Answer to Question #249493 in Microeconomics for learner

Question

An agent consumes quantity (x₁,x₂) of goods 1 and 2. Here is his utility function: 𝑈(𝑥1, 𝑥2) = min {𝑥1, 2 ∗ 𝑥2}.

(a) Draw a typical indifference curve for the agent.

(b) The agent has income m=10, prices are (p₁, p₂) = (1, 3). Derive Marshallian demand (x*₁, x*₂).

Answer 1

a.

b.

$m=10,\space (p_1,p_2)=(1,3)\\u(x_1, x_2)=min\{x_1,2x_2\}\\x_1=2x_2$

Budget constraint

$p_1x_1+p_2x_2=m$

$x_1+3x_2=10$

Substituting x1 in the equation

$2x2+3x_2=10\\5x_2=10\\x_2=2$

$x_1=4$

optimal bundle $(x*_1,x*_2)=(4,2)$

marshallian demand

$x_1=2x_2$

X1 put in the budget constraint

$p_1x_1+p_2x_2=m\\p_1(2x_2)p_2x_2=m\\2p_1x_2+p_2x_2=m\\x_2(2p_1+p_2)=m\\x*_2=\frac{m}{2p_1+p_2}.................marshallian\space demand\space x_2$

$x*_1=\frac{2m}{2p_1+p_2}...............marshallian\space demand \space x_1$