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Lecture Notes 5 - Elasticities and Taxation

Elasticities and Taxation
Module

Economics Principles 1 (L1099)

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Introduction to Economics

Lecture Five

Demand, Supply & Elasticities - A More Formal Treatment

5 Introduction

Graphs are not the only devices that are useful in the study of economics. Indeed, in complicated problems, mathematical equations are often not only necessary but also more illuminating. The example to be used here is not all that complicated but captures the essence of what the mathematical approach offers to economic modelling. This lecture will also introduce the concept of the elasticity, which plays a fundamental role in economic policy.

5 Demand and Supply Equations

In this example we are going to formally specify a demand and a supply equation, graph the corresponding curves and solve mathematically for the equilibrium price and quantity. We will use a hypothetical market for beef as the application. We specify the quantity of beef per day (in thousands of tons) as a function of the price per ton of beef (in £s). The following is a formal mathematical statement of the relationship between the demand for beef (Qd) and its price per ton (P):

Qd = 100 - 2P

If P=0 then Qd = 100; If P=50 then Qd = 0.

The price of £50 per ton is the ‘choke’ price.

We can take this information and translate it into figure 5 (a). The 50 represents the vertical intercept of the demand curve (i., where the demand curve cuts the price axis) and 100 represents the horizontal intercept (i., where the demand curve cuts the quantity axis). Indeed, in the equation above, 100 represents the intercept for the demand equation. Note that the curve captures the inverse relationship between price and quantity and is downward sloping. The number (-2) attached to the price variable (P), which is called the coefficient, conveys the fact that there is a negative relationship between demand (Qd) and price (P). What’s the interpretation of -2? This number shows how much the quantity of beef demanded (in thousands of tons per day) changes in response to a unit change in the price per ton of beef. The unit change in this case is a £1 change. This coefficient suggests that a unit increase in price (i., a £1 increase in the price of beef per ton) reduces the quantity demanded of beef by 2 units (i., 2,000 tons of beef). This can be illustrated very easily by reference to the mathematical equation of the demand curve. If the price of beef is £10 per ton, the quantity demanded can be calculated from the equation as:

Qd = 100 - 210 = 80

If the price rises from £10 to £11 per ton, the new quantity demanded is

Qd = 100 - 211 = 78

The increase in price from £10 to £11 per ton leads to a reduction in the daily demand for beef of 2 units (or 2,000 tons of beef).

The coefficient on the price variable (which is -2 in this case) is the slope or gradient of the demand curve. It is important to remember that the slope or gradient is dependent on the units of measurement

used for the price of beef (£’s) and the quantity (thousands of tons) of beef. If we changed the price to pence and the quantity units to 100s rather than 1000s, the slope or gradient would change in value. This poses problems when we wish to compare across commodities but fortunately economists have developed a device for dealing with this type of problem. We will discuss this device later.

We can now turn to the supply curve. The following is a formal mathematical statement of the relationship between the supply of beef (Qs) and its price per ton (P):

Qs = -10 + 0

If Qs = 0 then P = 20.

P=20 represents the vertical intercept for the supply curve. This is where it cuts the price axis. Because the supply curve is positively sloped (note the coefficient of 0 on P), the supply curve has no horizontal intercept. The price value at the vertical intercept gives an indication of the minimum price needed to induce beef farmers to supply their beef to the market. At £20 per ton or below, no beef is provided in the market. We can take this information and translate it into a graph of the supply curve as we have done in figure 5 (b).

The positive slope or gradient captures the positive relationship between the supply of beef and its price. If the price of beef rises by one unit (i., by £1 per ton), the supply of beef rises by 0 of a unit per day (i., by 500 tons of beef per day). We can demonstrate this using the supply equation. Assume the price of beef is £30; then the supply is obtained by using the equation for the supply curve as:

Qs = -10 + 030 = 5

If the price rises from £30 to £31 the supply response is

Qs = -10 + 031 = 5.

The increase in supply is of the order of 0 of a unit or 500 tons of beef per day.

It should again be stressed that the slope or gradient of the supply curve, as with the demand curve, is dependent on the units of measurement.

The equilibrium for this market can be found by reference to figure 5 (c). There is an intersection of the demand and the supply curve at point A (i., the market equilibrium). From this diagram, we can see that the equilibrium price lies somewhere between £40 and £50 per ton and the equilibrium quantity lies somewhere between 10 and 20 thousand tons of beef per day. Graph paper could help us find out approximately where but a more precise answer can be provided by using some basic mathematics. We have two equations (demand and supply) and two unknowns (price and quantity).

