linear equarions - economic applications

EC115 Methods of Economic AnalysisLecture 2: Linear equations 2 – Economic

Applications

Week 3, Autumn 2008

A Model of Demand and Supply

� Demand and Supply are the basic building blocks for economic analysis

� A model of demand and supply describes how prices affect the behaviour of both consumers and producers

� In general, the quantity demanded of a good sold in a market depends not only on its own price but also on other factors, such as the prices of other goods and consumers’ income.

� For example, if the price of the Pepsi increases while the one of the Coca-Cola remains constant, some consumers may switch and consume more Coca-Cola if they feel that the two products are close substitutes (they are very similar products).

Demand

� The concept of a demand function is based on consumer theory: the lower is the price of the good, the more the consumer wants to buy it (taking as given the other factors)

� Adding up all the individual demands gives us the market demand:qD = -ap + b,

where p is the market price and a and b are positive parameters.

� The demand curve is negatively sloped: The higher the price of the good, the lower is the quantity demanded of that good – Law of Demand

� The quantity demanded (qD) is the dependent variable, and we assume that, for each individual, the market price (p) is given so that she/he chooses how much to buy at that price.

Demand curveq

p

qD = -ap + b

Supply

� The concept of the supply function is based on the theory of firm: The higher is the market price of the good, the more firms will choose to supply that good

� Combining all firms’ supply gives us the market supply: qS = cp + d,

where c>0 and d is zero or negative (with zero price firms wish to supply nothing).

� The supply curve is positively sloped: The higher is the price of the good the higher is the quantity supplied of that good – Law of Supply

� Quantity supplied (qS) is the dependent variable and price (p) is assumed to be given.

Supply curve

p

q

qS = cp + d

Market Equilibrium

� An equilibrium is a situation where none of the agents have an incentive to change anything.

� The market is in the equilibrium when the quantity that the consumers are willing to buy equals the quantity the producers are willing to supply: qD = qS (equilibrium condition)

� If qD > qS, then some consumers would like to buy more than is supplied in the market by firms

� If qD < qS, then some producers would like to supply more than is demanded in the market by consumers

There is disequilibrium in the market.

Example (1)

Suppose the demand for books is specified as the following relationshipbetween quantity (Q) and price (P) : qD = 40 – 2p.

The supply of books is specified as another relationship: qS = 3p – 10.

The equilibrium condition is: qD = qS.

This is a system of three linear equations and three unknowns (qD, qS,p). To solve for the equilibrium, we set

40 – 2p = 3p – 10,

and solve for p: p* = 10.

Plug this into one of the equations above to get q* = 3(10) – 10 = 20.

Market Equilibrium

20

40

q

p

qD = 40 – 2p

-10

p* = 10

q* = 20 E

qS = 3p – 10

10/3

Disequilibrium (qD > qS)q

20

40

p

qD = 40 – 2p

-10

E

qS = 3p – 10

5

yD = 30

yD = 5

Excess demand

Comparative statics

� In addition to computing the equilibrium quantity and price of books, we can use the model to analyse the impact of changes in the circumstances under which the decisions are made.

� For instance, buyers’ income or production technology may change, or the government may intervene in the market (taxes etc.)

� Comparing the initial equilibrium with the new equilibrium that is induced by a disturbance or shock to the system is the basic idea of comparative statics.

Comparative statics – Example (1)

� Suppose the demand for books is now a function of price (P) and income (I): qD’ = –2P + 40 + I

� Supply of books is as before: qS = 3P – 10.

� In the equilibrium: qD’ = qS, i.e.,

40 – 2P + I = 3P – 10, and solving for P gives: P = 10 + I/5.

� If I = 50, then P* = 20, and q* = 50.

� Now suppose income increases to 100. The model tells us that thequantity sold in the market increases by 30 and the price increases by 10.

� From the graph, the two demand curves are parallel but the new curve has shifted upwards from the original one.


45

90

q

p

qD = –2P + 90

20

50 E0

qS = 3P – 10

qD’ = –2P + 140

E180

7030

140


� A demand for beef is given by qD = -7p + 1000, while the supply for beef is qS = 5p – 200.

� In the equilibrium: qD = qS, hence p* = 100 and q* = 300.

� Now suppose government introduces a specific tax of t pence per kilo sold. Suppose t = 25.The tax bill now depends on the quantity of beef sold, not the price.

