Question

Question 1 (1 point)

Suppose the consumer's utility function is given by

U(x1,x2)=xa1xb2{"version":"1.1","math":"U(x_1, x_2)=x_1^ax_2^b"}

, where a=9.5, and b=8.5. Suppose further, the price of good 1 is 3.0, price of good 2 is 2.7 and income is 69

Finally suppose the price of good 1 has decreased to 0.3

What is the substitution effect for good 1 Round your answer to two decimal places

Your Answer:

Answer 1

The substitution effect for good is calculated as the change in the consumption of that good when the price of that has changed keeping the money income constant. Let's do it

The utility function is given to us as,

U(X1,X2) = (X1)^a(X2)^b

Here a = 9.5 and b = 8.5

And price of X1, P1= 3

And price of X2, P2= 2.7

And consumers income is equal to 69, Y=69

The utility function to us is cobb douglas utility function and we directly calculate the amount of good 1 and good 2 that the consumer will consume,

X1* = a/a+b × Y/P1

And X2* = b/a+b × Y/P2

Putting the values in we get,

X1* = 9.5/9.5+8.5 × 69/3

X1* = 9.5/18 × 23

X1* = 218.5/18

X1* = 12.13

So the optimal amount of good 1,that is X1* that the consumer will consume is 12.13.

Ans note we just need to calculate the substitution effect for good 1, so we don't need to calculate the amount of good 2 that the consumer is consuming.

Now when price of good 1, P1'= 0.3

Notice that consumer before the price change the consumer was consuming 12.13 units of good 1. After the price change the consumer is saving money on the consumption of good 1, since the price of good 1 has decreased. Or in other words we can say that the real income of the consumer has increased. Increase in the income can be calculated as,

Y = X1*P1

Y = 12.13 × (3 - 0.3)

Y = 12.13 × (2.7)

Y = 32.75

So the real income of the consumer has gone up by 32.75 but to calculate the pure substitution effect we need to keep the income of the consumer constant. So we must be deduce 32.75 from 69 to keep the consumer's income constant. New income will be,

Y' = 69 - 32.75

Y' = 36.25

Let's calculate the amount of good 1 that the consumer will consume keeping his income constant at 36.25.Again using the formula,

X1' = a/a+b × Y'/P1'

Putting in the values we get,

X1' = 9.5/9.5+8.5 × 36.25/0.3

X1' = 9.5/18 × 120.83

X1' = 1,147.91/18

X1' = 63.77

So the consumer was initially consuming 12.13 units of good 1 but after the price change of good 1 the consumer is consuming 63.77 units keeping his income constant.

Substitution effect = X1' - X1*

Substitution effect = 63.77 - 12.13 = 51.64

So the substitution effect for good 1 is 51.64.