Question

Given the demand and supply function for the three goods

qd1=20-p1-p3 ;qs1=-10+p1

qd2=40-2p2-p3 ;qs2=2p2

qd3=10-p1+p2-p3 ;qs3=-5+3p3

A Write the equilibrium condition?

B find the equilibrium price?

Answer 1

$a)$

Equilibrium in the three commodity market occur when the supply for each of the 3 commodities equal to its demand.

That is,

$Qd_1=Qs_1\\Qd_2=Qs_2\\Qd_3=Qs_3$

This is the equilibrium condition.

$b)$

To find the equilibrium price we proceed as follows.

For commodity 1,

$Qd_1=Qs_1$

So,

$20-p_1-p_3=-10+p_1\\ 2p_1+p_3=30......(1)$

For commodity 2,

$Qd_2=Qs_2$

So,

$40-2p_2-p_3=2p_2\\ 4p_2+p_3=40...(2)$

For commodity 3,

$Qd_3=Qs_3$

So,

$10-p_1+p_2-p_3=-5+3p_3\\ p_1-p_2+4p_3=15.....(3)$

Equation (1)-Equation (2) gives,

$2p_1-4p_2=-10.....(4)$

$Equation(3)-(4\times Equation(2))$ gives,

$p_1-17p_2=-145.....(5)$

Solving equation (4) and (5),

$Equation(4)-(2\times Equation(5))\\ 30p2=280\implies p2={28\over3}$

From equation (4), we can obtain the value of $p_1$

$2p_1-(4\times{28\over3})=-10\\ 2p_1={82\over3}\implies p1={41\over3}$

From equation (1) we can obtain the value $p_3$

So,

$(2\times{41\over3})+p_3=30\\ p_3=30-{82\over3}\\ Therefore,\\ p_3={8\over3}$

Therefore, the equilibrium prices for commodities 1,2,3 are $p_1={41\over3}, p_2={28\over3}, and\space p_3={8\over3}$ respectively.