Question

There are 5,000 identical individuals in themarket for commodity X, each with demand function given by Qd=6-Px, and 1,000 identical suppliers of commodity X, each with function given by Qs=10Px.a)Find the market demandfunction and market supply function for commodity X.b)Construct the market demand and the market supply schedule of commodity X and from them, find the equilibrium price and equilibrium quantity. (Note: use Px=0-6)c)On one set of axes, plot the market demand curve and the market supply curve for commodity X and show the equilibrium point.d)Obtain the equilibrium price and the equilibrium quantity mathematically.e)At each price, determine if there is surplus, shortage and market equilibrium.

Answer 1

Solution:

Given,

Number of identical individuals in the market for commodity X = 5,000

Individual demand function: Q_d=6-P_x

Number of identical suppliers of commodity X = 1,000

Individual supply function: Q_s= 10P_x

a) The market demand function for X is the sum of all individual demand functions for the commodity. Here we have the individual demand function and the number of consumers, so the product of the number of cosumers and the individual demand function will give us the market demand fuction.

Similarly, the market supply function is obtained by multiplying the individual supply function and the number of suppliers of commodity X.

Let Q be the market Q_D be the market quantity demanded and Q_S be the market quantity supplied.

Therefore, the market demand function:

Q_D = 5000 x Q_d

=> Q_D = 5000 x (6-P_x)

=> Q_D = 30,000 - 5000P_x , is the market demand function for commodity X.

Similarly, market supply function:

Q_S = 1000 x Q_s

=> Q_S = 1000 x 10P_x

=> Q_S = 10000P_x , is the market supply function for commodity X.

b) The follwoing table shows the market demand schedule and the market supply schedule of commodity X, when price (P_X) ranges from 0 to 6. The area highlighted in yellow show the equilibrium quantity, 20,000, and the equilibrium price, $2.

c) The following graph is the plot for the market demand curve and the market supply supply curve. The point E is the equilibrium point.

d) The equilibrium price and the equilibrium quantity can be obtained mathematically in the following way:

We know that at the market equilibrium, the market quantity demanded is equal to the market quantity supplied, so we have,

  Q_D = Q_S

=> 30,000 - 5000P_x = 10000P_x

=> 30,000 = 10000P_x + 5000P_x

=> 30,000 = 15000P_x

=> P_x = 30,000 / 15,000

=> P_x^M = $2, is the market equilibrium price of commodity X.

Now using the equlibrium price in the market demand function, we get,

Q_D = 30,000 - 5000P_x^M

  => Q_D = 30,000 - (5000 x 2)

=> Q_D = 30,000 - 10,000

=> Q_D^M = 20,000, market equlibrium quantity of commodity X.

e) In order to determine that at each price level whether there is surplus, shortage or market equilibrium, we can use the following table, which is the combination of the market demand schedule and the market supply schedule.