Abstract
A simultaneous econometric model of supply and demand provides estimates of own-price effects and the effect of exogenous variables on supply or demand. Most of the time economists use elasticities derived from econometric analysis in a ceteris paribus context, more seldom in a total elasticity setting (Buse in J Farm Econ 40:881–891, 1958). Perhaps even more seldom net effects of exogenous changes on prices and quantities are determined in a Muth (Oxf Econ Pap 16:221–234, 1964) type model. In this paper, we replicate the econometric model by Epple and McCallum (Econ Inq 44:374–384, 2006) and use it to specify a comparative static model that quantifies the period-to-period net effects on price and quantity from observed changes in the exogenous variables. Furthermore, we extend (Brækkan et al. in Eur Rev Agric Econ 45:531–552, 2018) approach for computing unexplained demand shifts by also calculating unexplained shifts in supply. This bridges the gap between comparative statics and simultaneous econometric models. Unexplained supply and demand shifts account for the unexplained variation in the endogenous variables. The data used in this application are from the US broiler chicken market.
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Notes
In E&M’s article supply and demand are Eqs. (13) and (12).
We use E&Ms notation, where superscript a refers to aggregate.
While more sophisticated models of supply and (or) demand could have been applied to illustrate the procedure, we prefer E&M’s 2006 model since it focuses on an interesting market with substantial changes in price and quantities over time, and the data used to estimate the model are publicly available.
For a discussion and tests of potential econometric issues, please refer to (Brumm et al. 2008). The authors find that the model is satisfactory also when applied to various tests for exogeneity, the strength of instruments, and spurious regression. A reviewer also asked whether adding a time trend to the demand equation would impact the results. Adding a time trend causes demand to be slightly less elastic (from − 0.4 to − 0.33) and a smaller income elasticity (from 0.84 to 0.52). However, the time trend is not significant, and we therefore choose to continue with E&Ms specification.
An earlier version of the paper expressed the model in percentage change form. A reviewer and the editor noted that this implies that the demand and supply equations in levels are linear (see also Zhao et al. (1997) for a further discussion of expressing equations in percentage change or log differentials). Considering E&Ms express the supply and demand equation in logs, we here express the model in log differentials. Note that this introduces a small error in the equilibrium Eq. (6), caused by using the approximation \( \frac{{z_{t} - z_{t - 1} }}{{z_{t - 1} }} \approx \ln \left( {\frac{{z_{t} }}{{z_{t - 1} }}} \right) \).
To see why \( \kappa D \) and \( \kappa X \) refer to last period’s shares, note that: \( \frac{{{\text{QPROD}}_{t}^{a} - {\text{QPROD}}_{t - 1}^{a} }}{{{\text{QPROD}}_{t - 1}^{a} }} = \frac{{Q_{t - 1} \times {\text{POP}}_{t - 1} }}{{{\text{QPROD}}_{t - 1}^{a} }}\frac{{\left( {Q_{t} - Q_{t - 1} } \right) \times {\text{POP}}_{t - 1} }}{{Q_{t - 1} \times {\text{POP}}_{t - 1} }} + \frac{{Q_{t - 1} \times {\text{POP}}_{t - 1} }}{{{\text{QPROD}}_{t - 1}^{a} }}\frac{{\left( {{\text{POP}}_{t} - {\text{POP}}_{t - 1} } \right) \times Q_{t - 1} }}{{{\text{POP}}_{t - 1} \times Q_{t - 1} }} + \frac{{X_{t - 1} }}{{{\text{QPROD}}_{t - 1}^{a} }}\frac{{X_{t} - X_{t - 1} }}{{X_{t - 1} }} + \frac{{Q_{t - 1} \times {\text{POP}}_{t - 1} }}{{{\text{QPROD}}_{t - 1}^{a} }}\frac{{\left( {Q_{t} - Q_{t - 1} } \right) \times \left( {{\text{POP}}_{t} - {\text{POP}}_{t - 1} } \right)}}{{Q_{t - 1} \times {\text{POP}}_{t - 1} }} \), which simplifies to (6) after approximating the changes in variables by using log differentials.
We will henceforth ignore the last term.
If (3′) and (4′) had been estimated econometrically and the estimated coefficients had been used to compute the unexplained shifts, \( v_{t} \) and \( u_{t} \) would of course be identical to the residuals from the econometric model. While estimating (4′) provides results similar to the coefficients estimated in (2), estimating (3′) yields nonsensical results. (4′) and (2) are both expressed in log differences, while (1) and (3) are expressed in logs and log differences, respectively. This is probably why estimating (3) does not yield sensible results. Thus, we continue using the coefficients estimated from EM’s original model (Eqs. (1) and (2)).
\(Z_{t}^{H}\) is a transformation of \({\mathbf{Z}}_{{\mathbf{t}}}\) where each column represents the change in an exogenous variable in year t. The transpose of \({\mathbf{Z}}_{{\mathbf{t}}} \) is replicated three times (hence three rows) to capture the impacts on each of the three endogenous variables when (element-by-element) multiplying \(Z_{t}^{H}\) with the reduced form elasticity matrix \({\mathbf{E}}_{{\mathbf{t}}}{.} \)
Effects on production and consumption per capita can be found in the R code provided in the “Appendix”.
Remember our approximation and simplification in Eq. (6), which now should account for the remaining inaccuracy of our predicted endogenous variables.
The maximum relative difference for a previous version of the model with variables expressed in relative changes is 2%. The reduction in accuracy in the current (log differences) version of the model is due to the approximation we do in the equilibrium condition (6): \( \frac{{z_{t} - z_{t - 1} }}{{z_{t - 1} }} \approx \ln \left( {\frac{{z_{t} }}{{z_{t - 1} }}} \right) \).
Thanks to an anonymous reviewer for noting this.
Frisch (1934) argues that there is no reason to expect supply and demand shifts to be uncorrelated.
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Brækkan, E.H., Myrland, Ø. “All along the curves”: Bridging the gap between comparative statics and simultaneous econometric models. Empir Econ 60, 1559–1573 (2021). https://doi.org/10.1007/s00181-019-01793-3
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DOI: https://doi.org/10.1007/s00181-019-01793-3