Question

2. An individual consumes products X and Y and spends $25. The pries of the two goods are $3 per unit of X and $2 per unit of Y. The consumer in this case has a utility function expressed as:

U(X,Y)=0.5XY          MU_X=0.5Y     MU_Y=0.5X

1. Draw the indifference curve for this consumer at U=20.       (2 pts)
2. Does this consumer’s preference exhibit diminishing MRS?            (1 pt)
3. Express the budget equation mathematically.                        (2pts)
4. Determine the values of X and Y that will maximize utility in the consumption of X and Y. Assume X and Y can be consumed in fractions.            (2pts)
5. What is the utility level for this individual?   (2pts)
6. Suppose now the price of X is $2 per unit, determine the values of X and Y that will maximize utility in the consumption of X and Y. Is the utility level higher or lower for this consumer after the price change? What is the intuition behind this?  (1 pt)
7. Suppose the price of X remains as $3, but consumer’s income has increased to $50. What are the values of X and Y will maximize consumer’s utility? What is the consumer’s utility after the income change? What is your take on “higher income tends to be associated with more happiness” using consumer’s theory we’ve learned so far. (1 pt)
8. Derive the consumer’s demand for X and Y(using Px, Py, and I) to maximize his utility? (2 pts)
1. What fraction of incomes does this consumer spend on good X, good Y? Does this depend on the Px and Py? (2pts)

Answer 1

U = 0.5XY

(a)

When U = 20,

0.5XY = 20

XY = 40

Y = 40/X

Some bundles lying on this indifference curve will be: A (X = 1, Y = 40), B (X = 10, Y = 4) and C (X = 40, Y = 1). In following graph, IC0 is the indifference curve.

(b)

MRS = MUX/MUY = Y/X

When X increases, MRS decreases, so there is diminishing MRS.

(c)

Budget equation: Income = X.Px + Y.Py

25 = 3X + 2Y

(d)

Utility is maximized when MRS = Px/Py = 3/2

Y/X = 3/2

3X = 2Y

Substituting in budget equation,

25 = 3X + 3X = 6X

X = 4.17

Again,

25 = 2Y + 2Y = 4Y

Y = 6.25

NOTE: As per Answering Policy, 1st 4 parts are answered.