Question

Q: Suppose there are three types of Apples A, B, and C being sold and consumed. The demand and supply equations for each type are:

D_A= 20 - 2P_A + 4P_B + P_C , S_A= 4P_A - 5

D_B=10+3P_A - 5P_B + 2P_C , S_B=3P_B - 7

D_C = 70+4P_A +2P_B - 5P_C , S_C=5P_C - 16

• Determine equilibrium prices and quantities.
• Calculate the elasticity of demand for B with respect to prices of variety A, B and C and interpret the economic meaning of the results

Answer 1

$D_A= 20 – 2P_A + 4P_B + P_C \\ S_A=4P_A-5 \\ D_B=10+3P_A-5P_B+2P_C \\ S_B=3P_B-7 \\ D_C=70+4P_A+2P_B-5P_C \\ S_C=5P_C-16$

At equilibrium,

$D_A=S_A \\ =>20-2P_A+4P_B+P_C=4P_A-5 \\ =>4P_A+2P_A-4P_B- P_C=20+5$

$=>6P_A-4P_B- P_C=25$ ....(1)

$D_B=S_B \\ => 10+ 3P_A - 5P_B + 2P_C = 3P_B – 7 \\ =>-3P_A+3P_B+5P_B-2P_C=10+7 \\$

$=>-3P_A+8P_B-2P_C=17$ ....(2)

$D_C = S_C$

$=>70+4P_A+2P_B-5P_C=5P_C-16 \\ =>-4P_A-2P_B+5P_C+5P_C=70+16$

$=>-4P_A-2P_B+10P_C=86$ ....(3)

From the system of equations, we get the determinant as,

$|D|=\begin{vmatrix} 6 &-4 &-1 \\ -3& 8 &-2 \\ -4 &-2 & 10 \end{vmatrix}=6(80-4)+4(-30-8)-1(6+32)=456-152-38=266$

By Cramer's rule,

$P_A^*=\frac{\begin{vmatrix} 25 &-4 &-1 \\ 17& 8 &-2 \\ 86 &-2 & 10 \end{vmatrix}}{|D|}=\frac{25(80-4)+4(170+172)-1(-34-688)}{266} \\ P_A^*=\frac{25 \times 76+4 \times 342-1(-722)}{266}=\frac{1900+1368+722}{266}=\frac{3990}{266} \\ P_A^*=15 \\ Q_A^*=4 \times 15-5=60-5=55 \\ P_B^*=\frac{\begin{vmatrix} 6 &25 &-1 \\ -3& 17 &-2 \\ -4 &86 & 10 \end{vmatrix}}{|D|}=\frac{6(170+172)-25(-30-8)-1(-258+68)}{266} \\ P_B^*=\frac{6 \times 342+25 \times 38-1(-190)}{266}=\frac{2052+950+190}{266}=\frac{3192}{266} \\ P_B^*=12 \\ Q_B^*=3 \times 12-7=36-7=29 \\ P_C^*=\frac{\begin{vmatrix} 6 &-4 &25 \\ -3& 8 &17 \\ -4 &-2 & 86 \end{vmatrix}}{|D|}=\frac{6(688+34)+4(-258+68)+25(6+32)}{266} \\ P_C^*=\frac{6 \times 722-4 \times 190+25 \times 38}{266}=\frac{4332-760+950}{266}=\frac{4522}{266} \\ P_C^*=17 \\ Q_C^*=5 \times 17-16=85-16=69 \\ D_B=10+3P_A-5P_B+2P_C \\ \frac{\mathrm{d} D_B}{\mathrm{d} P_A}=3 \\ \frac{\mathrm{d} D_B}{\mathrm{d} P_B}=-5 \\ \frac{\mathrm{d} D_B}{\mathrm{d} P_C}=2$

At equilibrium,

$D_B=29 , \small P_A^*=15 , \small P_B^*=12 , \small P_C^*=17$

Elasticity of demand with respect to PA,

$\varepsilon _A=\frac{\mathrm{d} D_B}{\mathrm{d} P_A}\frac{P_A}{D_B}=3 \times\frac{15}{29}=1.55$

$\varepsilon _A>0$ implies A and B are substitute goods.

Elasticity of demand with respect to PB,

$\varepsilon _B=\frac{\mathrm{d} D_B}{\mathrm{d} P_B}\frac{P_B}{D_B}=-5 \times \frac{12}{29}=-2.0$

$|\varepsilon _B|>1$ implies the demand for good B is elastic.

Elasticity of demand with respect to PC,

$\varepsilon _C=\frac{\mathrm{d} D_B}{\mathrm{d} P_C}\frac{P_C}{D_B}=2 \times\frac{17}{29}=1.17$

$\varepsilon _C>0$ implies B and C are substitute goods.