2. The Simple Model of Price Determination
© 2000 Fernando Quijano Electronic Blackboard for Microeconomics

3. The Simple Model of Price Determination
The Simple Model of Price Determination is the algebraic explanation of a linear model of supply and demand.
The purpose of the model is to predict the values of equilibrium price and quantity in a market, based on a system of demand and supply equations.
Supply and demand are linear functions, each of which describes the respective relationship between price and quantity.
Among the objectives of this lesson is to demonstrate that, since supply and demand are linear functions, expressing quantity as a function of price Q=f(P) or price as a function of quantity P=f(Q), yields the same equilibrium results.
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4. The System of Equations
The system of equations that describes the behavior of demand and supply, when Q = f (P) consists of:
The Demand Function: QD = a - b P
The Supply Function: QS = - c + d P
The Equilibrium Condition: QD = QS
a = intercept of the demand function (parameter)
b = slope of the demand function (parameter)
c = intercept of the supply function (parameter)
d = slope of the supply function (parameter)
QD = quantity demanded (variable)
QS = quantity supplied (variable)
P = market price (variable)
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5. The Mathematical Composition of the System
QD = a - b P
QS = - c + d P
QD = QS
The signs of the slopes in the supply and demand equations describe the laws of supply and demand, respectively.
Quantity demanded varies inversely with changes in price, and quantity supplied varies directly with changes in price.
There are four parameters, or known values (a, b, c, d), and three (unknown) variables: QD, QS and P.
Mathematical note: In a system with n number of goods/markets, general equilibrium, or a system solution exists only if the system has:
3n number of equations,
3n unknowns, and
2n(n+1) number of parameters.
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6. Equilibrium Price and Quantity
To determine equilibrium price, set QD = QS and solve for P:
a - bP = - c + dP
a + c = bP + dP
a + c = P(b+d)
To determine equilibrium quantity (Q*), replace P* in either the demand equation or the supply equation. For example, using the demand equation:
QD* = a - b(P*)
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7. Graphical Presentation
When Q = f (P), quantity depends on price. Quantity is the dependent variable (placed on the vertical axis), and price is the independent variable (placed on the horizontal axis).
QD = a - bP
QS = - c + dP
QD = QS
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8. Numerical Example The Demand Function: QD = 100 - 4P
The Supply Function: QS = P
The Equilibrium Condition: QD = QS
Interpreting these values:
When P = 0, quantity demanded equals 100 units.
For each one-unit increase in price, quantity demanded decreases by four units.
When P = 0, quantity supplied equals There is, of course, no negative quantity supplied. Another interpretation is that price would have to be far greater than zero before quantity supplied becomes positive.
For each one-unit increase in price, quantity supplied increases by six units.
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9. Equilibrium Price and Quantity
To determine equilibrium price, set QD = QS and solve for P:
P = P
then: = 4P + 6P
= P(4+6)
To determine equilibrium quantity (Q*), replace P* in either the demand equation or the supply equation. For example, using the demand equation:
QD* = (11)
QD* =
QD* = 56
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10. Graphical Presentation
QD = P
QS = P
QD = QS
P* = 11
Q* = 56
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11. Finding the Inverse of Linear Supply and Demand Functions
In order to express P = f(Q), we must find the inverse of the linear functions previously expressed as Q = f (P).
When P = f(Q), quantity is the independent variable (placed on the horizontal axis) and price is the dependent variable (placed in the vertical axis).
Expressing either price as a function of quantity or quantity as a function of price does not affect the ultimate outcome of the analysis. Equilibrium price and quantity remain the same.
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12. The Original and Inverse Functions Graphically
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13. Finding the Inverse of the Demand Function
Finding the inverse of the demand function requires finding the inverse of the intercept and the inverse of the slope of the original function.
To find the inverse of the original intercept, set the demand equation equal to zero, and solve for P:
To find the inverse of the original slope, simply divide 1 by the old value:
Old value =
New value =
a - bP = 0
-bP = -a
(both sides x -1) bP = a
then
This value of P equals the value of the new intercept of the demand function.
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14. Finding the Inverse of the Supply Function
Finding the inverse of the supply function requires finding the inverse of the intercept and the inverse of the slope of the original function.
To find the inverse of the original slope, simply divide 1 by the old value:
Old value = + d
New value =
To find the inverse of the original intercept, set the supply equation equal to zero, and solve for P:
-c + dP = 0
dP = c
then
This value of P equals the value of the new intercept.
© 2000 Fernando Quijano Electronic Blackboard for Microeconomics

15. Exercise The Demand Function: QD = 100 - 4P
Find the inverse functions of the following system of equations:
The Demand Function: QD = P
The Supply Function: QS = P
The Equilibrium Condition: QD = QS
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16. Solution 100 - 4P = 0 100 = 4P then -10 + 6P = 0 6P = 10 then
New Intercept of Demand:
New Intercept of Supply:
P = 0
100 = 4P
then
P = 0
6P = 10
then
New Slope of Supply:
New Slope of Demand:
Old slope = - 4
Old slope = +6
New Slope =
New Slope =
New Demand Equation:
New Supply Equation:
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17. The New System of Equations
When P = f(Q):
The Demand Function:
The Supply Function:
The Equilibrium Condition: PD = PS
© 2000 Fernando Quijano Electronic Blackboard for Microeconomics

18. Determining Equilibrium Quantity and Price
To determine equilibrium quantity (Q*), set PD = PS and solve for Q:
Determine equilibrium price by replacing Q* in either the demand equation or the supply equation:
23.33 = Q
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19. Graphical Presentation
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20. The Original and Inverse Functions Graphically
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21. Price Floor: View when Q=f(P)
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22. Price Floor: View when P=f(Q)
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23. Price Ceiling: View when Q=f(P)
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24. Price Ceiling: View when P=f(Q)
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