Answer to Question #285949 in Microeconomics for Nancy

Question #285949

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

1
Expert's answer
2022-01-10T15:03:22-0500

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 \u2013 4P_1 + 3P_2 \u2013 4P_3=2P_1-20\\\\\u22126P_1+3P_2\u22124P_3=-130\\\\6P_1\u22123P_2+4P_3=130 \u2026(1)"


For commodity 2:

"Qd_2=Qs_2\\\\46 + 2P_1 \u22124P_2 + 4P_3=-14+2P_2\\\\2P_1\u22126P_2+4P_3=-60\\\\\u22122P_1+6P_2+4P_3= 60 \u2026(2)"


For commodity 3:

"Qd_3=Qs_3\\\\20\u2212 P_1 +4P_2 \u22122P_3=\u2013 10+2P_3\\\\\u2212P_1+4P_2\u22124P_3=-30\\\\P_1\u22124P_2+4P_3=30 \u2026(3)"


Subtracting (2) from (1) gives:

"6P_1\u22123P_2+4P_3=130 \u2026(1)\\\\\u22122P_1+6P_2+4P_3=60 \u2026(2)\\\\(2)\u2212(1):\\\\(6P_1\u22123P_2+4P_3)\u2212(\u22122P_1+6P_2+4P_3)=130\u221260\\\\6P_1+2P_1\u22123P_2\u22126P_2+4P_3\u22124P_3=70\\\\\u21d28P_1\u22129P_2=70 \u2026(4)"


Similarly, subtracting (3) from (2):

"\u22122P_1+6P_2+4P_3=60 \u2026(2)\\\\P_1\u22124P_2+4P_3=30 \u2026(3)\\\\(2)\u2212(3):\\\\(\u22122P_1+6P_2+4P_3)\u2212(P_1\u22124P_2+4P_3)=60\u221230\\\\ \u22123P_1+10P_2=30 \u2026(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\u00d73\\\\P_2=\\frac{450}{53}=8.49\\\\Since,\\\\P_1=\\frac{10P_2-30}{3 } \\\\ =\\frac{10\u00d78.49-30}{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 \u2013 4P_1 + 3P_2 \u2013 4P_3 \\\\ =110\u22124(18.3)+3(8.49)\u22124(11.415) \\\\ =53.01\\\\Q_2= 46 + 2P_1 \u2013 4P_2 + 4P_3\\\\Q_2=46+2(18.3)\u22124(8.49)+4(11.415)\\\\Q_2^*=94.3\\\\Q_3= 20 \u2013 P_1 + 4P_2 \u2013 2P_3 \\\\Q_3=20\u221218.3+4(8.49)\u22122(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}\u00d7\\frac{P_3}{Q_3}\\\\Given,\\\\ Qd_3 = 20 \u2013 P_1 + 4P_2 \u2013 2P_3\\\\E_d=\u22122\\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 Ed with respect to P1

"=\\frac{dQ_3}{P_1}\u00d7\\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 Ed with respect to P2

"=\\frac{dQ_3}{P_2}\u00d7\\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.



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