Download Demand, elasticities and Consumer theory

Chapter 3 – Demand,
Supply and the Market
Prepared by :
Takesh Luckho
What is a Market?
A market is a mechanism through which
buyers comes in contact with sellers in
order to complete a transaction.
 Market can take the form of

 Any
geographical location (for example
Portobello Road Market in London)
 Telecommunication Networks or Computer
Networks (e.g Forex Markets or Stocks
Markets)
Actual and Effective Demand

Actual Demand and Effective Demand
 Actual
demand is what you want (your ends)
 Effective demand is what you want and this is
backed by the desire/willingness to pay for it.

Hence in Economics when you talk about
demand it mean the effective demand
Types of Demands

Individual Demand – Demand of goods and
services by an individual/household
demand – demand for goods that are jointly
consumed (e.g Car and Petrol)
 Competitive demand – demand competing goods that
can provide the same satisfaction (e.g Tea and
coffee)
 Derived Demand – demand subjective to the demand
of the final product (demand for factors like labour are
derived demands)
 Joint

Market Demand – Demand for all Individuals/
households
Types of Demands

Non Durable and Durable Good demand
good – goods that are perishable (like
vegetables, milk)
 Durable good – goods that can last for long (furniture,
electronics)
 Non-durable

Company demand and Industry demand
demand – demand of a single firm
 Industry demand – demand of all firms in the industry
 Company


Demand for consumer goods (consumed
immediately) and producer goods (capital goods)
Short-Run and Long-Run demand
Demand: Definition



The demand for a good/service is the quantity
that a household wants to buy during a period
of time at a given price, providing this want is
backed by the ability and willingness to pay.
From these information we can write a
mathematical function to represent demand
Qxd = f (Px, Py, Pz, Y, Climate, Taste,
Preferences, Expectations, Population,Etc..)

X is the good, Y is a substitute
 Z is a complement and Y is household income
The Law of Demand



law of demand : Assuming Ceteris paribus
Condition (all other things remain equal), as
price of a good or service rises, its quantity
demanded falls or as the price of a good or
service falls, its quantity demanded increases.
== > There is an inverse relationship between
the price of a good and the quantity demanded
of that good
Demand Function = Qdx = f(Px)
i.e: Qdx = α - β Px
P Q
or P Q
What can explain the downwards sloping
shape of the demand curve?



Law of diminishing marginal utility
Income effect
 Assuming that the price of a good goes down,
you need to spend less money to buy the same
amount of that good
 Your real income goes up and with the same
money in the pocket you can buy more of the
good
Substitution effect
 When price of a good falls, it becomes cheaper
relative to its other competitors
 A rational consumers will shift to the cheaper
substitute
Exceptions to the law of demand
Giffen goods – a special type of inferior
good that does not respect the law of
demand. As price goes up, quantity also
goes up. E.g Discounted products
 Conspicuous consumption – like art or
diamonds (bought by the rich). Its only
when the price goes up that people buy
such product
 Expectation of a future rise in prices
 Emergencies – like in war time period

Demand Schedule
Price
1
5
Total Quantity
Demanded
1
4
2
3
3
2
4
1
5
Demand Curve
Also known as the inverse
demand curve as the diagram is
the inverse of the demand
function Qx =f(P)
6
5
Price
4
3
2
1
D
0
1
2
3
4
Quantity
demanded slope,
The demand curve has
a negative
consistent with the law of demand.
5
Effect of Changes in the
Determinants of Demand:
Movements and Shifts
Movement: Expansion of
demand
(Due to a fall in price)
6
5
A
Price
4
3
B
2
1
D
0
1
2
3
Quantity demanded
4
5
Movement: Contraction of
Demand
(due to a rise in Price)
6
5
A
Price
4
3
B
2
1
D
0
1
2
3
Quantity demanded
4
5
Shift in the Demand Curve
A change in any variable other than price
that influences quantity demanded produces
a shift in the demand curve.
 Factors that shift the demand curve include:

 Change
in consumer incomes
 Taste
 Climate
 Expectations
 Population
 Prices
of related Complements and Substitutes

Prices of related goods
 Complements
- an increase in the price of a
complement reduces the demand of the good, thus
shifting the demand curve to the left.
 Substitutes - an increase in the price of a substitute
increases the demand of the good, shifting the
demand curve to the right.