� To analyse the effects of the tax, we must first make a distinction between the market price, p, and the price actually received by the sellers, p’.

� A seller who sells his product at the market price, p, has to pay a tax of t pence and, therefore, for every unit he sells, he receives p’ = p –t.


� Given the tax, the willingness of the sellers to supply beef nowdepends on p’ rather than p.

� We rewrite the supply function as: qS’ = 5p’ – 200.

� Collecting all the equations of the modified model, we have:qD = -7p + 1000 (1)qS’ = 5p’ – 200 (2)qD = qS’ (3)p’ = p – t (4)t = 25 (5),

where t = 25 is the assumed tax rate per kilo of beef sold. We have a linear system of 5 equations and 5 unknowns (qD, qS’, p, p’, t).


� To solve this system, first replace (5) into (4):

p’ = p – t = p – 25 (4’)

� We can then substitute (4’) into (2):

qS’ = 5(p – 25) – 200 = 5p – 125 – 200 (2’)

Note that (2’) has the same slope as the original supply function qS = 5p – 200. However, we can see that the supply, given any price, is lower in (2’) than in the original equation.

The graphs are parallel but (2’) has a smaller y-intercept, and has shifted down.


It remains to set qD = qS’ and solve for p: p = 110,4 (price paid by consumers).

p’ = 110,4 – 25 = 85,4 (price received by suppliers)

The corresponding equilibrium quantity: q* = 227,1.


1000

227,1

-200

qS = 5p – 200

qS’ = 5p – 125 – 200

qD = -7p + 1000

p

q

142, 9

300

110,4

E0

E1

Tax revenue

85,4 100


� The tax shifts the supply function down by 125 units

� Quantity has fallen from 300 to 227,1 and market price has increased from 100 to 110,4.

� The tax burden is shared by the consumers, who pay more and producers, who receive less than before the tax was introduced.

� The shaded area indicates the tax revenue accruing to the government.

Solving for the Market Equilibrium: A

general case� Consider the three equations in a demand and supply

model:qD = -ap + b (1)

qS = cp + d (2)qD = qS (3)

� From the mathematical point of view, we have a system of three linear equations and three unknowns (qD, qS, p).

� To solve this is simple. When (3) holds, then we can set the right-hand sides of (1) and (2) equal to each other and solve for p.

Solving for the Market Equilibrium

� Set

-ap + b = cp + d (4)

Equation (4) is a simple linear equation with only one variable, p, and we can easily solve it:

ca

dbp

dbacp

bdapcp

+

−=⇒

−=+

−=−−

*

)(


� p* now denotes the equilibrium value of the price in the

market, that is, the price at which quantity demanded is

equal to the quantity supplied.

� To obtain the equilibrium quantity, substitute p* into the

demand (or supply) function:

( ) ( )

ca

bcadq

ca

bcbaadabq

ca

cab

ca

dbab

ca

dbaq

bapq

D

D

D

+

+=⇒

+

+++−=⇒

+

++

+

−−=+

+

−−=⇒

+−=

*


The solution to the system is:

ca

adbcq

ca

dbp

+

+=

+

−= *,*

We started with three equations and three variables. But one of

the equations in our system simply says that qD=qS. Thus, at the

solution of our system qD = qS = q*.

Graphically, the solution p* and q* represents the coordinates of the point where the demand and supply intersect.

Inverse Demand and Supply

� So far we have treated the quantity as the dependent variable and plotted it on the vertical axis of the graph.

� The convention in economics is generally to place p, the price, on the vertical axis when plotting demand and supply functions.

� Following the convention in economics, we need to rearrange the demand and supply functions so as to isolate p on the left-hand side of the equation.


Recall our example of the market for beef.

Rearranging the demand function qD = -7p + 1000 gives:

7

1000

7+−=

Dqp This function is called the inverse demand function.

405

+=S

qp

Rearranging the supply function, qS = 5p – 200, we obtain:

This function is called the inverse supply function.


� We can use the inverse demand and supply functions to

solve for the market equilibrium

� We have the two inverse functions,

405

,7

1000

7

+=

+−=

S

D

qp

qp

together with the familiar market equilibrium condition: qD = qS.

Hence, we have a system of three linear equations with three unknowns (qD, qD, p).


Since qD = qS in the equilibrium, we can replace one of the variables with the other. For example, rewrite the inverse supply function:

405

+=

Dq

p

We can then set: ,4057

1000

7+=+−

DD qq

and solve this linear equation with only one unknown.

