# Question 9.12: Given the supply and demand functions for two related goods,......

Given the supply and demand functions for two related goods, A and B,

Good A : $\begin{matrix} \{Q_{da} = 30 − 8P_a + 2P_b \\ \{Q_{sa} = −15 + 7P_a \end{matrix}$           Good B : $\begin{matrix} \{Q_{db} = 28 + 4P_a − 6P_b\\ \{Q_{sb} = 12 + 2P_b \end{matrix}$

(a) Write down the equilibrium condition for each good. Hence, deduce two equations in $P_a$ and $P_b$ .
(b) Use Cramer’s rule to find the equilibrium prices and quantities for goods A and B.

Step-by-Step
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(a) The equilibrium condition for each good is that $Q_s = Q_d$.

For good A:

$−15 + 7P_a = 30 − 8P_a + 2P_a$
$15P_a − 2P_b = 45$

For good B:

$12 + 2P_b = 28 + 4P_a − 6P_b$
$−4P_a + 8P_b = 16$

Therefore, the simultaneous equations are

(1) $15P_a – 2P_b = 45$
(2) $–4P_a + 8P_b = 16$

(b) Applying Cramer’s rule:

$P_{a}={\frac{\Delta_{P_{a}}}{\Delta}}= \frac {\begin{vmatrix} 45 & -2 \\ 16 & 8 \end{vmatrix} }{\begin{vmatrix} 15 & -2 \\ -4 & 8 \end{vmatrix} } = \frac {(45)(8) − (16)(−2)}{(15)(8) − (−4)(−2)} = \frac {360 − (−32)}{120 − 8} = \frac {392}{112} = 3.5$

$P_{b}={\frac{\Delta_{P_{b}}}{\Delta}}= \frac {\begin{vmatrix} 15 & 45 \\ -4 & 16 \end{vmatrix} }{\begin{vmatrix} 15 & -2 \\ -4 & 8 \end{vmatrix} } = \frac {(15)(16) − (−4)(45)}{(15)(8) − (−4)(−2)} = \frac {240 − (−180)}{120 − 8} = \frac {420}{112} = 3.75$

Substituting these values of $P_a$ and $P_b$ into any of the original equations, solve for $Q_a$ and $Q_b$ .
$Q_a$ = 9.5, $Q_b$ = 19.5.

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