Income - an increase in income shifts the
demand curve of normal goods to the right.
Number of potential buyers - an increase in
population or market size shifts the demand
curve to the right.
Demand curve shifts to the right
This demand curve has
shifted to the right.
Quantity demanded is
now higher at any
given price.
6
5
Price
4
3
2
1
D
0
1
2
3
Quantity demanded
4
5
Demand curve shifts to the left
This demand curve has
shifted to the left.
Quantity demanded is
now lower at any given
price.
6
5
Price
4
3
2
D
1
0
1
2
3
Quantity demanded
4
5
Important to Remember

A quick recap on Movement and Shift in
the demand curve
 A change
in the price of the good will cause a
movement along the demand curve
 A change in any other variable will lead to a
shift in the demand curve
Market Demand
To get the Market Demand: Horizontal Summation
across the individual Demand Curves
Consumer Surplus
Price
D

Consumer
Surplus
E
P
Actual
Expenditure
O
Consumer surplus is
the difference between
the maximum price a
consumer is willing to
pay and the actual
price he do pay when
buying the product
 CS
Q
Qty
= Total Willing to
pay – Actual Payment
= 0DEQ – OPEQ
= DEP
Supply: Definition



The supply of a good/service is the quantity that
a firm will offer for sale during a period of time
at a given price.
From these information we can write a
mathematical function to represent supply
Qxs = f (Px, Po, Pf, T, Govt, Scale, Objectives,
Expectations, Etc..)

X is the good, O is the price of other goods (rival or
jointly supplied)
 f are the factors of production, T is technology
 Govt = Taxes or subsidy
The Law of Supply
law of supply: Assuming Ceteris Paribus
Condition (other things remaining constant),
as the price of a good rises, its quantity
supplied will rise, and as the price of a good
falls, its quantity supplied will fall.
===>There is a direct relationship between the
price of the good and the quantity supplied of
that good
 In algebra, Qxs = f (Px)
i.e, Qxs = α + βPx

The Law of Supply - cond

Why do producers produce more output
when prices rise?
 They

make higher profits
Exception to the law of Supply
 Non-profit
maximising firms
 Subsistence farming – as price rises farmer can sell
less of the product to get the same revenue and keep
the excess balance for their own consumption

Calculate market supply in the same way
that we get market demand – sum of
supply from all individual firm
Supply Schedule
Price
1
1
Total Quantity
Supplied
1
2
2
3
3
4
4
5
5
Supply Curve
6
5
S
Price
4
3
2
1
0
1
2
3
4
5
Quantity supplied
The supply curve has a positive slope,
consistent with the law of supply.
Market Equilibrium: Determination
Market Price
In economics, an equilibrium is a situation
in which unconstraint variable do not
tend/want to change - that is they are in a
state of rest.
 Hence, market equilibrium occurs where
the free market price has no tendency to
change, assuming ceteris paribus.

 Let
show Market Equilibrium on a demand and
supply diagram and find the market price
Market Equilibrium: Case of a Surplus
The Market clearing process
Price of
tomatoes
(dollars per kg.)

40
Supply
Excess supply
2
3
30
Market equilibrium
25
Demand
0
170
200


Case 1 35
1
At $ 35,
300
Hence 200 unit is said to be the
equilibrium quantity and the market
(equilibrium) price is said to be 30
Qty Demanded: 170
Qty Supply: 300
 Excess supply (surplus in
production) of 130 unit
 To much of the good on
the market will cause the
price to fall.
 Demand will expand,
supply will contract until
both reach 200 unit.
 At $30, neither demand
nor supply want to
change (they have
reached a state of rest
Market Equilibrium: Case of a Shortage
The Market clearing process
Price of
tomatoes
(dollars per kg.)

40
Supply


35
30
Market equilibrium
4 5
6
Case 2 25
Excess demand
Demand
0
At $ 25,
100
200 220
Hence 200 unit is said to be the
equilibrium quantity and the market
(equilibrium) price is said to be 30
Qty Demanded: 220
Qty Supply: 100
 Excess demand
(shortage in production)
of 120
 Shortage of the good on
the market will cause the
price to rise.
 Demand will contact,
supply will expand until
both reach 200 unit.
 At $30, neither demand
nor supply want to
change (they have
reached a state of rest
WHAT IS AN ELASTICITY?