100*300*

360012

1400750005

=⇒=⇒

−=−

+=+−

pq

q

qq

D

DD


40

p

q

p = qS/5 + 40

p = -qD/7 + 1000/7

1000

142,9

300

100 E

Macroeconomic equilibrium

� Macroeconomics studies the economy as a whole

� In the following we will consider a linear macroeconomic model to find how the equilibrium level of income for the whole economy is determined

� The key relationships of the model are:

1) Y ≡ Q and Q ≡ E.

This means that households earn their incomes (Y) by producing output (Q), and so the aggregate household income, Y, must equalthe value of output, Q. In addition, The output (Q) must be bought by somebody and this creates the expenditure (E).


Combining Y ≡ Q and Q ≡ E gives our first equation

Y ≡ E (1)

Equation (1) is an identity. It means that the aggregate income

and aggregate expenditure are necessarily equal.

2) Aggregate expenditure that consists of households’

consumption expenditure, C, and firms’ investment

expenditure, I.

E ≡ C + I (2)


3) The planned or desired consumption expenditure of households.This is represented by a consumption function, where consumption is a function of income:

Ĉ = aY + b, (3)

where Ĉ is the planned consumption and a and b are positive parameters. Parameter a is called the marginal propensity to consume (MPC) and it is less than 1. The positive slope means that the higher is income, Y, the higher is consumption, Ĉ. As a specific consumption function, consider

Ĉ = 0.5Y + 200

The MPC is 0.5. This means that if the consumer earns an extra £1 he will spend half of it and save the rest.


The planned savings can then be represented by

Ŝ ≡ Y – Ĉ

This is an identity because what is not consumed will be necessarily saved.

Substituting Ĉ from above into the identity gives us the savings function:

Ŝ = Y – (aY + b) = Y(1 – a) – b

In our specific example, the savings function takes the form:

Ŝ = Y – (0.5Y + 200) = 0.5Y – 200


� C and Ĉ are not necessarily equal. C is the actual output and Ĉ is what the consumers would like to consume.

� Whenever C ≠ Ĉ, there will be changes in spending,

which will, in turn, cause changes in income

� Therefore, for an equilibrium, we need

C = Ĉ (4)


� Equations (1) – (4) constitute our macroeconomic model with four equations and five unknowns (Y, E, I, C, Ĉ).

� Because we have more unknowns than equations, the

solution to the system is not unique.

To solve this system, first replace (1) into (2):

Y ≡ C + I (5)


Then plug (4) into (5) to get:

C = 0.5Y + 200 (6)

Then substitute (6) into (5) to obtain:

Y = 0.5Y + 200 + I, and rearrange:

5.0

200

200)5.01(

2005.0

IY

IY

IYY

+=

+=−

+=−

(7)


� Equation (7) expresses Y as a function of I.

� Because there are two unknowns, it doesn’t give us a unique solution.

� Note that the denominator is equal to 1 – MPC = 1 – a.

If a = 1, the denominator is zero.

If a > 1, the denominator is negative.

� Hence, we want a<1.


To obtain a specific value for Y, we need to find additional

information about I.

For simplicity, suppose that I has a fixed value, say I = 500.

Then we have that:

14005.0

500200

5.0

200=

+=

+=

IY

This is the equilibrium level of income. The plans of households to

consume and firms to invest are consistent with the economy’s actual production of these goods.


� Here I is considered as an exogenous variable, because its value comes from outside the model.

� Variables whose values are determined within the

model, such as Y and C, are called endogenous.


Y

Ĉ, E = Ĉ + I

Y = E

Ĉ = 0.5Y + 200

200

700

E = Ĉ + I = 0.5Y + 700

1400

1400

Comparative Statics

Suppose that investment increases to I = 550. This

means that the income increases to

15005.0

550200

5.0

200=

+=

+=

IY

The increase of 50 in investment results in an increase of 100

for income. In fact, any given increase in investment will

increase income twice as much.

This effect is due to the investment multiplier.

Graphically, this corresponds to an upward shift of the

aggregate demand function, E = Ĉ + I.

Y

Ĉ, E = Ĉ + I

Y = E

Ĉ = 0.5Y + 200

200

700

E = Ĉ + I = 0.5Y + 700

1400

1400

E = Ĉ + I = 0.5Y + 750

750

1500

1500

linear equarions - economic applications

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