1. HEC 222 QUANTITATIVE ECONOMIC –
LECTURE 5, SIMULATANEOUS EQUATION
AND ITS APPLICATION
I AM GREAT FAN OF SCIENCE, BUT I CANNOT DO A SIMULATANEOUS
EQUATION.
TERRY PRATCHETT (1706 – 1787)
COMPILED BY: MR. MMUPI
2. Learning Outcomes
The method for solving simultaneous equation has already being introduced in
Lecture 2. These methods are used to determine equilibrium outcomes in various
markets such as the goods, labour and money markets. In addition, national
income equilibrium is analysed and the IS-LM model is introduced. This chapter is
divided into the following sections:
3.1 Equilibrium and break-even
3.2 Consumer and producer surplus
3.3 National income model and the IS-LM model
3. 3.1 Equilibrium and Break-even
At the end of this section you should be familiar with:
• Equilibrium in the goods market and labour market
• Price controls, government intervention in various markets
• Market equilibrium for substitute and complementary goods
• Taxes, subsidies and their distribution between producer and consumer
• Break-even analysis
Introduction
The method of simultaneous equations is now applied to the determination of
equilibrium conditions in various markets; for example the goods, labour and money
markets. In addition, situations are considered that prevent the occurrence of
equilibrium.
Furthermore, the analysis also considers factors that change the state of equilibrium
from one position to another.
Note: A state of equilibrium within a model is a situation that is characterised by a lack
of tendency to change.
4. 3.1.1 Equilibrium in the goods and labour markets
Goods market equilibrium (market equilibrium) occurs when the quantity demanded
(Qd) by consumers and the quantity supplied (QS) by producers of a good or service
are equal.
Equivalently, market equilibrium occurs when the price that a consumer is willing to
pay (Pd) is equal to the price that a producer is willing to accept (Ps). The equilibrium
condition, therefore, is expressed as
Note: In equilibrium problems, once the equilibrium condition is stated, Q and P are
used to refer to the equilibrium quantity and price respectively.
5. WORKED EXAMPLE - GOODS MARKET EQUILIBRIUM
The demand and supply functions for a good are given as
Calculate the equilibrium price and quantity algebraically and graphically.
Solution
Market equilibrium occurs when Qd = Qs and Pd = Ps. Since the functions are written in
the form P = f (Q) with P as the only variable on the LHS of each equation, it is easier to
equate prices, thereby reducing the system to an equation in Q only; hence, solve for Q:
Pd = Ps
100 – 0.5Q = 10 + 0.5Q equating equations (3.2) and (3.3)
100 – 10 = 0.5Q + 0.5Q
90 = Q equilibrium quantity
Now solve for the equilibrium price by substituting Q = 90 into either equation
(3.2) or (3.3):
P = 100 - 0.5(90) substituting Q = 90 into equation (3.2)
P = 55 equilibrium price
6. Cont.
Check the solution, Q = 90, P = 55 by
substituting these values into either
equation (3.2) or (3.3). This exercise
is left to the reader.
Figure 3.5 illustrates market
equilibrium at point E0 with
equilibrium quantity, 90, and
equilibrium price, K55.
The consumer pays K55 for the good
which is also the price that the
producer receives for the good.
There are no taxes (what a
wonderful thought!).
7. 3.1.2: Labour market equilibrium
Labour market equilibrium occurs when the labour demanded (Ld) by firms is equal to
the labour supplied (Ls) by workers or, equivalently, when the wage that a firm is
willing to offer (Ws) is equal to the wage that workers are willing to accept (Wd ).
Labour market equilibrium, therefore, is expressed as
Again, in solving for labour market equilibrium, once the equilibrium condition is
stated, L and w refer to the equilibrium number of labour units and the equilibrium
wage respectively.
8. WORKED EXAMPLE - LABOUR MARKET EQUILIBRIUM
The labour demand and supply functions are given as
Labour demand function: Wd = 9 - 0.6Ld
Labour supply function: Ws = 2 + 0.4Ls
Calculate the equilibrium wage and equilibrium number of workers algebraically and
graphically. (In this example 1 worker Ξ 1 unit of labour.)
