Question

Suppose the consumer's utility function is given by U(x1, x2) = xqxh , where a=7.0, and b=5.0. Suppose further, the price of good 1 is 1.2, price of good 2 is 0.5 and income is 53 Finally suppose the price of good 1 has decreased to 0.9 What is the substitution effect for good 1 Round your answer to two decimal places

Answer 1

Answer:

Given, $U(x_1,x_2)=x_1^{a}x_2^{b}=x_1^{7}x_2^{5}$

And price of good-1, P_x1=1.2 and price of good-2, P_x2=0.5

Budget constraint of the consumer, $1.2x_1+0.5x_2=53$

Now price of good-1 has decreased to 0.9, thus $0.9x_1+0.5x_2=53$

We know that,

PI MUT MRS Pr MU

Thus, $\frac{MU_1}{MU_2}=\frac{Px_1}{Px_2}=>\frac{7x_1^{6}x_2{5}}{5x_1^{7}x_2{4}}=\frac{1.2}{0.5}=>\frac{7x_2}{5x_1}=2.4=>x_2=\frac{12x_1}{7}$

Plugging this value in the budget constraint we get, $1.2x_1+0.5*\frac{12x_1}{7}=53=>x_1=\frac{53}{2.0571}=25.7644$

Thus, $x_2=\frac{12*25.7644}{7}=44.1676$

But when the price decrease to 0.9 we get, $\frac{MU_1}{MU_2}=\frac{Px_1}{Px_2}=>\frac{7x_1^{6}x_2{5}}{5x_1^{7}x_2{4}}=\frac{0.9}{0.5}=>\frac{7x_2}{5x_1}=1.8=>x_2=\frac{9x_1}{7}$

Plugging this value in the budget constraint we get, $0.9x_1+0.5*\frac{9x_1}{7}=53=>x_1=\frac{53}{1.5429}=34.3509$

Thus, $x_2=\frac{9*34.3509}{7}=44.1654$

Now the real change in income=25.7644*(0.9-1.2)=-7.7293

Thus the real income=53-7.7293=45.2707=45 (rounded)

Thus the new real income in the budget constraint we get, $0.9x_1+0.5*\frac{9x_1}{7}=45=>x_1=\frac{45}{1.5429}=29.1659$

Thus, the substitution effect=New consumption-old consumption=29.1659-25.7644=3.4015=3.40 units

Therefore when the price of good-1 decreases to 0.9 the consumption of x₁ increases to 3.4 units.