An elasticity is a measure of the sensitivity of
demand or supply to changes in their
determinants.
 In
other words elasticity are said to be measuring the
responsiveness of one variable (i.e one determinant)
to changes in the quantity of the good being
demanded/supplied.


The higher the value of elasticity, the greater will
be the responsiveness of consumers to a
change in the determinant.
Elasticities are often used to show the
steepness/flatness of the demand or supply
curve
Elasticity
3 basic types of demand elasticities:
Price elasticity of demand
 Income elasticity of demand
 Cross elasticity of demand

Price Elasticity of Demand (PED)
Price elasticity of demand measures the
responsiveness of the quantity demanded
of a good to a change in price of that
good, assuming ceteris paribus conditions.
 That is, it measures the rapidity and
volume of the change in the quantity
demand of that good as a response to
change in its selling price.

Computing the Price Elasticity of Demand

The price elasticity of demand is obtained by
dividing the percentage change in quantity
demanded by the percentage change in prices.
PED =
Percentage change in quantity demanded
Percentage change in the price
%  Qty Demanded of good X
PED =
%  in the Price of good X
PED =
Q1  Q0 /Q 0
P1  P0 /P0
x 100
Examples of Own Price
Demand Elasticities
When the price of gasoline rises by 1% the
quantity demanded falls by 0.2%, so
gasoline demand is not very price
sensitive.
Price elasticity of demand is -0.2 .
 When the price of gold jewelry rises by 1%
the quantity demanded falls by 2.6%, so
jewelry demand is very price sensitive.
Price elasticity of demand is -2.6 .

Sign of Price Elasticity
According to the law of demand, whenever
the price rises, the quantity demanded
falls. Thus the price elasticity of
demand is always negative.
 Because PED is always negative,
economists usually state the value without
the sign.


Range of PED: - ∞ ≤ PED ≤ 0 or
0 ≤ PED ≤ ∞
Classifying Demand and Supply
as
Elastic
or
Inelastic
 Demand is said to be elastic if the percentage
change in quantity is greater than the percentage
change in price. (i.e E < -1 or E > 1)
 This
means that the change in the quantity demanded is
more than proportionate to the change in the price level

Demand is inelastic if the percentage change in
quantity is less than the percentage change in
price. (i.e E > -1 or E < 1)
 This
means that the change in the quantity demanded is
less than proportionate to the change in the price level

An Inelastic Demand curve is steeper than an
Elastic Demand Curve.
Elastic Demand
- Elasticity is less than -1
Price
1. A 25% $5
increase
in price... $4
Quantity
50
100
2. ...leads to a 50% decrease in quantity.
Inelastic Demand
- Elasticity is greater than -1
Price
$5
1. A 25%
increase
in price... $4
Quantity
90 100
2. ...leads to a 10% decrease in quantity.
Special cases of Demand Curves

Perfectly Inelastic: PED = 0
 Quantity
demanded does not respond to price
changes.

Perfectly Elastic: PED = - ∞
 Quantity
demanded changes infinitely with any
change in price.

Unit Elastic: PED = - 1
 Quantity
demanded changes by the same
percentage as the price.
Perfectly Inelastic Demand
- Elasticity equals 0
Price
Demand
$5
1. An
increase
in price... 4
Quantity
100
2. ...leaves the quantity demanded unchanged.
Perfectly Elastic Demand
- Elasticity equals minus infinity
Price
At any price
above $4, quantity
demanded is zero.
Demand
$4
At a price of $4,
quantity demanded is infinite.
Quantity
Unit Elastic Demand (Rectangular Hyperbola)
- Elasticity equals 1
Price
1. A 25% $5
increase
in price... $4
Demand
75
100
2. ...leads to a 25% decrease in quantity.
Quantity
Income Elasticity of Demand
(YED)
Income elasticity of demand measures
the responsiveness of demand to a
change in income. That is,how much
the quantity demanded of a good
responds to a change in consumers’
income.
 It is computed as the percentage
change in the quantity demanded of a
good divided by the percentage change
in household/individual income.

