Question

An agent consumes quantity (x₁,x₂) of goods 1 and 2. Here is his utility function: 𝑈(𝑥1, 𝑥2) = 2 ∗ 1 ∗ 𝑥2 + 1, 𝑝1 = 𝑝2 = $1, ℎ𝑖𝑠 𝑖𝑛𝑐𝑜𝑚𝑒 𝑚 = 20.

 (a) Calculate the agent’s Marshallian demand (x^*₁ , x*₂ ).

(b) If the government put a $1 tax on x1, which increase p1 to $2, assume p2 and m do not change, what is the demand for x1?

(c) If the government collect tax on the agent’s income, the amount of tax is the same as the tax revenue collected in (b), what is the agent’s utility? Compare it with the agent’s utility in (b). 

Answer 1

Utility function is $x_1^3x_2$ with budget constraint $p_1x_1+p_2x_2=m$

At optimum,  $\frac{MU X_1}{ MU X_2} =\frac{p_1}{ p_2}$

a) Therefore,

$MUx_1=3x_1^2x_2\\MUx_2=x_1^3\\At\space optimum,\space \frac{3x_1^2x_2}{x_1^3}=\frac{p_1}{p_2}⇒\frac{3x^2}{x_1}=\frac{p_1}{p_2}\\⇒x_2=\frac{p_1}{3p_2}x1$

Putting value of x₂ in x₁, we get

$⇒p_1x_1+p_2(\frac{p_1}{3p_2})x1=m\\⇒p_1x_1+\frac{p_1}{3}x1=m\\⇒3p_1x_1+p_1x_1=3m\\⇒4p_1x_1=3m\\⇒x1*=\frac{3m}{4p_1}$

b) Similarly,

$x2^*=\frac{p_1}{3p_2}(\frac{3m}{4p_1})\\x2^*=\frac{m}{4p_2}$

Therefore, the agent's Marshallian demand $(x1^*,x2^*)= (\frac{3m}{4p_1},\frac{m}{4p_2})$

c) Hence, indirect utility function is derived by putting x1* and x2* in utility function is

$V(m,p1,p2)=(\frac{3m}{4p_1})^3(\frac{m}{4p_2})\\V(m,p1,p2)=\frac{3m^4}{16p_1p_2}$