Question

An agent consumes quantity (x₁,x2) of goods 1 and 2. Here is his utility function: 𝑈(𝑥1, 𝑥2) = 𝑥1³ 𝑥2, his budget constraint is: p1x1+p2x2=m.

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

 (b) Calculate the agent’s indirect utility function.

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

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

Similarly,

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

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

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

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