### Supply and Demand

#### 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 through d will always be given to you with values assigned, e.g. a = 5.

#### Interpreting the equations:

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.

#### Finding the equilibrium price and quantity:

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 ).

#### The effect of a Tax

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

_____  D = a - bx

_____  S = c + dy ________________(8)

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, of (P - T), where T = the amount of the tax. The system now becomes:

_____  D = 12 - P

_____  S = P - T __________________(9)

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

_____  D = 12 - P

_____  S = P - T = P - 2 ___________(10)

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 ______________(11)

Add 2 to each side  14 - P = P Add P to each side  14 = 2P Dividing each side by 2

_____  P = 7 _____________________ (12)

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 supply curve has shifted to the left). 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.

#### 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 actually paid by the consumer is now (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 _______________________(14)

setting S = D:

_____  P = 12 - P + 2

_____  2P = 14

_____  P = 7 _______________________ (15)

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 _______________________(16)

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 (shift the demand curve to the right) increasing the equilibrium price and quantity given that supply remains constant.

#### 6. Price Dynamics: A simple example

In most cases suppliers do not know the demand curve they face. This example illustrates how a market might move to the equilibrium, given that suppliers adjust their price by a simple rule based on unsold inventories (something they can measure). Consider the following dynamic supply and demand model:

_____  Dt = a - bPt

_____  St = dPt _____________________ (17)

Where a = 12, b = 1, c = 0, d = 1, P1 = 9 and Dt represents the demand in t time period. Assume that: (a) unsold inventories spoil, (b) suppliers set the price in period t as the price in period t - 1 minus 1/3 of the value of the unsold inventory, that is:

_____  Pt = Pt-1 - 1/3[St-1-Dt-1] ___(18)

Given the initial condition that P1 = 9 , suppliers will supply 9 units and demanders will buy 3: 6 units will rot in inventory.

_____  D1 = 12 - P1

________  = 12 - 9

________  = 3

_____  S1 = P1

________  = 9 _______________________(19)

Now, unsold inventories are: S1 - D1 = 9 - 3 = 6
In period 2, suppliers will set a different price, according to the quantity of unsold inventories in period 1,

_____  P2 = P1 - 1/3(S1 - D1)

_____  P2 = 9 - 1/3(9-3) = 9 - 6/3 = 9 -2

_____  P2 = 7 ________________________(20)

At this price, S2 = P2 = 7; but consumers will demand a lower quantity:

_____  D2 = 12 - P2

________  = 5

_____  S2 - D2 = 7 - 5 = 2 ______________(21)

Since period 2 also showed unsold inventories suppliers will reduce their price even more in period 3:

_____  P3 = P2 - 1/3(S2 - D2)

_____  P3 = 7 - 1/3(7 - 5)

________  = 6 1/3 _____________________(22)

Thus, through successive approximations, suppliers will eventually set a price at which S = D, and since at that price unsold inventories will be zero, the price will cease to change.