Answer to Question #212597 in Microeconomics for PAUL ONDIGO

Question

Given below are the demand and 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

price and cross elasticities of demand for the third market and interpret 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.