Calibration and Parameter Estimation for a Melitz Sector in a CGE Model

Abstract

This chapter is about giving numbers to parameters and unobservable variables in a Melitz CGE model. We start by describing how a Melitz model can be calibrated. This is the process by which unobservable variables (preference and fixed cost variables, δ’s, F’s and H’s) are evaluated so that for given parameter values (inter-variety substitution elasticities and productivity distribution parameters, σ and α) the model reproduces base-year data. We find that for a Melitz model there are multiple legitimate calibration possibilities but that the choice between these does not affect simulation results. Then, we review the method that Balistreri et al. (2011) have pioneered for estimating parameters in a Melitz model. This method combines calibration and estimation. Rather than setting initial values for unobservable variables to reproduce base-year data, Balistreri et al. impose theoretically preferred structures on the unobservable variables. These structures are incompatible with precise calibration, but pave the way for estimation. Parameters can be estimated by choosing the values that allow calibration to base-year data that is as close as possible subject to meeting the preferred structural constraints on the unobservable variables. Balistreri et al.’s approach is likely to be a starting point for many potentially fruitful calibration/estimation efforts.

Appendices

Appendix 4.1: Relating Observables to Melitz Concepts, and Demonstrating the Fixity of the Shares of Production, Link and Establishment Costs in an Industry’s Total Costs

In this appendix we derive Eqs. (4.1)–(4.3). Then we show that the Melitz model with a Pareto specification of firm productivities implies a constant split of industry costs between variable, link and establishment costs.

• Derivation of ( 4.1 )

In (4.1) we assume that the landed-duty-paid value of trade on the sd-link, V(s, d), is the value for the typical firm, ${\text{P}}_{{ \bullet {\text{sd}}}} {\text{Q}}_{{ \bullet {\text{sd}}}}$, times the number of trading firms, N_sd. To derive (4.1) we start from

$$ {\text{V}}({\text{s}},{\text{d}}) = \sum\limits_{{{\text{k}} \in {\text{S}}({\text{s}},{\text{d}})}} {{\text{N}}_{\text{s}} {\text{g}}(\Phi_{\text{k}}^{{}} ){\text{P}}_{\text{ksd}} {\text{Q}}_{\text{ksd}} } , $$

(4.25)

that is, the landed-duty-paid value of widgets sent from s to d is the value, ${\text{P}}_{\text{ksd}} {\text{Q}}_{\text{ksd}}$, sent by a k-class firm times the number of such firms, ${\text{N}}_{\text{s}} {\text{g}}(\Phi_{\text{k}}^{{}} )$, aggregated over all k in S(s, d). Using the AKME versions of (T2.1) and (T2.3) and assuming that $\upgamma_{\text{ksd}} = 1$ for all k, we find that

$$ \frac{{{\text{P}}_{\text{ksd}} }}{{{\text{P}}_{{ \bullet {\text{sd}}}} }} = \frac{{{{\Phi }}_{{ \bullet {\text{sd}}}} }}{{{{\Phi}}_{\text{ksd}} }}. $$

(4.26)

and

$$ \frac{{{\text{Q}}_{\text{ksd}} }}{{{\text{Q}}_{{ \bullet {\text{sd}}}} }} = \left( {\frac{{{\text{P}}_{{ \bullet {\text{sd}}}} }}{{{\text{P}}_{\text{ksd}} }}} \right)^{\upsigma } . $$

(4.27)

Combining (4.26) and (4.27) gives

$$ {\text{Q}}_{\text{ksd}} {\text{P}}_{\text{ksd}} = \left( {\frac{{\Phi_{\text{ksd}} }}{{\Phi_{{ \bullet {\text{sd}}}} }}} \right)^{\upsigma - 1} *{\text{Q}}_{{ \bullet {\text{sd}}}} {\text{P}}_{{ \bullet {\text{sd}}}} . $$

(4.28)

