The Simple Model of Price Determination © 2000 Fernando QuijanoElectronic Blackboard for Microeconomics.

The Simple Model of Price Determination

© 2000 Fernando Quijano© 2000 Fernando Quijano Electronic Blackboard for MicroeconomicsElectronic Blackboard for Microeconomics

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.

The Simple Model of Price Determination is the algebraic explanation of a linear model of supply and demand.

The Simple Model of Price Determination

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.


The system of equations that describes the behavior of demand and supply, when Q = f (P) consists of:

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)

The System of Equations

The Demand Function: QD = a - b PThe Supply Function: QS = - c + d P


QD = a - b PQS = - c + d PQD = QS

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, and2n(n+1) number of parameters.

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.

The Mathematical Composition of the System


To determine equilibrium quantity (Q*), replace P* in either the demand equation or the supply equation. For example, using the demand equation:

Pa c

b d* =

++

To determine equilibrium price, set QD = QS and solve for P:

a - bP = - c + dPa + c = bP + dPa + c = P(b+d)

Equilibrium Price and Quantity

QD* = a - b(P*)


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 = QS

Graphical Presentation

QD = a - bPQS = - c + dP


The Demand Function: QD = 100 - 4PThe Supply Function: QS = - 10 + 6PThe Equilibrium Condition: QD = QS

When P = 0, quantity demanded equals 100 units.For each one-unit increase in price, quantity demanded decreases by four units.

Interpreting these values:

When P = 0, quantity supplied equals -10. 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.

Numerical Example


To determine equilibrium quantity (Q*), replace P* in either the demand equation or the supply equation. For example, using the demand equation:

To determine equilibrium price, set QD = QS and solve for P:

Equilibrium Price and Quantity

100 - 4 P = - 10 + 6 Pthen: 100 + 10 = 4P + 6P 100 + 10 = P(4+6)

P*

1 0 0 1 0

4 611

QD* = 100 - 4(11)QD* = 100 - 44QD* = 56



QD = 100 - 4PQS = - 10 + 6PQD = QS

P* = 11

Q* = 56


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.

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).

Finding the Inverse of Linear Supply and Demand Functions


The Original and Inverse Functions Graphically

Q a bPD Q c dPS

P a b QD ' 'P c d QS ' '

Q QD S P PD S© 2000 Fernando Quijano© 2000 Fernando Quijano Electronic Blackboard for MicroeconomicsElectronic Blackboard for Microeconomics

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 slope, simply divide 1 by the old value:

Old value =

New value =

Pa

b=

To find the inverse of the original intercept, set the demand equation equal to zero, and solve for P:

a - bP = 0-bP = -a(both sides x -1) bP = athen

This value of P equals the value of the new intercept of the demand function.

Finding the Inverse of the Demand Function

1

b

b


To find the inverse of the original intercept, set the supply 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 = + d

New value = +1

d

Finding the inverse of the supply function requires finding the inverse of the intercept and the inverse of the slope of the original function.

Pc

d=+

-c + dP = 0dP = cthen

This value of P equals the value of the new intercept.

Finding the Inverse of the Supply Function


Find the inverse functions of the following system of equations:

The Demand Function: QD = 100 - 4PThe Supply Function: QS = - 10 + 6PThe Equilibrium Condition: QD = QS

Exercise


100 - 4P = 0100 = 4Pthen P = =

100

425

New Slope of Demand:Old slope = - 4

New Intercept of Demand:

New Demand Equation:

Solution

P QD 2 51

4

1

4New Slope =

New Supply Equation:

P 1 0

61 6 7.

-10 + 6P = 06P = 10then

Old slope = +6

New Intercept of Supply:

New Slope of Supply:

New Slope = 1

6

P QS 1 6 71

6.


When P = f(Q):

The New System of Equations

The Demand Function: P QD 2 51

4

The Supply Function: P QS 1 6 71

6.

The Equilibrium Condition: PD = PS


To determine equilibrium quantity (Q*), set PD = PS and solve for Q:

2 3 3 31

6

1

4. Q Q

2 5 1 6 73

1 2

2

1 2 . Q Q

Q * .

.

2 3 3 3

0 4 1 65 6

Determining Equilibrium Quantity and Price

23.33 = 0.416 Q

2 51

41 6 7

1

6 Q Q.

PD* ( ) 2 5

1

45 6

Determine equilibrium price by replacing Q* in either the demand equation or the supply equation:

PD* 2 5 1 4

PD* 11


P QD 2 51

4

P QS 1 6 71

6.

P PD S



The Original and Inverse Functions Graphically


Price Floor: View when Q=f(P)


Price Floor: View when P=f(Q)


Price Ceiling: View when Q=f(P)


Price Ceiling: View when P=f(Q)


The Simple Model of Price Determination © 2000 Fernando QuijanoElectronic Blackboard for Microeconomics.

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