If Jacob receives by consuming ice-cream cone (X) and hamburger (Y) at Burger Ki

Question

If Jacob receives by consuming ice-cream cone (X) and hamburger (Y) at Burger King is represented by the utility function U(X;Y) = XY. In addition, the per-unit prices of
ice-cream cone and hamburger are $1 and $3 respectively. Jacob has an income of $24 to spend on the two goods.

Derive Jabob’s budget constraint equation.
What is the slope of the budget line?
Derive the marginal utility of good X (MUX) using partial differentiation
Derive the marginal utility of good Y (MUY) using partial differentiation.
Using the equlibrium condition –MUX/MUY = -PX/PY to derive the quantity of each good
consumed at the utility-maximizing equilibrium.
What is the utility level?
If income now doubles, while there is no change to the per-unit price of the two goods,
derive the quantity of each good consumed as well as the utility level at the new
equilibrium. What type of goods are X and Y?

Explanation / Answer

$1*X + $3*Y = $24 is your budget constraint The slope is derived by subtracting by X and dividing by 3 to get 1/3. MUX = Y and MUY = X -Y/X = -1/3 ===> X = 3Y ===> plugging this in we get 3Y + 3Y = 24 ===> Y =4 ===> X = 24 - 3*4 = 12. At this point, Utility = 4 * 12 = 48 If income doubles to $48, then 3Y + 3Y = 48 ===> Y = 8 ===> X = 24 ===> essentially, Jacob just gets double the amount he got before, with a new Utility of = 8 * 24 =192. The goods are complementary.

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