What is the demanded bundle when the price of good X is 10 and the price of good Y is 10? (b) Redo part (a) when the price of good X is doubled? (c) Redo part (a) when the price of good Y is...

A consumer purchases two goods, x and y and has utility function U(x; y) = ln(x)...

A consumer purchases two goods, x and y and has utility function U(x; y) = ln(x) + 3y. For this utility function MUx =1/x and MUy = 3. The price of x is px = 4 and the price of y is py = 2. The consumer has M units of income to spend on the two goods and wishes to maximize utility, given the budget. Draw the budget line for this consumer when M=50 and the budget line when...

Suppose that a consumer has utility given by U(X, Y ) = XY + 10Y and...

Suppose that a consumer has utility given by U(X, Y ) = XY + 10Y and income of $100 to spend on goods X and Y. The prices of X and Y are both $1 per unit. Use a Lagrangian to solve for the optimal basket of goods. Suppose that the price of X increases to $5 per unit. Use a Lagrangian to solve for the new optimal basket of goods. Find the total effect of the price change on...

Suppose a consumer’s utility function is given by U(X,Y) = X*Y. Also, the consumer has $200...

Suppose a consumer’s utility function is given by U(X,Y) = X*Y. Also, the consumer has $200 to spend, and the price of X, PX = 5, and the price of Y, PY = 2. a) How much X and Y should the consumer purchase in order to maximize her utility? b) How much total utility does the consumer receive? c) Now suppose PX decreases to 1.25. What is the new bundle of X and Y that the consumer will demand?...

Let assume that a consumer has a utility function u(x, y) = xy, and px =...

Let assume that a consumer has a utility function u(x, y) = xy, and px = 1 dollar, py = 2 dollars and budget=50. Derive the followings. (3 points each) 1) Marshallian demands of x and y 2) Hicksian demands of x and y 3) Indirect utility function 4) Expenditure function 5) Engel curve

"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...

For one purpose, the utility function of the consumer is u(x,y)=4?x+2y for maximum utility. 1. For...

For one purpose, the utility function of the consumer is u(x,y)=4?x+2y for maximum utility. 1. For one purpose, the customer's income is I, and the price of X is Px and the price of Y is Py. Obtain the demand function of this person's Y ash through Px, Py, I. 2. For one purpose, the consumer has an income I = 40, and initially the price of Px=1, and the price is Py=1. (1) What is the difference between the...

Subjects