4.3: Using Elasticities to Model an Equilibrium
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Recall from Chapter 3, that elasticities are expressed as ratios of percentages. Consequently, to use elasticities to model a market equilibrium, the exogenous demand and supply shocks must also be expressed in percentages. With this in mind, let \(\% \Delta S_{i}\) and \(\% \Delta D_{i}\) be the percentage exogenous shock to demand and supply in market \(i\), respectively. The demand and supply side of the markets can be expressed in change form as follows:
\(\% \Delta Q_{i}^{D} = \varepsilon_{ii} \% \Delta P_{i} + \% \Delta D_{i}, \)
and
\(\% \Delta Q_{i}^{S} = \phi_{ii} \% \Delta P_{i} + \% \Delta S_{i} .\)
The convention introduced in Chapter 3 is being followed here, where \(\phi\) and \(\varepsilon\) are used for supply and demand elasticities, respectively. When there is a change from one market equilibrium to another, it must be true that the change in the equilibrium quantity supplied is equal to the change in equilibrium quantity demanded. Thus, it is possible find the percentage change in equilibrium price that would result from the demand and supply shocks by setting \(\% \Delta Q_{i}^{S} = \% \Delta Q_{i}^{D}\) and solving for \(\% \Delta P_{i}\). Doing this provides
\[\% \Delta P_{i}^{E} = \dfrac{\% \Delta D_{i} - \% \Delta S_{i}}{\phi_{ii} -\varepsilon_{ii}}.\]
Substituting Equation \(\PageIndex{1}\) back into \(\% \Delta Q_{i}^{S}\) or \(\% \Delta Q_{i}^{D}\) provides the percentage change in equilibrium quantity:
\[\% \Delta Q_{i}^{E} = \dfrac{\phi_{ii} \% \Delta D_{i} - \varepsilon_{ii} \% \Delta S_{i}}{\phi_{ii} - \varepsilon_{ii}}.\]
Use of elasticities in modeling equilibria has some clear advantages. First, there is no need to assume that demand and supply are linear. The equilibrium model is a system of equations that is linear in the elasticities even if the underlying supply or demand equations are non-linear. Second, while it is true that equilibrium models in elasticity form provide percentage changes in equilibrium price and quantity, the prevailing equilibrium price and quantity are commonly known. Thus, percentage impacts from these models contain the information needed to assess the economic effects of an exogenous shock on the market. Third, the approach described below can accommodate any number of exogenous shocks and these shocks can occur simultaneously. Finally, the modeling approaches of this section can be extended to real-world situations involving numerous inter-related markets.
Using Elasticities of Supply or Demand from Exogenous Shift Variables
Recall from Chapter 3, that the general definition of an elasticity enables it to be used to forecast a quantity change. This means that if you have the elasticity of demand or supply with respect to an exogenous variable, it can be used to predict a shock to demand or supply as follows:
\(\% \Delta S_{i} = \phi_{iZ} \% \Delta Z\)
and
\(\% \Delta D_{i} = \varepsilon_{iX} \% \Delta X.\)
Thus, one way to obtain exogenous supply or demand shocks, \(\% \Delta S_{i}\) or \(\% \Delta D_{i}\), is to use projected changes in exogenous variables into demand/supply shocks using the elasticities for these exogenous variables.
Convert Potential Quantity Shocks to Percentages
Unfortunately, some shocks do correspond to exogenous variables for which you have elasticity estimates. In these cases it may be feasible to estimate the supply or demand shock in terms of units and then convert those units into a percentage of market quantity. For instance, in the early 2000’s, there was an outbreak of Hoof and Mouth Disease (HMD) in the United Kingdom. According to the BBC, the outbreak resulted in the destruction of more than six million sheep, cattle, and pigs (Bates 2016) . The last confirmed case of HMD in the United States was 1929, but HMD is endemic in parts of the world and there is always a risk that it could again become a problem in the United States as well (USDA-APHIS 2013a). In fact, the USDA has an entire agency, The Animal Plant Health Inspection Service, that works closely with Customs and Border Protection to prevent the introduction of HMD and other diseases or pests that could harm US Agriculture (USDA-APHIS 2013b).
