Question

In an exchange economy, there are two people, A and B, and two goods, 𝑥1and 𝑥2. Their respective utility functions and endowments are

𝑢𝐴(𝑥𝐴1,𝑥𝐴2)=min{𝑥𝐴1,2𝑥𝐴2}

𝑢𝐵(𝑥𝐵1,𝑥𝐵2)=𝑥𝐵1+𝑥𝐵2

𝜔𝐴=(9,19)

𝜔𝐵=(9,0)

Note: The As and Bs in the equations above are not powers but rather superscripts to indicate each agent.

The minimum and maximum possible amounts of 𝑥𝐴2 on the contract curve are

Minimum 𝑥𝐴2

Maximum 𝑥𝐴2=

The minimum and maximum possible amounts of 𝑥𝐴1 on the contract curve are

Minimum 𝑥𝐴1=    

Maximum 𝑥𝐴1=

  

 

Answer 1

Consumers will always choose a bundle where: $aXA_1=bXA_2$ Where a and b are constants.

Therefore $XA_2$ Is maximum when:

$XA_2=(\frac{a}{b})XA_1=\frac{1}{2}XA_1$

Maximum$XA_2=\frac{1}{2}\times19=9.5$

Minimum$XA_2=\frac{1}{2}\times9=4.5$

Where as $XA_1$ Is maximum when:

$XA_1=(\frac{b}{a})XA_2=2XA_2$

Maximum $XA_1=2\times19=38$

Minimum$XA_1=2\times9=18$