Solution
Labour market equilibrium occurs when Ld = Ls and wd = ws. Since the functions are
written in the form w = f(L), equate wages, thereby reducing the system to an equation in L
only; hence, solve for L:
Wd = Ws
9 - 0.6L = 2 + 0.4L equating equations (3.5) and (3.6)
9-2 = L
7 = L equilibrium number of workers
Now solve for w by substituting L = 7 into either equation (3.5) or (3.6):
W - 9 - 0.6(7) substituting L = 7 into equation (3.5)
W = 4.8 equilibrium wage
9. Cont.
Figure 3.6 illustrates labour market equilibrium at point E0 with equilibrium number of
workers, 7, and equilibrium wage, K4.80. Each worker receives K4.80 per hour for his
or her labour services which is also the wage that the firm is willing to pay.
10. 3.2.3 Price controls, government intervention in
various markets
In reality, markets may fail to achieve market equilibrium due to a number of
factors; for example, the intervention of governments, the existence of firms with
monopoly power.
Government intervention in the market through the use of price controls is now
analysed.
Price ceilings
Price ceilings are used by governments in cases where they believe that the
equilibrium price is too high for the consumer to pay. Thus, price ceilings operate
below market equilibrium and are aimed at protecting consumers. Price ceilings
are also known as maximum price controls, where the price is not allowed to go
above the maximum or 'ceiling' price (for example, rent controls or maximum price
orders).
11. WORKED EXAMPLE - GOODS MARKET EQUILIBRIUM AND PRICE CEILINGS
The demand and supply functions for a good are given by
(a) Analyse the effect of the introduction of a price ceiling of K40 in this market.
(b) Calculate the profit made by black marketeers if a black market operated in this
market.
Solution
(a) The demand and supply functions are the same as those in Worked Example 3.7
where the equilibrium price and quantity were K55 and 90 units respectively. The price
ceiling of K40 is below the equilibrium price of K55. Its effect is analysed by comparing
the levels of quantity demanded and supplied at P = K40.
13. Cont.
Since the quantity demanded (Qd = 120) is greater than the quantity supplied (QS = 60),
there is an excess demand (XD) of: XD = Qd – Qs = 120 – 60 = 60.
This is also referred to as a shortage in the market. It is illustrated in Figure 3.7. (b) The
existence of price ceilings often leads to the establishment of black markets where goods
are sold illegally at prices above the legal maximum.
Black marketeers would buy the 60 units supplied at the controlled price of £40 per unit.
However, as there is a shortage of goods, consumers are willing to pay a higher price for
these 60 units. The price that consumers are willing to pay is calculated from the demand
function for Q = 60. Substitute Q = 60 into the demand function:
So, Pd = 70 is the price consumers are willing to pay. Therefore, black marketeers buy the 60
units at the maximum price of £40 per unit, costing them 60 x £40 = £2400, and then sell these
60 units at £70 per unit, generating revenue of 60 x £70 = £4200. Their profits (TT) is the
difference between revenue and costs:
15. Price Floors
Price floors are used by governments in cases where they believe that the equilibrium price
is too low for the producer to receive. Thus, price floors operate above market equilibrium
and are aimed at protecting producers. Price floors are also known as minimum prices,
where the price is not allowed to go below the minimum or 'floor' price (for example, the
Common Agricultural Policy (CAP) in the European Union and minimum wage laws).
WORKED EXAMPLE - LABOUR MARKET EQUILIBRIUM AND PRICE FLOORS
Given the labour demand and supply functions as:
Labour demand function: Wd = 9 - 0.6Ld (3.9)
Labour supply function: Ws = 2 + 0.4Ls (3.10)
Analyse the effect on the labour market if the government introduces a minimum wage law of
£6 per hour.
Solution
The labour demand and supply functions are the same as those in Worked Example 3.8
where the equilibrium wage and units of labour were £4.80 per hour and 7 labour units
respectively. The minimum wage law (price floor) of £6 is above market equilibrium. Its effect
is analysed by comparing the levels of labour demanded and supplied at w = 6.
16. Cont.
Since labour supplied (Ls = 10) is greater than labour demanded (Ld = 5), ther is an
excess supply of labour of XS = Ls - Ld = 10 - 5 = 5. This is also referred to as a surplus,
that is, there is unemployment in the labour market. The graphical illustration of this
result is left to the reader.
