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

Learn more on how do we answer questions.

(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.

Question: 9.7

All three equations must be written in the same fo...

Question: 9.20

Step 1: Use the underlying assumption total input=...

Question: 9.18

(a) Use the definition of the inverse given in equ...

Question: 9.17

(a) Write out the augmented matrix consisting of t...

Question: 9.9

Rearrange the equations to have variables on the L...

Question: 9.8

The equations are already written in the required ...

Question: 9.2

(a) For x type I gates and y type II gates, the nu...

Question: 9.21

(a) Set up the augmented matrix, consisting of the...

Question: 9.19

Step 1: Write all the equations in the same order:...

Question: 9.16

(a) The variable terms containing Y, C and T are a...