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:

_____ $$ 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:

_____ $$ 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

_____ $$ 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

_____ $$ 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

_____ $$ D_t = a - bP_t

_____ $$ S_t = dP_t _____________________ (17)

Where $$a = 12, b = 1, c = 0, d = 1, P₁ = 9 and $$D_t 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:

_____ $$ P_t = P_t-1 - 1/3[S_t-1-D_t-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.

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