17. 3.2.4 Market equilibrium for substitute and
complementary goods
Complementary goods are goods that are consumed together (for example, cars
and petrol, computer hardware and computer software). One good cannot
function without the other.
On the other hand, substitute goods are consumed instead of each other (for
example, coffee versus tea; bus versus train on given routes).
The general demand function is now written as,
That is, the quantity demanded of a good is a function of the price of the good itself
and the prices of those goods that are substitutes and complements to it.
Note: In this case, Ps refers to the price of substitute goods, not to be confused with Ps
which is used to refer to the supply price of a good.
Consider two goods, X and Y. The demand function for good X is written differently
depending on whether good Y is a substitute to X or a complement to X .
19. WORKED EXAMPLE - EQUILIBRIUM FOR TWO SUBSTITUTE GOODS
Find the equilibrium price and quantity for two substitute goods X and Y given their
respective demand and supply equations as.
Solution
The equilibrium condition for this two-goods market is
23. 3.2.5 Taxes, subsidies and their distribution
Taxes and subsidies are another example of government intervention in the market. A tax
on a good is known as an indirect tax. Indirect taxes may be either:
• A fixed amount per unit of output (excise tax); for example, the tax imposed on petrol and
alcohol. This will translate the supply function vertically upwards by the amount of the tax.
• A percentage of the price of the good; for example, value added tax. This will change the
slope of the supply function. The slope will become steeper since a given percentage tax will
be a larger absolute amount the higher the price.
Fixed tax per unit of output
When a tax is imposed on a good, two issues of concern arise:
• How does the imposition of the tax affect the equilibrium price and quantity of the good?
• What is the distribution (incidence) of the tax, that is, what percentage of the tax is paid by
consumers and producers respectively?
In these calculations
• The consumer always pays the equilibrium price.
• The supplier receives the equilibrium price minus the tax.
24. WORKED EXAMPLE - TAXES AND THEIR DISTRIBUTION
The demand and supply functions for a good are given as
Demand function: Pd = 100 - 0.5Qd (3.17)
Supply function: Ps = 10 + 0.5Qs (3.18)
(a) Calculate the equilibrium price and quantity.
(b) Assume that the government imposes a fixed tax of £6 per unit sold.
(i) Write down the equation of the supply function, adjusted for tax.
(ii) Find the new equilibrium price and quantity algebraically and graphically.
(iii) Outline the distribution of the tax, that is, calculate the tax paid by the consumer
and the producer.
Solution
(a) The equilibrium quantity and price are 90 units and £55 respectively.
Remember: the equilibrium price of £55 (with no taxes) means that the price the
consumer pays is equal to the price that the producer receives.
25. Cont.
(i) Remember: Translations, Chapter 2.
The tax of £6 per unit sold means that the effective price received by the producer is (Ps —
6). The equation of the supply function adjusted for tax is
Ps – 6 = 10 + 0.5Q
Ps =16 + 0.5Q (3.19)
The supply function is translated vertically upwards by 6 units (with a corresponding
horizontal leftward shift). This is illustrated in Figure 3.8 as a line parallel to the original supply
function.
(ii) The new equilibrium price and quantity are calculated by equating the original demand
function, equation (3.17), and the supply function adjusted for tax, equation (3.19):
Pd = Ps
100 - 0.5Q= 16 + 0.5Q equating equations (3.17) and (3.19)
Q = 84
Substitute the new equilibrium quantity, Q = 84 into either equation (3.17) or equation
(3.19) and solve for the new equilibrium price:
P = 100 - 0.5(84) substituting Q = 84 into equation (3.17
P = 58
26. Cont.
The point (84, 58) is shown as point E1 in Figure 3.8.
(iii) The consumer always pays the equilibrium price, therefore the consumer pays £58, an
increase of £3 on the original equilibrium price with no tax, which was £55.
This means that the consumer pays 50% of the tax. The producer receives the new
equilibrium price, minus the tax, so the producer receives £58 — £6 = £52, a reduction
of £3 on the original equilibrium price of £55. This also means that the producer pays
50% of the tax.
