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Mathematics lectures
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- Mathematics
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- Erasmus Universiteit Rotterdam (EUR)
This document contains all notes of all mathematics lectures and will prepare you for your exam. However, practice is key!
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Notes mathematics International Business AdministrationDemand and supply in economic applications
Demand:
- If the price is €5 per unit, then the demand is 510 units.
- An increase in price of €1 per unit will lead to a decrease in demand of 10 units.
Question: Find mathematical models of the relationships between price and demand
P Q
5 510
6 500
Demand as function of price
Q (P) = mP + c
510 = m × 5 + c -> × -1 -> -510 = -5m -c + -10 =m
500 = m × 6 + c c = 500 – 6m -> c = 560
So: Q (P) = -10P + 560
Price as function of demand
P (Q) = mQ + c
5 = m × 510 + c + -1 = 10m -> m = - 1/10
6 = m × 500 + c -> × -1 -> -6m = -500m -c c = 500 – 6m -> c = 560
So: P (Q) = -1/10 Q + 56
Also:
Q (P) = -10P + 560
-10P = -Q + 560
P = -1/10Q +56
Supply:
- If the price is €5 per unit, then the supply is 510 units.
- An increase in price of €1 per unit will lead to an increase in supply of 10 units.
Question: Find mathematical models of the relationships between price and supply
P Q
5 35
6 50
Demand as function of price
Q (P) = mP + c
35 = m × 5 + c -> × -1 -> -35 = -5m -c + 15 =m
50 = m × 6 + c c = 35 – 5m -> c = -40
So: Q (P) = 15P - 40
Price as function of demand
P (Q) = mQ + c
5 = m × 35 + c + -1 = -15m -> m = 1/15
6 = m × 50 + c -> × -1 -> -6m = -50m -c c = 5 – 7/3 -> c = 8/3
So: P (Q) = 1/15 Q + 8/3
,Also:
Q (P) = 15P – 40
-15P = -Q – 40
P = 1/15Q + 8/3
For further reading: 2.2 mathematical modelling
2.3 Applications: Demand, Supply, Cost, Revenue
For further practicing: progress exercises 2.3, 2.4, 2.5:3
Price Elasticity for linear demand functions
Companies often use a numerical value that represents the sensitivity of the demand for changes in
the price at different price levels.
Price Elasticity of demand: Percentage change in demand : percentage change in price
Or: %ΔQ : %ΔP
Or: (ΔQ : Q) × (P : ΔP)
Or: (ΔQ : ΔP) × (P : Q)
P = Price level of the product
Q = Corresponding demand
ΔP = Change in price level
ΔQ = Corresponding change in demand
Example: Demand function: Q (P) = 560 – 10P
If the price of the product is €30, does 1% increase in price then lead to 1% decrease in demand or to
a larger or smaller percentage.
ΔQ= -10 (because the slope is -10)
ΔP = 1
P = 30
Q (30) = 560 – 10 × 30 = 260
(-10 : 1) × (30 : 260) = -1,15 so a larger percentage
For further reading: Bookchapter 2.6 Elasticity of demand, supply and income
For further practicing: Progress exercises 2.7
Equilibria of linear functions and Governmental Interventions
Demand: Qd (P) = -10P + 560 (P(Q) = -1/10Q + 56)
Supply: Qs (P) = 15P – 40 (P(Q) – 1/15Q + 8/3
It is usual in the economy that the price of a commodity fluctuates and tends to adjust such that the
quantity the supplier is willing to supply is equal to the quantity customers are willing to buy:
When this occurs, we say that the market is in equilibrium.
Supply = demand (Qs = Qd)
Price supplied = buying price (P = P)
,Question: Find the equilibrium price and the equilibrium quantity
-10P + 560 = 15P – 40 -1/10Q + 56 = 1/15Q + 8/3
-25P = -600 -1/6Q = -160/3
P = 24 Q = 320
-10 × 24 + 560 = 320 -1/10 × 320 +56 = 24
(Q, P) = (320, 24) (Q, P) = (320, 24)
Governmental interventions:
Examples of governmental interventions are taxes and subsidies.
Qd = -10P +560
Qs = 15P – 40
Questions:
1. If the government requires a fixed tax of €3 per unit sold, calculate then the new equilibrium price
and the new equilibrium quantity.
2. Which percentage of the tax is paid by the customers and which percentage by the supplier?
3. The government gives a subsidy of €2 per unit sold. How will this influence the market
equilibrium?
1. Qs = 15P + 40 -> P = 1/15Q + 8/3 -> P = 1/15Q + 17/3 (+3)
Qd = -10P + 560 -> p = -1/10Q + 56 (the demand function does not change)
1/15Q + 8/3 = -1/10Q +56
1/6Q = 151/3
Q = 302
1/15 × 302 + 17/3 = 25,8
New equilibrium P = 25,80 and Q = 302
2. Customers pay the equilibrium price:
Without tax: €24
With tax: €25,80
25,80 – 24 = 1,80
(1,80 : 3) × 100% = 60% of the tax
The supplier pays 40% of the tax (€1,20)
3. Qd = -10P + 560 -> p = -1/10Q + 56 (the demand function does not change)
Qs = 15P + 40 -> P = 1/15Q + 8/3 -> P = 1/15Q + 2/3 (-2)
1/15Q + 2/3 = -1/10Q + 56
1/6Q = 166/3
Q = 332
1/15 × 332 + 2/3 = 22,80
Without subsidy: €24
With subsidy: €22,80
24 – 22,80 = 1,20
(1,20 : 2) × 100% = 60%
So the supplier gets 40% of the subsidy and the customer get 60% of the subsidy.
For further reading: bookchapter: 3.2 Equilibrium and break-even
For further practicing: 3.2, 3.3
, Revenue, break-even points and profit in economic
A company produces a product that is sold directly to the customers. Fixed production costs €6000
and variable production costs €30
The demand function of the product is given by: P(Q) = 100 – 0,1Q with P denoting price per unit in
euros. Q denoting the demand.
Revenue = What we earn by selling
TR = total revenue
TC = total costs
TP = total profit
Questions:
1.Give the price per unit of the product giving maximum total revenue, and compute the maximum
total revenue.
2.Give the break-even points
3. Give the price per unit giving maximum profit, and give maximum profit.
1. TR = Q × (100 – 0,1Q)
-0,1Q² + 100Q
Calculate Q-intercepts (red points)
Q² - 1000Q = 0
Q × (Q – 1000) = 0
Q = 0 v Q = 1000
(1000 + 0) : 2 = 500 (green point)
-0,1 × 500² + 100 × 500 = 25.000
2. Break-even points (red points): total revenue = total costs (TR = TC)
TR = -0,1Q² + 100Q and TC = 6000 + 30Q
-0,1Q² + 100Q = 30Q + 6000
-0,1Q² + 70Q – 6000 = 0
Q² - 700Q + 60000 = 0
(Q – 100) × (Q – 600) = 0
Q = 100 v Q = 600
TC and TR are 9000 v TC and TR are 24.000
3. TP = TR – TC
TP = (-0,1Q² + 100Q) – (6000 + 30Q)
TP = -0,1Q² + 70Q- 6000
-0,1Q² + 70Q- 6000 = 0 (red points)
Q² - 700Q + 60000 = 0
(Q – 100) × (Q – 600) = 0
Q = 100 v Q = 600
(100 + 600) : 2 = 350 (green point)
-0,1 × 350² + 70 × 350 – 6000 = 6250
TP = 6250
Price: 100 – 0,1 × 350 = 65
Fur further reading: Bookchapter 4.1:3
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