Substituting into (4.25) we obtain

$$ {\text{V}}({\text{s}},{\text{d}}) = \frac{{{\text{P}}_{{ \bullet {\text{sd}}}} {\text{Q}}_{{ \bullet {\text{sd}}}} }}{{\Phi_{{ \bullet {\text{sd}}}}^{\upsigma - 1} }}\sum\limits_{{{\text{k}} \in {\text{S}}({\text{s}},{\text{d}})}} {{\text{N}}_{\text{s}} {\text{g}}(\Phi_{\text{k}}^{{}} )} {\kern 1pt}\Phi_{\text{ksd}}^{\upsigma - 1} . $$

(4.29)

Then applying (2.19) we arrive at (4.1):

$$ {\text{V}}({\text{s}},{\text{d}}) = {\text{N}}_{\text{sd}} {\text{P}}_{{ \bullet {\text{sd}}}} {\text{Q}}_{{ \bullet {\text{sd}}}} . $$

(4.30)

• Derivation of ( 4.2 )

Starting from the Melitz version of (T2.3) we have

$$ {\text{P}}_{{ \bullet {\text{sd}}}} {\text{Q}}_{{ \bullet {\text{sd}}}} {\text{N}}_{\text{sd}} = {\text{Q}}_{\text{d}} \updelta_{\text{sd}}^{\upsigma } {\text{P}}_{{ \bullet {\text{sd}}}}^{1 - \upsigma } {\text{P}}_{\text{d}}^{\upsigma } {\text{N}}_{\text{sd}} . $$

(4.31)

Adding over s and using (4.30) gives

$$ \sum\limits_{\text{s}} {{\text{V}}({\text{s}},{\text{d}})} = {\text{Q}}_{\text{d}} {\text{P}}_{\text{d}}^{\upsigma } \sum\limits_{\text{s}} {\updelta_{\text{sd}}^{\upsigma } {\text{P}}_{{ \bullet {\text{sd}}}}^{1 - \upsigma } } {\text{N}}_{\text{sd}} . $$

(4.32)

Now using (T2.2) we obtain

$$ \sum\limits_{\text{s}} {{\text{V}}({\text{s}},{\text{d}})} = {\text{Q}}_{\text{d}} {\text{P}}_{\text{d}}^{\upsigma } {\text{P}}_{\text{d}}^{1 - \upsigma } $$

(4.33)

which leads to (4.2).

• Derivation of ( 4.3 )

We start by multiplying the Melitz version of (T2.7) through by W_s:

$$ {\text{W}}_{\text{s}} {\text{L}}_{\text{s}} = \sum\limits_{\text{d}} {\frac{{{\text{W}}_{\text{s}} {\text{N}}_{\text{sd}} {\text{Q}}_{{ \bullet {\text{sd}}}} }}{{\Phi_{{ \bullet {\text{sd}}}} }}} + \sum\limits_{\text{d}} {{\text{N}}_{\text{sd}} } {\text{F}}_{\text{sd}} {\text{W}}_{\text{s}} + {\text{N}}_{\text{s}} {\text{H}}_{\text{s}} {\text{W}}_{\text{s}} . $$

(4.34)

The left hand side is total costs in the widget industry of country s. The first term on the right hand side is the industry’s total variable costs. The second term is trade-link set-up costs incurred by the industry. The third term is establishment costs incurred by firms in the industry. We work separately on each of these three terms.

Substituting from (T2.1) into the variable-cost term gives

$$ \sum\limits_{\text{d}} {\frac{{{\text{W}}_{\text{s}} {\text{N}}_{\text{sd}} {\text{Q}}_{{ \bullet {\text{sd}}}} }}{{\Phi _{{ \bullet {\text{sd}}}} }}} = \frac{\upsigma - 1}{\upsigma }*\sum\limits_{\text{d}} {\frac{{{\text{P}}_{{ \bullet {\text{sd}}}} {\text{N}}_{\text{sd}} {\text{Q}}_{{ \bullet {\text{sd}}}} }}{{{\text{T}}_{\text{sd}} }}} . $$

(4.35)

