There are occasions in this course where you will need to compute elasticities. There are two formulas used to do this. One is the point formula. The point formula will be most important for this course and is also most commonly in published studies on food demand. The other formula is called the arc or average formula. Let us spend some time on each.
The Point Formula for Demand Elasticities
The general formula for an elasticity can be rearranged algebraically to arrive at the point formula. To illustrate, rewrite the own-price elasticity formula shown above as follows:
\[\epsilon_{11} = \dfrac{\% \Delta Q_{1}}{\% \Delta P_{1}} = \dfrac{(\Delta Q_{1} / Q_{1}) \times 100}{(\Delta P_{1} / P_{1}) \times 100} = \dfrac{\Delta Q_{1}}{\Delta P_{1}} \times \dfrac {P_{1}}{Q_{1}}\]
Point formulas for different kinds of demand elasticities are reported in Table \(\PageIndex{1}\). Note the similarity in the formula for each kind of elasticity. As a practical matter, you need three terms (numbers) to compute an elasticity using the point formula.
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Depending on the type of elasticity, the first number you need is \(\dfrac{\Delta Q_{i}}{\Delta P_{i}}, \dfrac{\Delta Q_{i}}{\Delta P_{j}}, \dfrac{\Delta Q_{i}}{\Delta M}, \: or \: \dfrac{\Delta Q_{i}}{\Delta X}\). These numbers are slope terms from a direct demand relationship. You might remember from algebra class that the slope is the rise over the run. In this case the rise is \(\Delta Q_{i}\) and the run is \(\Delta P_{i}, \Delta P_{j}, \Delta M, \: or \: \Delta X\).
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The second number you need is the value of the demand variable in question \(P_{i}, P_{j}, M, \: or \: X\).
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The third and final number you need is the value of \(Q_{i}\).
Table \(\PageIndex{1}\). Point formulas for demand elasticities.
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Type
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Formula
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Own-price elasticity
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\(\epsilon_{ii} = \dfrac{\Delta Q_{i}}{\Delta P_{i}} \times \dfrac{P_{i}}{Q_{i}}\)
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Cross-price elasticity
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\(\epsilon_{ij}= \dfrac{\Delta Q_{i}}{\Delta P_{j}} \times \dfrac{P_{j}}{Q_{i}}, i \neq j\)
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Income elasticity
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\(\epsilon_{iM} = \dfrac{\Delta Q_{i}}{\Delta M} \times \dfrac{M}{Q_{i}}\)
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Elasticity for other demand shift variable \(X\)
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\(\epsilon_{iX} = \dfrac{\Delta Q_{i}}{\Delta X} \times \dfrac{X}{Q_{i}}\)
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To illustrate the implementation of the point formula, consider the following direct demand equation for good 1:
\[Q_{1} = 1.5A + 0.01M + 2P_{2} -4P_{1}\]
In this equation, \(Q_{1}\) is the quantity of good 1 in thousands of units, \(A\) is advertising, \(M\) is disposable income in dollars, \(P_{2}\) is the price of good 2 in dollars, and \(P_{1}\) is the price of good 1 in dollars. Let us first compute the quantity demanded for the following data point: \((A^{0} = 40, M^{0} = 30000, \(P_{2}^{0} = 20, \: and \: P_{1}^{0} = 60)\). Entering these values into the demand equation provides a quantity demanded of
\[Q_{1}^{0} = 1.5 \times 40 + 0.01 \times 30000 + 2 \times 20 \times 60 = 160 \: thousand \: units\]
You can now use the point formula to obtain elasticity measures. These elasticities are reported in Table \(\PageIndex{2}\) under the heading “Data Point 0”. Take some time to look at the elasticity computations. Make sure you see where each value used in the elasticity computation originates.
Now, repeat the elasticity calculation at another, slightly different data point: \((A^{1} = 40, M^{1} = 30000, P_{2}^{1} = 20, \: and \: P_{1}^{1} = 50)\). At this new data point, let us call it “Data Point 1”, we get a new quantity demanded of
\(Q_{1}^{1} = 1.5 \times 40 + 0.01 \times 30000 + 2 \times 20 - 4 \times 50 = 200 \: thousand \: units\)
Elasticities for this new data point are also reported in Table \(\PageIndex{2}\). Notice that all the point elasticities are different even though the data points themselves differ only in the value of \(P_{1}\). The takeaway here is that the point formula gives an elasticity number at a specific data point and elasticities can depend on point of location along the demand equation. This is one of the reason why caution should be used when using an elasticity to forecast the effects of a large change in one of the demand variables. The elasticity values could be very different at the new point and the old.
Table \(\PageIndex{2}\). Computing elasticities with the point formula.
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Type
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Data Point 0
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Data Point 1
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Own-price elasticity
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\(\epsilon_{11} = -4 \times \dfrac{60}{160} = -1.5\)
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\(\epsilon_{11}= -4 \times \dfrac{50}{200} = 1.0\)
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Cross-price elasticity
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\(\epsilon_{12} = 2 \times \dfrac{20}{160} = 0.25\)
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\(\epsilon_{12} = 2 \times \dfrac{20}{200} = 0.20\)
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Income elasticity
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\(\epsilon_{1M} = 0.01 \times \dfrac{30000}{160} = 1.875\)
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\(\epsilon_{1M} = 0.01 \times \dfrac{30000}{200} = 1.5\)
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Advertising elasticity
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\(\epsilon_{1A} = 1.5 \times \dfrac{40}{160} = 0.375\)
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\(\epsilon_{1A} = 1.5 \times \dfrac{40}{200} = 0.30\)
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In this course, you will primarily be using linear demand specifications, such as the one we used to get the elasticity numbers in Table 3. With linear demands, the elasticity will depend on the point of location along the demand schedule. This is illustrated below in Panel A of Demonstration 3. Panel A shows that demand is elastic at high prices and inelastic at low prices.