Computing Income Elasticity
Percentage change in quantity demanded
YED =
Percentage change in Income
%  Qty Demanded of good X
YED =
%  in Household or Individual Income

Q1  Q 0 /Q 0
YED =
Y1  Y0 /Y0


x 100
If YED > 1 demand is income elastic (Luxury/Superior)
If YED > 0 and < 1 demand is income inelastic
Income Elasticity
- Types of Goods 
Normal Goods (YED > 0)
 Income Elasticity is positive.
 YED > 1  Luxury Good
 YED = between 0 and 1  Necessity

Inferior Goods (YED < 0)
 Income

Elasticity is negative.
Higher income raises the quantity demanded
for normal goods but lowers the quantity
demanded for inferior goods.
Determinants of Income Elasticity
• Goods consumers regard as necessities
tend to be income inelastic
Examples include food, fuel, clothing,
utilities, and medical services.
• Goods consumers regard as luxuries tend
to be income elastic.
Examples include sports cars, furs, and
expensive foods.
• Level of Income the consumer wants to
spend on the good
Cross Elasticity of Demand
(XED)

A measure of the degree of
responsiveness of the demand
for one good (X) to a change
in the price of another good
(Y)
Computing Cross Price Elasticity Of
Demand Between Good X And
Good Y
Percentage change in quantity demanded of good X
XED =
Percentage change in the price of good Y
%  Qty Demanded of good X
XED =
%  in the Price of good Y

Q
XED =
P


 Q x 0 /Q x 0
x 100
y
y
y
1  P 0 /P 0
x
1
Goods which are complements:
Cross Elasticity will have negative
sign (inverse relationship between
the two)
Goods which are substitutes:
Cross Elasticity will have a
positive sign (positive relationship
between the two)
Question: How would you interpret an Xed = 0 ? Any example you have
in mind?
How to calculate Elasticities from a given Equation??

If the demand for A is given by:
QA = 100 - 2PA +PB + 0.5Y

Where PA = 15, PB = 5, Y = 100, Find whether demand is elastic
or not.
•
•
Step 1: Calculate the Value of QA
QA = 100 – 2(15) + 5 + 0.5 (100) = 125
Step 2:Recall: PED = %Change in Qty A/ % Change in
Price A

But when facing an equation, we can rewrite the formula as
follows:
PED = (dQA/QA) / (dPA/PA) = dQA/dPA . PA/QA
•
Step 3: dQA/dPA = the gradient w.r.t Price A = -2
 Hence, PED = dQA/dPA . PA/QA
= -2 . (15/125) = -0.24  Demand is inelastic
Question: Find the Income and Cross Elasticity of Demand and say what
it means??
How can elasticities be useful??
Elasticity and Total Revenue
Total revenue is the amount paid by
buyers and received by sellers of a
good.
 Computed as the price of the good
times the quantity sold.

TR = P x Q
Elasticity and Total Revenue
Price
Total revenue is price x
quantity sold. In this
example, TR = £5 x 100,000
= £500,000.
£5
This value is represented by
the grey shaded rectangle.
Total Revenue
D
100
Quantity Demanded (000s)
Elasticity and Total Revenue
If the firm decides to
decrease price to (say)
£3, the degree of price
elasticity of the demand
curve would determine
the extent of the
increase in demand and
the change therefore in
total revenue.
Price
£5
£3
Total Revenue
D
100
140
Quantity Demanded (000s)
Elasticity and Total Revenue
Producer decides to lower price to attract sales
Price (£)
% Δ Price = -50%
10
Ped = -0.4 (Inelastic)
% Δ Quantity Demanded = +20%
Total Revenue would fall
5
Not a good move!
D
5 6
Quantity Demanded
Rule No 1: If demand is price inelastic, then a price change will affect
total revenue directly – that is, in the same direction as that price
change
Elasticity and Total Revenue
Price (£)
10
Producer decides to reduce price to increase sales
% Δ in Price = - 30%
% Δ in Demand = + 300%
Ped = - 10 (Elastic)
Total Revenue rises
Good Move!
7
D
5
Quantity Demanded
20
Rule No 2: If demand is price elastic, then a price change will affect
total revenue inversely – that is, in the opposite direction to that price
change.
Quick Recap
Inelastic
Demand
Elastic
Demand
Increase in
Price
TR Increases
TR falls
Decrease in
Price
TR Falls
TR Increases
Question: What will happened to Total Revenue following a price rise/fall
when the elasticity of demand is unity (Ped =1)??
DETERMINANTS OF DEMAND
ELASTICITY

Existence of substitutes


The closer the substitutes and the
more substitutes there are, the more elastic is demand is likely to
be.
The Degree of Necessity/addiction

Goods that are terms as necessities tend to be more inelastic

Addictive also tend to have inelastic demand curve

Advertising – The aim of advertising is to show that the good as
being a necessity (when if fact its not).

