Question

Solving the following system of equations, which is a model of price and quantity determination in a one-good, widget, market:

Q_d = 100– 10P + 2G …..(1)

Q_s = 60 + 20P - 3N……….(2)

Q = Q_d = Q_{s………………… ..(3)}

                           G= 11 …………………………(4)

                           N= 6 …………………………..(5)

(i)                     Find the equilibrium values of P and Q.  

(ii)               If G is up by 2 and other things are equal, What will be the new P and Q.

(iii)             If N is up by 2 and other things are equal, What will be the new P and Q.

Answer 1

Answer

(i) Q_d = 100– 10P + 2G …..(1)

Q_s = 60 + 20P - 3N……….(2)

Now, given G = 11 , and N = 6

Putting the value of G in equation (1), we get,

Q_d = 100 – 10P + 2 * 11

Or, Q_d = 100 + 22 - 10P

Or, Q_d = 122 - 10P.....(1^')

Putting the value N in equation(2), we get,

Q_s =  60 + 20P - 3 * 6

Or, Q_s = 60 - 36 + 20P

Or, Q_s = 24 + 20P........(2^')

At the equilibrium point, Q_d = Q_s

122 - 10P = 24 + 20P

Or, -10P - 20P = 24 - 122

Or, -30P = -98

Or, 30P = 98

Or, P = 98 / 30 = 3.27

So, P = $3.27

Now, putting the value of P in equation (1^'), we get,

Q_d = 122 - 10 * (3.27)

Or, Q_d = 122 - 32.7

Or,  Q_d = 89.3

The equilibrium quantity, Q_d = Q_s = 89.3 units

So, the equilibrium price is $3.27, and the equilibrium quantity is 89.3 units.

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(ii) If G is up by 2 and other things are equal, the value of G would be, G = 11 + 2 = 13

Now, putting the value of G in equation(1), we get,

Q_d = 100 – 10P + 2 * (13)

Or, Q_d = 100 – 10P + 26

Or, Q_d = 126 - 10P.....(1^'')

Now, as at the equilibrium point, Q_d = Q_s ,

126 - 10P = 24 + 20P

Or, - 10P - 20P = 24 - 126

Or, - 30P = -102

Or, P = -102 / -30 = 3.4

Or, P = $3.4

The new equilibrium price would be $3.4.

Now, putting the value of P in equation (1^''), we get,

Q_d = 126 - 10 * (3.4)

Q_d = 126 - 34

Or, Q_d = 92

The new equilibrium quantity is 92 units.

So, If G is up by 2 and other things are equal, the new equilibrium price will be $3.4, and the new equilibrium quantity will be 92 units.

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(iii) If N is up by 2 and other things are equal, the value of N would be, N = 6 + 2 = 8

Now, putting the value of N in equation(2), we get,

Q_s = 60 + 20P - 3 * 8

Or, Q_s = 60 + 20P - 24

Or, Q_s = 36 + 20P......(2^'')

Now, as at the equilibrium point, Q_d = Q_s ,

122 - 10P = 36 + 20P

Or, - 10P - 20P = 36 - 122

Or, - 30P = -86

Or, P = -86 / -30 = 2.87

Or, P = $2.87

Now, putting the value of price in equation (2^"), we get,

Q_s = 36 + 20 * (2.87)

Or, Q_s = 36 + 57.4

Q_s = 93.4

So, If N is up by 2 and other things are equal, the new equilibrium price will be $2.87, and the new equilibrium quantity will be 93.4 units.

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