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

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

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