Some prices from 1987
The Mathematics of Finance
MEI conference 2012
© MEI
Introduction to the Retail Prices
Index (RPI)
A. Mars bar 20p
B. Pint of milk 25p
C. Litre of petrol 38p
D. Paperback book £2.50
E. Glastonbury Festival ticket £21
F. Average house £44 000
• How much do you think these items cost
today?
From the 1956 RPI proposal
• The Retail Prices Index looks at changes in the
price of a shopping basket of goods.
• RPI is short for Retail Prices Index.
• In 1987 the shopping basket cost £100.
• In January 2011 the cost of the same shopping
basket was £229.
• If all prices increased by the same rate from
1987 to 2011, what would each item above
cost?
The weightings in the RPI are based on
household spending
• In 1956, the average household spent
£350 out of every £1000 on food.
• The weightings in the RPI are changed on
a regular basis and new categories
introduced.
• The actual items priced are also kept
under regular review to keep the RPI
representative of average spending.
Collection of prices
• Prices are collected from a sample of
shops.
• Collection takes place on the second or
third Tuesday of each month.
• To allow all the prices to be collected,
some are collected on the Monday or
Wednesday.
1
Calculating the RPI
• The total price in a base year is taken to
be 100.
• The current base year is 1987.
• Indices are calculated in this order
– Item
– Section
– All items
Item index
• January 1987 Mars bar price 20p
• July 2011 Mars bar price 49p
• Price now is 2.45 times higher than in
January 1987.
• Taking the index in Jan 1987 as 100, the
Mars bar index now is 245.
1
A Problem
Mathematical Modelling in
Economics
2
Make assumptions to
allow work to begin
3
Represent the problem
in mathematical form
8
Review assumptions
No
4
Solve to produce
theoretical results
5
Select information from
experiment, experience
or observation
Barack Obama
Only government can break the cycles that are
crippling our economy -- where a lack of spending
leads to lost jobs which leads to even less
spending; where an inability to lend and borrow
stops growth and leads to even less credit.
That's why we need to act boldly and act now to
reverse these cycles. That's why we need to put
money in the pockets of the American people,
create new jobs, and invest in our future.
7
Is the solution
satisfactory?
6
Compare with
theoretical
results
Yes
9
Present findings
The problem
• In an economic crisis, is it better to spend
or to save?
2
Some assumptions
• When a person has money, he either
spends or saves it.
• Each person spends the same fraction of
their money.
A simple example
Suppose everyone spends half their income and
saves half.
Person A earns £100 and spends £50 with
person B.
£50
£100
A
B
£25
C
£12.50
D
The total amount of income is more than A’s
original £100.
Marginal propensity to consume
• The fraction of their income which each
person spends is called the marginal
propensity to consume.
• Suppose the marginal propensity to
consume is 0.5 for each person.
• The next slides show the total income
building.
What will happen to the total
income eventually?
This square represents the initial
increase in income
spent
Income
for others
If marginal propensity to consume
is 0.5; half is spent
If the propensity to consume is 0.5
• Eventually the total income is twice the
intial income.
It is more difficult to see this visually for
other values of the propensity to
consume.
• What happens if the propensity to
consume is some other value?
3
The idea in a nutshell
• Money circulates in an economy.
• If the amount spent in the economy increases,
some of the money spent will be paid to other
people and then be spent by them. So the total
increase in spending is more than the initial
increase in spending.
• The injection is the initial increase in spending.
Multiplying the injection by the multiplier gives
the total increase in spending.
Comparing to reality
• Economists vary in their estimates of the
size of the multiplier.
• It is also very difficult to measure how
much of an impact government spending
has on the economy
A quote
Another problem
“All models are wrong, but some are useful.”
George Box, statistician
• How does demand for a product change
when the price changes?
• Is the economic multiplier model useful?
UK consumption of petrol and diesel (tonnes millions)
How do you think the graph looks?
• For petrol?
• For beer?
• For a product of your
choice?
2006
2007
2008
2009
2010
Petrol
18.14
17.59
16.68
15.76
14.99
Diesel
20.15
21.07
20.61
20.06
20.87
Total
38.29
38.66
37.29
35.82
35.86
Year
2006
Petrol price
(p per litre)
88.9
2007
87.9
2008
103.9
2009
89.9
2010
111.9
4
UK Petrol 2006-2010
A simple demand curve
20 Demand (million tonnes)
19
18
17
16
15
14
13
12
11
10
80
Price (p per litre)
90
100
110
120
130
140
150
Price Elasticity of Demand (PED)
PED =
% change in quantity demanded
Price elasticity of demand =
% change in price
PEDdemand
Geogebra file
% change in quantity demanded
% change in price
In general, the equation of a line of this form is
P = a – bQ .
Can you find a formula for the PED at point A
with coordinates (P, Q), in terms of a, b, P and Q?
HMRC research on price elasticity of
demand for alcohol
UK Alcohol Studies
Beer
Wine
Spirits
Median
Mean
‐0.40
‐0.56
‐0.86
‐0.90
‐0.72
‐0.75
Maximum
Minimum
Std Deviation
‐0.09
‐3.20
0.57
‐0.14
‐2.42
0.57
‐0.20
‐1.60
0.37
A challenge
• What would the demand curve look like if
PED was constant?
• How would revenue depend on price for
such a curve?
5
Maximising Revenue
R = Q(a – bQ) where R is the total revenue.
Why does the graph of R against Q have this shape?
What are the coordinates of A, B and C on the Q-axis?
What value of Q gives a maximum value of R?
P = a − bQ
What is the value of P when R is a maximum?
PED = −
1P
bQ
What is the value of PED when revenue is a maximum?
© MEI 2011