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Empirical Economics

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17.07.2017

Welfare analysis in a two-stage inverse demand model: an application to harvest changes in the Chesapeake Bay

verfasst von: Chris Moore, Charles Griffiths

Erschienen in: Empirical Economics | Ausgabe 3/2018

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Abstract

Like many agricultural commodities, fish and shellfish are highly perishable and producers cannot easily adjust supply in the short run to respond to changes in demand. In these cases it is more appropriate to conduct welfare analysis using inverse demand models that take quantities as given and allow prices to adjust to clear the market. One challenge faced by economists conducting demand analysis is how to limit the number of commodities in the analysis while accounting for the relevant substitutability and complementarity among goods. A common approach in direct demand modeling is to assume weak separability of the utility function and apply a multi-stage budgeting approach. This approach has not, however, been applied to an inverse demand system or the associated welfare analysis. This paper develops a two-stage inverse demand model and derives the total quantity flexibilities which describe how market clearing prices respond to supply changes in other commodity groups. The model provides the means to estimate consumer welfare impacts of an increase in finfish and shellfish harvest from the Chesapeake Bay while recognizing that harvests from other regions are potential substitutes. Comparing the two-stage results with single-stage analysis of the same data shows that ignoring differentiation of harvests from different regions, or the availability of substitutes not affected by a supply shock, can bias welfare estimates.

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In this specific application, the first stage inverse demand system does not estimate expenditure shares of total consumption for broad commodity groups but rather focuses on one such commodity group and finds the market clearing prices of the elementary commodities in the second stage. The estimation approach and expression derived for flexibilities, however, would apply to a multi-stage budgeting approach in which the first stage estimates a demand system for total expenditures.

This is different from the specification of Eales and Unnevehr (1994), \(\ln [D(U,\mathbf{X})] = (1-U)\cdot \ln [{a}(\mathbf{X})] + {U}\cdot \ln [{b}(\mathbf{X})]\). Footnote 1 in the appendix explains how the distance function could be written to resolve the difference between the two specifications.

In this model, a change in quantity affects the group expenditure through its impact on the normalized prices. This is because the quantity index, ln Q, in this model is only a function of quantities and not expenditures. As was pointed by our reviewer, a differential version of an inverse demand system (e.g., Brown et al. 1995) could represent real expenditures through d ln Q. In this case, consumers would directly allocate expenditures across commodity groups.

To simulate distributions for each flexibility we take 1000 draws from a multivariate normal distribution using the vector of point estimates of the inverse demand parameters \(\gamma \) and \(\beta \) and full covariance matrix for each inverse demand system. The draws are then plugged into expressions (14), (15), and (16) to simulate a distribution for each flexibility. The means and standard errors from these distributions provide the results shown in Tables 2, 3, and 4. See Krinsky and Robb (1986) for more details on using numerical simulations to estimate statistical properties of elasticities and flexibilities.

This is different from the specification of Eales and Unnevehr (1994), \(\ln [D(U,\mathbf{X})] = (1-U)\cdot \ln [a(\mathbf{X})] + U\cdot \ln [b(\mathbf{X})]\), where \(\ln [a(\mathbf{X})] = {\alpha }_{0} + \sum \nolimits _{\mathbf{i }}\, {\alpha }_{{i}} \ln (X_{{i}}) + (1/2) \sum \nolimits _{\mathbf{i }} \sum \nolimits _{\mathbf{j }} \,{\gamma }_{{ij}} \ln ({X}_{{i}}) \ln ({X}_{{j}})\), and \(\ln [{b}(\mathbf X )] = {\beta }_{0} \,\Pi _{\mathbf{i }}{X}_{{i}}^{-{\beta } {i}} + \ln [{a}(\mathbf{X})]\). This specification would require that \({\beta }_{0}<0\) in order to meet the requirement that a distance function is increasing in X and decreasing in U (Deaton 1979). The Moschini and Vissa (1992) specification, with a “–U”, emphasizes the fact that this function is decreasing in U. One way to make the Moshini and Vissa function comport with the Eales and Unnevehr specification is to write the distance function as \(\ln [{D}({U},\mathbf{X})] = (1-U)\cdot {a}(\mathbf{X}) - {U}\cdot {b}^{'}(\mathbf{X})\), where \({b}^{'}(\mathbf{X}) = {\beta }_{0} \,\Pi _{{i}}{X}_{{i}}^{{\beta } {i}} -{a}(\mathbf{X})\), which then reduces to our specification. In this case, if utility is scaled to \(0 \le U \le 1\), then a(X) is the value by which a quantity vector, X, must be divided to reach subsistence, \(U=0,\) and \(b'(\mathbf{X})\) is the value by which a quantity vector must be divided to reach bliss, \(U=1\).

In an effort to avoid excessive notation, i and j can serve as both indices over all species and to indicate an individual species.

This utilizes the fact that \(\partial {f(x)}/\partial {\ln (x)} = [\partial {f(x)}/\partial {x}]\cdot {x}\) and that \({\gamma }_{{ij}}={\gamma }_{{ji}}\).

Similar to what was done for the first stage, r and s can serve as both indices over all regions and to indicate a specific region.

The agricultural economics literature refers to this relationship as a “price flexibility” (for example, see Houck 1965). We use the term “quantity flexibility” here to emphasize the fact that this is an inverse model and that we are measuring the response due to a change in our independent variable, quantity.

This uses the results that \({Y}_{{i}}/({x}_{{(i)r}}\cdot {p}_{{(i)r}}) = 1/{w}_{{(i)r}}\) and \({w}_{{(i)r}}+{\beta }_{{(i)r}}\, {\ln Q}_{{(i)}}={\alpha }_{{(i)r}} + \sum \nolimits _{{s}}{\gamma }_{{(i)rs}}\, {\ln }({x}_{{(i)s}})\).

See Park and Thurman (1999) for a discussion of the link between scale flexibilities and income elasticities.

This uses the fact that \(\sum \nolimits _{{i}}{\alpha }_{{i}} = 1\), \(\sum \nolimits _{{j}}{\gamma }_{{ij}}=0\), and \({Y}/({M}_{{i}}^{*}\lambda {Xi}) = 1/{W}_{{i}}\).

Eales and Unnevehr (1994) derive a scale flexibility equal to \(-1+{\beta }_{{i}}/{W}_{{i}}^{*}\), but recall that their specification requires that \({\beta }_{{i}}<0\) (see appendix footnote 1), so our specification is equivalent to Eales and Unnevehr if \({\beta }_{{i}}>0\).

This is the inverse demand analog to \(\partial {P}_{{s}}/\partial {p}_{{sj}}\) in Edgerton’s (1997) direct demand model.

Note that \(\partial {\ln }({X}_{{i}})/\partial \ln ({x}_{{(j)s}})\ne 0\) if \(i=j\) (i.e., if we are evaluating the effect of a regional quantity change on the aggregate quantity of the same species) so it is set equal to \({\delta }_{{ij}}{\xi }_{{(j),(j)r}}\).

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Titel: Welfare analysis in a two-stage inverse demand model: an application to harvest changes in the Chesapeake Bay
verfasst von: Chris Moore
Charles Griffiths
Publikationsdatum: 17.07.2017
Verlag: Springer Berlin Heidelberg
Erschienen in: Empirical Economics / Ausgabe 3/2018
Print ISSN: 0377-7332
Elektronische ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-017-1309-3

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