The Share of Income Spent on the good – if you are spending a
lower proportion of you income of the good then changes in price is
expected to have a little effect on the amount of good being
purchased. Hence PED for these goods are very low.
DETERMINANTS OF
DEMAND ELASTICITY

The length of time allowed
for adjustment

The longer any price change persists, the greater is the elasticity of
demand.

Price elasticity is greater in the long run than in the short run.

Habit forming goods or Brand Loyalty

How to define the short run and the long run

The short run is a time period too short
for consumers to fully adjust to a
price change.

The long run is a time period long enough for consumers to fully
adjust to a change in price, other things constant.
The Theory of Consumer
Behaviour



Consumer theory gives a logical explanation of
how rational consumers behave.
In the real world: Limited Income that forces us
to choose between our unlimited wants/ends
How to choose:
 Cost/Benefits Analysis
 Priority List or Wish List
 Project Appraisal methods

However, these methods depends a lot on value
judgement and knowledge of future inflows of
income.
The Theory of Consumer
Behaviour


In Economics, we make use of a more scientific
approach. We make use of the theory of consumer’s
satisfaction as a decision making criteria.
A Rational Consumer will want to consumer a
product (or bundle of products) that maximises
his/her satisfaction.
 A rational
consumer is one who has clear aims and
acts in the best way to achieve them. He uses his
logic and available information to make the best
decision.

Hence, in order to know the bundle of products to
buy, it is important to measure the level of
consumer’s satisfaction.
How can we measure consumer
satisfaction?


The technical term used in economics for
consumer satisfaction is known as utility
Utility is the satisfaction, pleasure or
need-fulfilment gained from the
consumption of goods or services. Two
ways of measuring utility:

The cardinal approach.
 The ordinal approach.
The Cardinal Approach: Utility
Theory Approach



The cardinal measure approach assumes that
straightforward numerical values (one, two,
three, etc) can be given to levels of satisfaction.
Each good consumed is assumed to generate a
number of units of satisfaction.
So the consumption of a particular good may be
said to be worth, say, nine unit of utility to a
given individual.
The important feature of such cardinal measures
is that they can be subjected to arithmetic
manipulation. Values can be added together,
subtracted, multiplied, etc.
The Ordinal Approach: The
Indifference Curve Approach

The ordinal measure approach involves
utility values taking the form of ‘first best’,
‘second best’, ‘third best’, ‘worst’, etc. In
other words, they can be put in order but
do not carry a value that implies how much
better one item of consumption is
compared to another.
The Utility Theory Approach




Suppose there is a consumer who likes drinking
cola. The amount of satisfaction that he get from
each successive can of the drink is likely to diminish
the more he drink per day.
This occurs as he quench his thirst, enjoy the taste,
but then eventually get sick of more of the same
product.
Let’s say that the first can of cola gives him 20 u of
unit of satisfaction, the second may give him 16 unit,
the third 10 units and so on. The seventh can of the
day could give him negative utility (called disutility)
of –2 utils.
From these information, we can draw up the
following utility schedule:
The Utility Theory Approach
Quantity Consumed per
day
Total
Utility
Marginal
Utility
1
20
20
2
36
16
3
46
10
4
52
6
5
54
2
6
54
0
7
52
-2
(Can of Cola)
Diagram: Total and Marginal Utility
Important to note that Marginal values are plotted
against the midpoint of unit consumed


Total utility is the aggregated units of satisfaction
generated from several units of a particular product
consumed in a given time period; whereas:
Marginal utility is the extra units of satisfaction
generated from one extra unit of a particular product
consumed.