Since we know that in equilibrium demand equals supply, we could set the demand equation and the supply equation equal to each other and solve for the resultant price. Thus

Qd = Qs which implies

100 - 2P = -10 + 0P

procedure adopted by economists is to take proportionate or percentage changes for each response. Accordingly, the responsiveness of demand to a price change is given by:

Responsiveness =

Propor tiona te Cha nge in Q ua ntit y Dema nded Propor tiona te Cha nge in P rice

Responsiveness =

Percentage Cha nge in Q ua ntit y Dema nded Percentage Cha nge in P rice

Responsiveness =

Q Q

P P

This responsiveness measure is known as the price elasticity of demand. In formal notational terms, this can be written as:

 = -

Q Q

P P

 = -

Q

P

P

Q

The negative sign occurs because in demand analysis quantity and price move in opposite directions. However, economists sometimes report the elasticity measure in absolute terms. This responsiveness measure represents what is called the point elasticity of demand.

More frequently we measure responsiveness using what is called the arc elasticity of demand. This uses two points on the demand curve. If we use the conventional responsiveness formula above, we would get different results depending on whether we induced a price rise or a price fall. The absolute changes in price and quantity are the same size regardless of direction. However, the original price is higher when moving down the demand curve than when moving up the demand curve. And the original quantity is greater when moving up the demand curve than when moving down the demand curve. For example, assume we were at a price of £10 and we move down to £8. The percentage change in prices is affected by from where we start. In this case, the percentage change is:

[[8 – 10]/10]100 = –20%.

However, if we start at £8 and increase the price to £10, the percentage change is

[[10 – 8]/8]100 = + 25%.

Thus, although the absolute size of the price change (£2) is the same, the percentage change in price is different depending on whether you are starting at £8 and rising to £10, or starting at £10 and falling to £8. A similar argument obviously attaches to quantity changes.

To avoid this outcome in the calculation we use the average of the two prices and the average of the two quantities. Assume the two prices are P1 and P2 and the corresponding quantities are Q1 and Q2. The formula for arc elasticity of demand is given by:

 =

Q

P

(P1+P2)/

(Q1+Q2)/2=

Q

P

(P1+P2)

(Q1+Q2)

as the 2s cancel out.

In a linear demand curve, the price elasticity of demand is different at different points on the demand curve. It can range from zero to a very large or infinite value. For instance, in figure 5(c) at P=50, the price elasticity of demand is infinite and at P=0 it is zero. Thus, as we move down the linear (or straight-line) demand curve, the price elasticity of demand falls in absolute terms. In addition, if  > 1, demand is said to be elastic, if  < 1 demand is said to be inelastic, and if  = 1 demand is said to be unitary elastic.

We can take the foregoing outcomes in turn. If demand is elastic, this means that quantity demanded rises more than the proportionate change in price. An example of this could be demand rising by 10% in response to a 5% fall in price. If demand is inelastic, this means that the quantity demanded rises less than the proportionate change in price. An example of this could be demand rising by 5% in response to a 10% fall in price. If demand is unitary elastic, this means that the quantity demanded rises by the same proportionate change as the price. An example of this could be demand rising by 5% in response to a 5% fall in price.

consumers to notice the price change and respond to it. Thus, the higher the share, the more elastic the response; and the smaller the share the less elastic the response.

Applied econometricians have devoted a lot of effort to computing demand and supply elasticities for a large number of commodities. These estimates provide an important insight into the relationship between price and quantity and are of importance to consumers, producers and the government who, as has already been noted, use such estimates to formulate tax policy. For example, the following long- run estimates for the price elasticity of demand were obtained for a number of commodities in the United Kingdom:

Dairy Produce -0. Bread & Cereals -0. Entertainment -1. Foreign Travel -1. Catering -2. Alcohol -0.

In this case we can see that entertainment and foreign travel are very responsive to price changes but that bread & cereals and dairy produce are not. The reason for this is that close substitutes are available for the first two and more imperfect substitutes only available for the last two.

The price elasticities of supply are determined by a number of factors. These include

a) whether the firm is operating below full capacity or not;

b) whether the firm has available stocks;

c) whether the production process is characterized by factor mobility and factor substitution;

d) the length of time in which producers can adjust to price changes.

The greater a), b) and c) are the more elastic the response. The longer the time allowed for adjustment (point d)), the greater the price elasticity of supply. This is because it takes time for producers to bid resources away from other activities to use and expand in their own production. Thus, the longer the time period, the longer existing firms have to devise new ways of increasing production. In addition, over a longer time period, more resources can flow into an industry through the expansion of existing firms or through the entry of new firms. The rented accommodation market provides a good illustration of this. If the price of renting accommodation increases, there is no time allowed for adjustment in the short-run. Over time landlords can find ways of increasing the amount of accommodation available (e., re-developing existing units to allow more accommodation units to be built) thus leading to a greater supply response in the long-run relative to the short-run.