The idea of demand being more elastic at higher prices makes intuitive sense and is probably true for most real-world cases. For instance, consumers will be more responsive when the gasoline price increases from $4.00 per gallon to $4.40 per gallon than when the gasoline price increases from $2.00 to $2.20 per gallon. In each case, there is a 10 percent increase in price, but the increase causes more pain at $4.00 than at $2.00. Nevertheless, the fact that elasticities vary along a linear demand schedule, such as that presented in Panel A of Demonstration \(\PageIndex{1}\), is a consequence of using a linear specification for demand.
Panel B of Demonstration \(\PageIndex{1}\), shows an alternative demand specification that provides a constant elasticity. This is a log-log demand specification. The term “log-log”" is used because this type of demand schedule is linear if we take its logarithmic transformation (it is linear in logarithms). With the linear demand curve, the slope does not change. In other words, \(\dfrac{\Delta Q_{1}}{\Delta P_{1}}\) is always the same regardless of location on the demand schedule. However, as shown in Panel A of Demonstration \(\PageIndex{1}\), the own-price elasticity is not constant and will vary between being elastic, unitary elastic, and inelastic as we move from high to low prices. With log-log demand, the slope of the demand schedule changes as we move from high to low prices, but the elasticity is always constant.
The Arc Formula for Elasticities
An alternative formula to compute elasticities is the the arc formula. The arc formula returns the elasticity at the average of two points on the demand equation. Table 4 provides arc elasticity formulas for different demand elasticities.
Table \(\PageIndex{3}\). Arc formulas for demand elasticities.
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Type
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Formula
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Own-price elasticity
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\(\epsilon_{ii} = \dfrac{Q_{i}^{1} - Q_{i}^{0}}{P_{i}^{1} - P_{i}^{0}} \times \dfrac{P_{i}^{1} + P_{i}^{0}}{Q_{i}^{1} + Q_{i}^{0}}\)
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Cross-price elasticity
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\(\epsilon_{ij} = \dfrac{Q_{i}^{1} - Q_{i}^{0}}{P_{j}^{1} - P_{j}^{0}} \times \dfrac{P_{j}^{1} + P_{j}^{0}}{Q_{i}^{1} + Q_{i}^{0}}, i \neq j\)
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Income elasticity
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\(\epsilon_{iM} = \dfrac{Q_{i}^{1}-Q_{i}^{0}}{M^{1}-M^{0}} \times \dfrac{M^{1} + M^{0}}{Q_{i}^{1} + Q_{i}^{0}}\)
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Elasticity for other demand shift variable \(X\)
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\(\epsilon_{iX} = \dfrac{Q_{i}^{1}- Q_{i}^{0}}{X^{1}-X^{0}} \times \dfrac{X^{1} + X^{0}}{Q_{i}^{1} + Q_{i}^{0}}\)
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Let us compute the own-price arc elasticity over the two data points we have already considered from the demand equation above. For the own price elasticity, the two price quantity pairs are (60,160) and (50,200), respectively. Using these, the arc elasticity formula provides an elasticity at the average of these two points on the demand schedule
\(\epsilon_{11} = \dfrac{200-160}{50-60} \times \dfrac{50 + 60}{200 + 160} = -1.22.\)
Notice that it does not matter which point is designated with a superscript 1 or superscript 0, but one must keep the points straight in the first term of the arc formula. Reverse the ordering of the points so that (60,160) is assigned the superscript 0 and (50,200) is assigned the superscript 1, you get
\(\epsilon_{11} = \dfrac{160-200}{60-50} \times \dfrac{60 + 50}{160 + 200} = -1.22.\)
Using the point formula, you found the own-price elasticity at \((P_{1}= 60, Q_{1} = 160)\) to be \(\epsilon_{11} = 1.5\). At \((P_{1} = 50, Q_{1} = 200)\), it was \(\epsilon_{11} =-1\). The average of these two points is \((P_{1}= 55, Q_{1}=180)\). Using the point formula to measure elasticity at this average point provides \(\epsilon_{11}=-1.22\), which is what was obtained from the arc formula. This should help you get some intuition about the arc formula. The arc formula provides an elasticity at the average of the two points used in its computation.
A compelling feature of the arc formula is that you only need to know two points to compute an elasticity. However, it is important to emphasize the “all else held constant” provision. In the example above, this provision was met because income, the price of good 2, and advertising were held constant. The only difference in the two data points was the price of good 1. Application of the arc formula requires you to assume that the entire change from \(Q_{1}^{0}\) to \(Q_{1}^{1}\) is attributable only to an observed change in \(P_{1}\). In the real world, it may be hard to defend this assumption.
As an exercise, use the arc formula to replicate the elasticity ranges reported above in Demonstration 3.2.2 after you increase or decrease the price. In this case you know that all else has remained constant because you are comparing two points on the same demand schedule. There may be a small bit of rounding error in the demonstration, but the arc formula should give you something very close to the same number when you compute it from a price increase or decrease.