Consider a consumer with the utility function given by U(X, Y) = XY where X and Y represent the two goods of consumption priced at Px and Py, respectively. The income of this consumer is assumed Rs 120, Px = Rs. 3 and Py = Rs.1. Suppose price of good X falls to Rs. 2.50, what will be its impact on consumption quantities of both the goods.

 To determine the impact of the price change on the consumption quantities of both goods, we need to analyze the substitution and income effects separately using the Hicksian approach.

First, we will calculate the initial consumption bundle using the given prices and income:

M = PxX + PyY = 3X + Y = 120

Solving for Y, we get:

Y = 120 - 3X

Using the utility function, we can express the consumer's indifference curve as:

U = XY = X(120 - 3X) = 120X - 3X^2

Taking the partial derivative of U with respect to X, we get:

MUx = 120 - 6X

Now, we can use the Hicksian approach to decompose the price change into substitution and income effects. Holding real income constant at Rs. 120, the consumer will adjust their consumption of goods X and Y in response to the change in the relative price of good X.

Substitution effect:

When the price of good X falls from Rs. 3 to Rs. 2.50, the relative price of good X decreases, making it relatively cheaper compared to good Y. This will induce the consumer to substitute towards good X from good Y. Holding real income constant, the consumer will maximize utility subject to the new relative prices.

To calculate the substitution effect, we need to calculate the new consumption bundle that maximizes utility subject to the new relative prices. The new relative price of good X is:

Px' = 2.50

We can write the new budget constraint as:

M = Px'X + Py*Y

Substituting the given values, we get:

120 = 2.50X + Y

Solving for Y, we get:

Y = 120 - 2.50X

Using the utility function and the new budget constraint, we can form the Lagrangian:

L = XY + λ(120 - 2.50X - Y)

Taking the partial derivatives of L with respect to X, Y, and λ and setting them equal to zero, we get:

MUx = λPx'

MUy = λPy

M = Px'X + Py*Y

Substituting the values, we get:

120 - 3X = λ(2.50)

X = 48 - 0.8333Y

120 = 2.50X + Y

Solving these equations simultaneously, we get:

X = 53.33

Y = 26.67

Therefore, the substitution effect leads to an increase in the quantity demanded of good X from 40 to 53.33 and a decrease in the quantity demanded of good Y from 26.67 to 13.33.

Income effect:

The decrease in the price of good X also leads to an increase in real income, since the consumer can purchase more of both goods with the same amount of money. Holding relative prices constant, the consumer will adjust their consumption of goods X and Y in response to the increase in real income.

To calculate the income effect, we need to compare the initial consumption bundle with the consumption bundle that the consumer would choose if the decrease in price of good X had not occurred, but the increase in real income had occurred instead. We can calculate the new budget constraint using the initial prices and the new real income of:

M' = 120/3 * 2.5 = Rs. 100

The new budget constraint is:

M' = PxX + PyY = 2.5X + Y

Substituting the value of Y from the initial budget constraint, we get:

M' = 2.5X + (120 - 3X)

Simplifying this expression, we get:

M' = 120 - 0.5X

Using the utility function and the new budget constraint, we can form the Lagrangian:

L = XY + λ(120 - 0.5X - Y)

Taking the partial derivatives of L with respect to X, Y, and λ and setting them equal to zero, we get:

MUx = λPx

MUy = λPy

M' = PxX + PyY

Substituting the values, we get:

120 - 3X = λ(2.5)

X = 40 - 0.8333Y

120 - 0.5X = 2.5X + Y

Solving these equations simultaneously, we get:

X = 50

Y = 20

Therefore, the income effect leads to an increase in the quantity demanded of good X from 40 to 50 and an increase in the quantity demanded of good Y from 26.67 to 30.

Overall effect:

The overall effect is the combination of the substitution effect and the income effect. The decrease in the price of good X leads to an increase in the quantity demanded of good X from 40 to 53.33 due to the substitution effect, and an increase in the quantity demanded of good X from 40 to 50 due to the income effect. Therefore, the overall effect is an increase in the quantity demanded of good X from 40 to 53.33. Similarly, the overall effect is a decrease in the quantity demanded of good Y from 26.67 to 13.33 due to the substitution effect, and an increase in the quantity demanded of good Y from 26.67 to 30 due to the income effect. Therefore, the overall effect is a decrease in the quantity demanded of good Y from 26.67 to 13.33.

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