Question

Given below are the demand and the supply functions for three interdependent commodities.

Qd1=110 - 4P1+ 3P2 -4P3;Qs1= 2P1 -20

Qd2= 46+ 2P1 -4P2+4P3; Qs2= -14+ 2P2

Qd3= 20 -P1 + 4P2 - 2P3 ;Qs3=2P3 -10

Determine the equilibrium prices and quantities for the three commodity market model. Then compute the prices and cross elasticities of demand for all the three markets and interprete their coefficients

Answer 1

Given:

Qd1 = 110 – 4P1 + 3P2 – 4P3 ; Qs1 = 2P1 – 20

Qd2 = 46 + 2P1 – 4P2 + 4P3 ; Qs2 = –14 + 2P2

Qd3 = 20 – P1 + 4P2 – 2P3 ; Qs3 = 2P3 – 10

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

That is,

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

For commodity 1:

$Qd_1=QS_1\\110 – 4P_1 + 3P_2 – 4P_3=2P_1-20\\−6P_1+3P_2−4P_3=-130\\6P_1−3P_2+4P_3=130 …(1)$

For commodity 2:

$Qd_2=Qs_2\\46 + 2P_1 −4P_2 + 4P_3=-14+2P_2\\2P_1−6P_2+4P_3=-60\\−2P_1+6P_2+4P_3= 60 …(2)$

For commodity 3:

$Qd_3=Qs_3\\20− P_1 +4P_2 −2P_3=– 10+2P_3\\−P_1+4P_2−4P_3=-30\\P_1−4P_2+4P_3=30 …(3)$

Subtracting (2) from (1) gives:

$6P_1−3P_2+4P_3=130 …(1)\\−2P_1+6P_2+4P_3=60 …(2)\\(2)−(1):\\(6P_1−3P_2+4P_3)−(−2P_1+6P_2+4P_3)=130−60\\6P_1+2P_1−3P_2−6P_2+4P_3−4P_3=70\\⇒8P_1−9P_2=70 …(4)$

Similarly, subtracting (3) from (2):

$−2P_1+6P_2+4P_3=60 …(2)\\P_1−4P_2+4P_3=30 …(3)\\(2)−(3):\\(−2P_1+6P_2+4P_3)−(P_1−4P_2+4P_3)=60−30\\ −3P_1+10P_2=30 …(5)$

Solving (4) and (5) gives:

equation (5) gives:

$-3P_1+10P_2=30\\-3P_1=30-10P_2\\P1=\frac{10P_2-30}{3}$

Substituting in (4)

$8P_1-9P_2=70\\8(\frac{10P_2-30}{3})-9P_2=70\\\frac{80P_2}{3}-80-9P_2=70\\\frac{80P_2-27P_2}3=70+80\\53P_2=150×3\\P_2=\frac{450}{53}=8.49\\Since,\\P_1=\frac{10P_2-30}{3 } \\ =\frac{10×8.49-30}{3}\\ =\frac{84.9-30}{3}\\ =18.3$

Substituting P1=18.3 and P2=8.49 in (1) gives P3:

$-2P_1+6P_2+4P_3=60\\-2(18.3)+6(8.49)+4P_3=60\\-36.6+50.94+4P_3=60\\14.34+4P_3=60\\P_3=\frac{60-14.34}{4}=11.415$

Substituting P1, P2 and P3 in the three demand equation gives Q1, Q2 and Q3.

$Q_1 = 110 – 4P_1 + 3P_2 – 4P_3 \\ =110−4(18.3)+3(8.49)−4(11.415) \\ =53.01\\Q_2= 46 + 2P_1 – 4P_2 + 4P_3\\Q_2=46+2(18.3)−4(8.49)+4(11.415)\\Q_2^*=94.3\\Q_3= 20 – P_1 + 4P_2 – 2P_3 \\Q_3=20−18.3+4(8.49)−2(11.415)\\Q_3^*=12.83$

Therefore, we get the following equilibrium combinations for the three goods,

 $(Q_1^*,P_1^*)=(53.01,18.3)\\(Q_2^*,P_2^*)=(94.3,8.49)\\(Q_3^*,P_3^*)=(12.83,11.415)$

(b) Price elasticity of demand is calculated as :

$Ed=\frac{dQ_3}{P_3}×\frac{P_3}{Q_3}\\Given,\\ Qd_3 = 20 – P_1 + 4P_2 – 2P_3\\E_d=−2\frac{P_3}{Q_3}$

The coefficient dQ₃/dP₃ implies that as the price of the 3rd commodity increases by 1 unit, its quantity demanded falls by 2 units.

Cross price elasticity is calculated as:

Cross E_d with respect to P₁

$=\frac{dQ_3}{P_1}×\frac{P_1}{Q_3}\\=-1\times \frac{P_1}{Q_3}\\=-\frac{P_1}{Q_3}$

The coefficient here, dQ₃/dP₁ implies that as the price of commodity 1 increases by 1 unit, the quantity demanded of commodity 3 falls by 1 unit.

Cross E_d with respect to P₂

$=\frac{dQ_3}{P_2}×\frac{P_2}{Q_3}\\=4\times \frac{P_1}{Q_3}\\=-\frac{P_1}{Q_3}$

The coefficient here, dQ₃/dP₁ implies that as the price of commodity 2 increases by 1 unit, the quantity demanded of commodity 3 increases by four units. A positive coefficient here implies that commodity 2 and 3 are substitutes.