In this example, the tax is evenly distributed between the consumer and producer.
The reason for the 50 : 50 distribution is due to the fact that the slope of the demand
function is equal to the slope of the supply function (ignoring signs). This suggests that
changes in the slope of either the demand or supply functions will alter this
distribution.
28. Subsidies and their distribution
Similar ideas may be analysed with respect to subsidies and their distribution. In the
case of subsidies, one would be interested in analysing how the benefit of the subsidy
is distributed between the producer and consumer.
In the analysis of subsidies, a number of important points need to be highlighted:
• A subsidy per unit sold will translate the supply function vertically downwards, that is,
the price received by the producer is (P + subsidy).
• The equilibrium price will decrease (the consumer pays the new lower equilibrium
price). The price that the producer receives is the new equilibrium price plus the subsidy.
• The equilibrium quantity increases.
29. WORKED EXAMPLE - SUBSIDIES AND THEIR DISTRIBUTION
The demand and supply functions for a good (f P per ton of potatoes) are given as
Demand function: Pd = 450 - 2Qd (3.20)
Supply function: Ps = 100 + 5Qs (3.21)
(a) Calculate the equilibrium price and quantity.
(b) The government provides a subsidy of £70 per unit (ton) sold:
(i) Write down the equation of the supply function, adjusted for the subsidy,
(ii) Find the new equilibrium price and quantity algebraically and graphically,
(iii) Outline the distribution of the subsidy, that is, calculate how much of the subsidy is
received by the consumer and the supplier.
Solution
(a) The solution to this part is given over to the reader. Show that the equilibrium quantity
and price are 50 units and £350 respectively.
30. Cont.
(b) (i) With a subsidy of £70 per unit sold, the producer receives (Ps + 70). The equation of the
supply function adjusted for subsidy is
Ps + 70 = 100 + 5Q
Ps = 30 + 5Q (3.22)
The supply function is translated vertically downwards by 70 units. This is illustrated in Figure
3.9 as a line parallel to the original supply function. (ii) The new equilibrium price and
quantity are calculated by equating the original demand function, equation (3.20), and the
supply function adjusted for the subsidy, equation (3.22):
Pd = (Ps + subsidy)
450 - 2Q = 30 + 5Q equating equations (3.20) and (3.22)
Q = 60
Substitute the new equilibrium quantity Q = 60 into either equation (3.20) or equation (3.22)
and solve for the new equilibrium price:
P = 450 - 2Q
P = 450 - 2(60) substituting Q = 60 into equation (3.20)
P = 330
31. Cont.
The point (P = 330, Q = 60) is shown as
point E1 in Figure 3.9.
(iii) The consumer always pays the
equilibrium price, therefore, the consumer
pays £330, a decrease of £20 on the
equilibrium price with no subsidy (£350).
This means that the consumer receives
20/70 of the subsidy. The producer
receives the equilibrium price, plus the
subsidy, so the producer receives £330 +
£70 = £400, an increase of £50 on the
original price of £350. The producer
receives 50/70 of the subsidy.
In this case, the subsidy is not evenly
distributed between the consumer and
producer; the producer receives a
greater fraction of the subsidy than the
consumer. The reason? The slope of the
supply function is greater than the slope
of the demand function (ignoring signs).
33. WORKED EXAMPLE - CALCULATING THE BREAK-EVEN POINT
The total revenue and total cost functions are given as follows:
TR = 3Q (3.23)
TC = 10 + 2Q (3.24)
(a) Calculate the equilibrium quantity algebraically and graphically at the breakeven point,
(b) Calculate the value of total revenue and total cost at the break-even point.
Solution
(a) The break-even point is algebraically solved by equating total revenue, equation (3.23), and total
cost, equation (3.24):
3Q= 10 + 20
Q= 10
The equilibrium quantity at the break-even point is Q = 10. This is illustrated in Figure 3.10.
(b) The value of total revenue and total cost at the break-even point is calculated by substituting Q=10
into the respective revenue and cost functions:
TR = 3Q - 3(10) = 30
TC= 10 + 2Q = 10+ 2(10) = 30
At Q = 10, TR = TC = 30.
