I. Supply and Demand I. Supply and Demand

1. Definitions of linear supply and demand:

D
=
a - bP
S
=
c + dP
(1)
where D = quantity demanded, S = quantity supplied, P = price per unit and a,b,c, and d are constants. Note: In this course the constants a thru d will always be given to you with values assigned, e.g. a = 5. When given a value for the price, the quantities supplied and demanded can be obtained by plugging the value for P into the equations and solving for D and S. For example, let a = 12, b = 1, c = 0, and d = 1 such that:
D
=
12 - P
S
=
P
(2)
When given a value for price, say $4, the values for D and S can now be found,
D
=
12 - 1(4) = 8
S
=
4
(3)
Notice that the quantity supplied does not equal the quantity demanded when P = 4. Only at the equilibrium price will they be equal. The equilibrium price is that price at which the quantity supplied equals the quantity demanded, or where D = S. To find the equilibrium price we first set the demand equation equal to the supply equation:

D
=
S
12-P
=
P
(4)

We now solve this equation for P to obtain the equilibrium price. The first step is to add P to each side, eliminating the P from the left side:

12 - P
=
 P
   +P
=
 +P
12 - 0
=
2P
12
=
2P
(5)

The next step is to divide each side by 2 in order to get the equilibrium value for P,

12/2 = 2P/2     or     6 = P
(6)
The equilibrium price in this case is P = $6. The equilibrium quantity can now be found by substituting the equilibrium value for P into either the original supply or demand equation:

D
=
12 - 6 = 6
S
=
6
(7)

The equilibrium quantity is D = S or 6 units. (It is a good idea to substitute the equilibrium value for P into both equations to make sure that D = S ).
Practice Problem: Consider the following system:

D
=
64 - 8P
S
=
8P
(8)

1. Given P = 5, what is the quantity demanded?, supplied?

2. What is the equilibrium price? the equilibrium quantity?

To consider the effect of a tax in this section and the effect of a subsidy in the next section, we need to revise the basic supply and demand model to:

D
=
a - bx
S
=
c + dy
(9)

where D = quantity demanded, S = quantity supplied, x = amount demander pays out of his pocket, y is the amount the supplier has to finance production, and a,b,c, and d are constants.

Suppose that the government imposes a tax upon the supplier. The price that the supplier now receives will not be the market price but the market (equilibrium) price minus the amount of the tax, that is y = (P - T), where T = the amount of the tax. The system now becomes:

D
=
12 - P
S
=
P - T
(10)

Example: Suppose that a tax = $2 per unit is levied on the supplier. Then:

D
=
12 - P
S
=
P - T = P - 2
(11)

To find the new equilibrium price with the tax we again set the demand equation to the supply and solve for P.

D
=
S
12 - P
=
P - 2
(12)

Add 2 to each side

14 - P = P
Add P to each side
14 = 2P
Dividing each side by 2
P = 7
The new equilibrium price is now P = 7. Notice that while this is the actual price paid by the consumer, the price that is actually received by the supplier is P - T or 7 - 2 = 5. to find the equilibrium quantity we substitute P = 7 back into the original equations:

D
=
12 - 7 = 5
S
=
7 - 2 = 5
(13)

The new equilibrium quantity D = S is 5. Notice that this is one unit less than before the tax was imposed. since the supplier is actually receiving $2 less per unit than before the tax, he will not offer as many units for sale at each market price. The result of the tax has been a decrease in supply. The result of this is a higher equilibrium price and a lower equilibrium quantity given that demand remains constant. Practice Problem: Consider the following system:

D
=
72 - 6P
S
=
6P
(14)

1. Given P = 4, what is the quantity demanded?, supplied?

2. What is the equilibrium price? the equilibrium quantity?

3. suppose that the government imposes a $4 per unit tax on the supplier. What is the new equilibrium price? What is the price actually received by the supplier? What is the new equilibrium quantity?

5. Effect of a subsidy

Suppose that the government now wishes to increase the demand for a product and thus decides to give a subsidy to the consumers of that product. The price paid out of pocket by the consumer is now x = (P - s) where s is the amount of the subsidy per unit. We would suspect that the lower price paid by the consumer will increase demand thus increasing the equilibrium price and quantity, assuming that supply remains constant. Example: Let s = $2 To find the equilibrium price we again set S = D and solve for P. The new system is:

D
=
12 - (P - s)
D
=
12 - P + 2
S
=
P
(15)

setting S = D:

P
=
12 - P + 2
2P
=
14
P
=
7
(16)

The new equilibrium price is P = 7. Plugging this back into the supply and demand equations we obtain equilibrium quantity:

D
=
12 - 7 + 2 = 7
S
=
7
(17)

The equilibrium quantity is now D = S or 7. Notice that the price actually paid by the consumer is only 7 - 2 or 5 and the supplier receives $7. The effect of the subsidy has been to increase demand increasing the equilibrium price and quantity given that supply remains constant. Practice Problem: Consider the following system:

D
=
64 - 8P
S
=
8P
(18)

1. Given P = 4, what is the quantity demanded?, supplied?

2. What is the equilibrium price? the equilibrium quantity?

3.Suppose instead that a $4 per unit subsidy is given to the consumer. What is the equilibrium price? Quantity? The actual price paid by the consumer?

 

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On 23 Jan 2000, 19:32.