Paarlberg, Lee, and Seitzinger (2002) considered the potential effects of an HMD outbreak on meat, poultry and dairy markets in the US. They used supply shocks of -5 percent for beef cattle and milk production, -2 percent for swine production, and -9 percent for lamb and sheep production. These estimates were based on actual livestock reductions observed in the UK outbreak. They also considered demand shocks to account for potential taste changes away from red meats and to account for the fact that discovery of HMD in the US would result in closure of export markets for livestock products. The point to be made here is that in many settings, such as a possible HMD outbreak, modelers may be able to come up with reasonable estimates for supply and demand shocks that can be used for \(\% \Delta S\) and/or \(\% \Delta D\) and then assess the effect on an equilibrium. A more involved analysis of the potential US market response to an HMD outbreak is provided in Paarlberg et al. (2008). This study provides background on the markets being examined and includes an appendix containing baselines and elasticities used in the equilibrium models. Moreover, the study measures market equilibrium responses to HMD outbreaks of differing levels of severity.
In the Paarlberg, Lee, and Seitzinger (2002) and Paarlberg et al. (2008) studies, the exogenous event was an HMD outbreak and the analyses examined how equilibrium prices and quantities would respond to this event given existing estimates of market elasticities. It is worth pointing out that these authors included markets for feedstuffs in their models. This is because an adjustment to livestock markets would have ramifications on markets for feedstuffs, which would then ripple back into the markets for livestock products. For this reason, it was important to include feedstuff prices and quantities as endogenous variables in the model.
Use Own-Price Elasticities to Convert Changes in Costs or Willingness to Pay Into Quantity Shocks
In other cases, the exogenous shock that needs to be modeled may not have an elasticity nor be easily represented in terms of a percentage of current market quantity. However, if you can estimate the effect as a percentage of the current equilibrium price you can use own-price elasticities to find the shock. This is because an increase in cost is analogous to price decrease for producers. Similarly, an improvement in a quality attribute valued by consumers is analogous to a price decrease for the consumer.
A Supply Shock Problem
Let us consider a supply shock problem. Suppose new biosecurity regulations are proposed to prevent spread of HMD on feedlot operations. Suppose further that the fed cattle market is at an initial equilibrium with a price of $1.25 per pound. It is estimated that it will cost producers $0.03 per pound to comply with the regulations. How will this affect market equilibrium? You know that the regulations will raise costs and this will shift the supply curve to the left resulting in a new equilibrium with a higher price and low quantity (see Table 1 above). The question is, how much higher and lower? To find out, let us first note that the $0.03 cost increase is 2.4 percent (0.03/1.25 = 0.024) of the current market price. Consequently, the costs resulting from the regulation are analogous to a 2.4 percent reduction in the price that producers receive. Suppose that farm-level supply and demand elasticities are as follows: \(\phi_{11} = 2\) and \(\varepsilon_{11} = -0.8\). The increase in per-unit costs translates into a supply shock of \(\% \Delta S = 2(-2.4) = -4.8 \%\). You can now use Equation \(\PageIndex{1}\) to predict the equilibrium price response as follows:
\(\% \Delta P_{1}^{E} = \dfrac{0-(-4.8)}{2-(-0.8)} \approx 1.71.\)
From Equation \(\PageIndex{2}\), you get the equilibrium quantity response to be
\(\% \Delta Q_{1}^{E} = \dfrac{(2)(0) - (-0.8)(-4.8)}{2-(-0.8)} \approx -1.37.\)
A Demand Shock Problem
Suppose that you have access to a study showing that high protein diet fads are growing in popularity and that within a year, consumers will, on average, be willing to pay 3 percent more for chicken. If this study is correct, how would it affect the equilibrium price and quantity of chicken? The study suggests that consumer preferences for chicken with strengthen. Thus, you would expect demand to shift outwards, and the equilibrium price and quantity should rise (see Table 1). The question is by how much? Since consumers are projected to obtain 3 percent more value from chicken, this is analogous to a 3 percent reduction in the price that consumers pay. Suppose that retail level supply and demand elasticities for chicken are as follows: \(\phi_{11}= 1.75 \: and \: \varepsilon_{11}= -1.1\). The demand shock is \(\% \Delta D = -1.1(-3) = 3.3 \%\). Using Equation \(\PageIndex{1}\), the effect on the equilibrium price is
\(\% \Delta P_{1}^{E} = \dfrac{3.3-0}{1.75 - (-1.1)} \approx 1.16.\)
From Equation \(\PageIndex{2}\), you get the equilibrium quantity response to be
\(\% \Delta Q_{1}^{E} = \dfrac{(1.75)(3.3) -(-1.1)(0)}{1.75- (-1.1)} \approx 2.03.\)