US econometricians estimated the following two supply elasticities:

Green Peppers 0. Spinach 4.

In the above case, land resources used in the production of spinach are less specific than those used in the production of green peppers. In particular, spinach can be grown on almost all kinds of land, green

peppers can’t. This partly explains the dramatic difference in values.

5 Income and Cross-Price Elasticities of Demand

a) We can also compute an income elasticity of demand. This is the percentage change in quantity demanded divided by the percentage change in income and captures the responsiveness of demand to income changes. This can be expressed as:

 = Income Elasticity of demand =

Percentage Change in Quantity Demanded

Percentage Change in Income

Goods can be classified on the basis of this elasticity. For example, if  > 0, the good is a normal

good; if  < 0, the good is an inferior good; if  > 1, the good is a luxury good, and if 0 <   1, the

good is a necessary good. The following empirical estimates have been obtained for income elasticities of demand using data from the United Kingdom:

Foreign Travel 1. Wines and Spirits 2. Dairy Produce 0. Recreational Goods 1.

b) The cross price elasticity of demand for good Y with respect to good X provides us with an indication of how the quantity demanded of good Y responds to a price change of good X.

This is defined as:

 = Cross-Price Elasticity of Demand =

Percentage Change in Quantity Demanded of Good Y

Percentage Change in Price of Good X

This elasticity allows us to classify goods into substitutes ( > 0) or complements ( < 0). The following empirical estimates have been obtained for cross-price elasticities of demand using data from the United Kingdom:

Food

Clothing & Footwear 0. Travel 0.

These estimates give the responsiveness of the demand for clothing & footwear, and travel to changes in food prices. The estimates appear to suggest that clothing & footwear and food, and travel and food are substitutes – albeit weak substitutes given the magnitude of the elasticity.

Figure 5 (c): The Market for Beef

Figure 5(a): Price Elasticity of Demand

Figure 5(b): Price Elasticity of Demand

Figure 5(c): The Market for Beef

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Lecture Notes 5 - Elasticities and Taxation

Module: Economics Principles 1 (L1099)

28 Documents
Students shared 28 documents in this course

University: University of Sussex

Was this document helpful?
Introduction to Economics
Lecture Five
Demand, Supply & Elasticities - A More Formal Treatment
5.1 Introduction
Graphs are not the only devices that are useful in the study of economics. Indeed, in complicated
problems, mathematical equations are often not only necessary but also more illuminating. The
example to be used here is not all that complicated but captures the essence of what the mathematical
approach offers to economic modelling. This lecture will also introduce the concept of the elasticity,
which plays a fundamental role in economic policy.
5.2 Demand and Supply Equations
In this example we are going to formally specify a demand and a supply equation, graph the
corresponding curves and solve mathematically for the equilibrium price and quantity. We will use a
hypothetical market for beef as the application. We specify the quantity of beef per day (in thousands
of tons) as a function of the price per ton of beef (in £s). The following is a formal mathematical
statement of the relationship between the demand for beef (Qd) and its price per ton (P):
Qd = 100 - 2P
If P=0 then Qd = 100;
If P=50 then Qd = 0.
The price of £50 per ton is the ‘choke’ price.
We can take this information and translate it into figure 5.1 (a). The 50 represents the vertical intercept
of the demand curve (i.e., where the demand curve cuts the price axis) and 100 represents the
horizontal intercept (i.e., where the demand curve cuts the quantity axis). Indeed, in the equation
above, 100 represents the intercept for the demand equation. Note that the curve captures the inverse
relationship between price and quantity and is downward sloping. The number (-2) attached to the
price variable (P), which is called the coefficient, conveys the fact that there is a negative relationship
between demand (Qd) and price (P). What’s the interpretation of -2? This number shows how much
the quantity of beef demanded (in thousands of tons per day) changes in response to a unit change in
the price per ton of beef. The unit change in this case is a £1 change. This coefficient suggests that a
unit increase in price (i.e., a £1 increase in the price of beef per ton) reduces the quantity demanded of
beef by 2 units (i.e., 2,000 tons of beef). This can be illustrated very easily by reference to the
mathematical equation of the demand curve. If the price of beef is £10 per ton, the quantity demanded
can be calculated from the equation as:
Qd = 100 - 210 = 80
If the price rises from £10 to £11 per ton, the new quantity demanded is
Qd = 100 - 211 = 78
The increase in price from £10 to £11 per ton leads to a reduction in the daily demand for beef of 2
units (or 2,000 tons of beef).
The coefficient on the price variable (which is -2 in this case) is the slope or gradient of the demand
curve. It is important to remember that the slope or gradient is dependent on the units of measurement
1

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