As more and more unit extra unit of the good is being
consumed over this period of time, marginal utility of
consuming this extra unit tend to fall, this is known as the
law of diminishing marginal utility. This explains the
downward sloping nature of the MU curve. (MU = dTU/dX)
The TU curve starts at the origin. Zero consumption
yields zero utility.
TU Curve reaches a peak when marginal utility is
zero
Utility Curves: Two or more goods


How to calculate maximum satisfaction in the
case of two or more goods :
Assuming that the country produced two goods
(let say good X and good Y at respective price
of Px and Py).
 To
get maximum satisfaction, we make use of the
equi-marginal condition
Mu y
Mu x

(assuming that all disposable income is used)
Px
Py
Indifference Curve Approach





Even though the multi-commodity version of marginal
utility theory is useful in demonstrating the underlying
logic of consumer choice, it still has a major weakness.
Utility cannot be measured in any absolute sense. We
cannot really say, therefore, by how much the marginal
utility of one good exceeds another.
An alternative approach is to use indifference analysis.
This does not involve measuring the amount of utility a
person gains, but merely ranking various combinations
of goods in order of preference.
Economist prefer the use of ordinal approach of
consumer’s decision making against the cardinal
approach.
Indifference analysis involves the use of indifference
curves and budget lines
Indifference Curve Approach


An Indifference curve is a line showing all those
combinations of two goods between which a consumer is
indifferent.
For example, let say that in a week, Clive likes to eat 10
pears and 13 oranges. However, he would not mind
giving up 1 or 2 unit of pear to consume more orange
and derive the same satisfaction level. The table below
shows all the combination of pear and orange that Clive
might consumer and attain the same level of satisfaction.

Some assumptions when drawing an
Indifference
Curve
Preferences are transitive

If a persons prefer good x to good y but at the same time prefers
good y to good z, we can say that the consumer prefers good x to
good y

i.e, X > Y (> meaning preferred to) and Y > Z

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More is always preferred to less

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Suppose that with a given amount of Money you can buy two
different bundle/combination of good x and good y as follows:
A(6,4) and B (7,5). You will always buy the higher bundles as its
expected to bring more utility.
Consumers have diminishing marginal utility


then, X > Y and X > Z or X > Y > Z
The Law of diminishing marginal utility helps to determine the
shape of the curve – the IC is convex
Indifferences curves never cross or intercept each other.

This would violate the second assumption where more is always
preferred to less.
An Indifference Curve

All these combinations can be represented
on a simple graph
Slope of the Indifference Curve

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
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The slope of the indifference curve shows the rate at which a consumer is
willing to exchange one good for the other, holding his or her level of
satisfaction the same.
For example, consider the move from point a to point b. Clive gives up 6
units of pears and requires 1 orange to compensate for the loss. The slope
of the indifference curve is thus −6/1 = −6.
Now move from point e to point f, Clive is willing to gives up 2 pears and
requires 2 oranges to compensate. Thus along this section of the curve, the
slope is −2/2 = −1
Slope along the Indifference curve is know as the as the marginal rate of
substitution (MRS). MRSxy = MUx/MUy
Ignoring the negative sign on the slope, it can be seem that the MRS
decrease as we are consuming more and more of pear and less of orange.
As Clive consumes more pears and fewer oranges, his marginal utility from
pears will diminish, while that from oranges will increase. He will be
prepared to give up fewer and fewer pears for each additional orange.
Hence, MRS diminishes.
The law of diminishing marginal rate of substitution states that
individuals will gain less and less additional satisfaction the more of a good
that they consume.
The decreasing MRS gives the shape of a convex Indifference curve.
Indifference Map
An indifference map is the collection of all indifference curves possessed
by an individual. Higher curves on the map represent higher utility levels
e.g: I5>I4>I3…….>I1
How to choose the optimal
Consumption point from an indifference
map?

First we will need a budget line

A budget line shows the income constraint binding on
individual/household at a given price level and Income. The
budget line is given as follows:
PxX  PyY  m


Where m = money income allocated to consumption (after
saving and borrowing)
Px = the price of a specific good
Py = the price of all other goods
x = amount purchased of a specific good
y = amount purchased of all other goods
Px can be re-written as:
The Budget
Line
Y m
X
Py

Second, the optimal consumption bundle
happens at a point where the slope of the
Indifference curve equals to the slope of
the budget line
 Slope
of indifference curve =
MRS = - MUx/MUy
 Slope of budget line = -Px/Py

Hence equilibrium occurs where
 -MUx/MUy = -Px/Py
hence: MUx/MUy = Px/Py
You will note that the marginal
utility theory and the indifference
curve theory give the same results
in terms of equilibrium output
The Optimal Consumption point
At point t
MUx/MUy = Px/Py
THANK YOU !