Question

An agent has utility u(x1, x2) = (x −1 1 + x −1 2 ) −1 for goods x1 and x2. The prices of the goods are p1 and p2. The agent has income m. a) Show preferences are convex. You can do this graphically or by showing that MRS is decreasing in x1. b) Solve for the agent’s optimal choice of (x1, x2). c) Show the agent’s indirect utility function is given by: V = ( m (p 1/2 1 +p 1/2 2 ) 2 

Answer 1

Given:

An agent has a utility of:

$U(x_1,x_2)=(x_1^{-1}+x_2^{-1})^{-1}$

The prices of goods are p₁ and p₂

agent income = m

Agent's indirect utility function:

$V=(\frac{m}{p_1^\frac{1}{2}+p_2^\frac{1}{2}})^2$

To find:

a)

$MRS=\frac{d_x/d_{x1}}{d_x/d_{x2}}=x_1-2(x_1-1+x_2-1)-2d_ud_{x2}$

$=-(x_1-1+x_2-1)-2\times(-1)(x_2)-2$

$\frac{d_x}{d_{x1}}=-(x_1^{-1}+x_2^{-1})^{-2}\times(-1)(x_1)^{-1}\frac{d_x}{d_u}$

$d_ud_{x2}=(x_2)-2(x_1-1+x_2-1)-2$

Now MRS will be:

$MRS=\frac{x_1^{-2}(x_1^{-1}+x_2^{-1})^{-2}}{x_2^{-2}(x_1^{-1}+x_2^{-1})^{-2}}$

$MRS=(\frac{x_2}{x_1})^2$

$\frac{\delta MRS}{\delta x_1} = -2 \times \frac{(x_1)^2}{(x_1)^3}<0$

b)

• The agent’s optimal choice of (x₁, x₂).

$u = (x_1^{-1} +x_2^{-1}B.C=x_1p_1=x_2p_2=m$

$L=U+(B.C)$

$L=(x_1^{-1}+x_2^{-1}+(m-x_1p_1-x_2p_2)$

Now,

$\frac{\delta L}{\delta x_1}=-1(x_1^{-1}+x_2^{-1})^{-2} \times (-1)(x_1)^{-2}-p_1$

$\frac{\delta L}{\delta x_1}=(x_1)^{-2}(x_1^{-1}+x_2^{-1})^{-2}=p_1\frac{\delta L}{\delta _x}$

optimal choice will be

$m-x_1p_1-p_2x_2=0$

$=p_1x_1+p_2\times \sqrt{\frac{p_1}{p_2}}\times x_1=m$

$x_1=\frac{m}{p_1+\sqrt{p_1p_2}}$

$x_2=\frac{m}{\sqrt{}p_1p_2+p_2}$

c)

Agent's indirect

$V=(x_1^{-1}+x_2^{-1})^{-1}$

$V=(\frac{m}{(\sqrt{p_1+\sqrt{}p_2})})$