Working with (T2.10) we find that link set-up costs are given by:

$$ \sum\limits_{\text{d}} {{\text{N}}_{\text{sd}} {\text{F}}_{\text{sd}} {\text{W}}_{\text{s}} } = \frac{1}{\upsigma - 1}*\sum\limits_{\text{d}} {\frac{{{\text{W}}_{\text{s}} {\text{Q}}_{{{ \hbox{min} }({\text{s}},{\text{d}})}} {\text{N}}_{\text{sd}} }}{{\Phi _{{{ \hbox{min} }({\text{s}},{\text{d}})}} }}} . $$

(4.36)

Using (T2.11) and (T2.12) we have ${{\text{Q}_{{\hbox{min} ({\text{s}},{\text{d}})}} } \mathord{\left/ {\vphantom {{\text{Q}_{{\hbox{min} ({\text{s}},{\text{d}})}} } {\Phi_{{\hbox{min} ({\text{s}},{\text{d}})}} }}} \right. \kern-0pt} {\Phi_{{\hbox{min} ({\text{s}},{\text{d}})}} }} = \upbeta^{1 - \upsigma } *\left[ {{{{\text{Q}}_{{ \bullet {\text{sd}}}} } \mathord{\left/ {\vphantom {{{\text{Q}}_{{ \bullet {\text{sd}}}} } {\Phi_{{ \bullet {\text{sd}}}} }}} \right. \kern-0pt} {\Phi_{{ \bullet {\text{sd}}}} }}} \right]$. Substituting this into (4.36) gives

$$ \sum\limits_{\text{d}} {{\text{N}}_{\text{sd}} {\text{F}}_{\text{sd}} {\text{W}}_{\text{s}} } = \frac{{\upbeta^{1 - \upsigma } }}{\upsigma - 1}*\sum\limits_{\text{d}} {\frac{{{\text{W}}_{\text{s}} {\text{Q}}_{{ \bullet {\text{sd}}}} {\text{N}}_{\text{sd}} }}{{\Phi _{{ \bullet {\text{sd}}}} }}} . $$

(4.37)

Then using (T2.1) we obtain

$$ \sum\limits_{\text{d}} {{\text{N}}_{\text{sd}} {\text{F}}_{\text{sd}} {\text{W}}_{\text{s}} } = \frac{{\upbeta^{1 - \upsigma } }}{\upsigma }*\sum\limits_{\text{d}} {\frac{{{\text{P}}_{{ \bullet {\text{sd}}}} {\text{Q}}_{{ \bullet {\text{sd}}}} {\text{N}}_{\text{sd}} }}{{{\text{T}}_{\text{sd}} }}} . $$

(4.38)

From (T2.6) and (T2.9) we find that establishment costs are given by

$$ {\text{N}}_{\text{s}} {\text{H}}_{\text{s}} {\text{W}}_{\text{s}} = \sum\limits_{\text{d}} {{\text{N}}_{\text{sd}} {\Pi}_{{ \bullet {\text{sd}}}} } . $$

(4.39)

Then using (T2.5) and (T2.1) we obtain

$$ {\text{N}}_{\text{s}} {\text{H}}_{\text{s}} {\text{W}}_{\text{s}} = \frac{1}{\upsigma }*\sum\limits_{\text{d}} {\frac{{{\text{N}}_{\text{sd}} {\text{P}}_{{ \bullet {\text{sd}}}} {\text{Q}}_{{ \bullet {\text{sd}}}} }}{{{\text{T}}_{\text{sd}} }} - \sum\limits_{\text{d}} {{\text{N}}_{\text{sd}} {\text{F}}_{\text{sd}} {\text{W}}_{\text{s}} } } , $$

(4.40)

which via (4.38) becomes

$$ {\text{N}}_{\text{s}} {\text{H}}_{\text{s}} {\text{W}}_{\text{s}} = \frac{{1 - \upbeta^{1 - \upsigma } }}{\upsigma }*\sum\limits_{\text{d}} {\frac{{{\text{N}}_{\text{sd}} {\text{P}}_{{ \bullet {\text{sd}}}} {\text{Q}}_{{ \bullet {\text{sd}}}} }}{{{\text{T}}_{\text{sd}} }}} . $$

(4.41)

Finally we add over (4.35), (4.38) and (4.41) and use (4.30) and (4.34). This gives

$$ {\text{W}}_{\text{s}} {\text{L}}_{\text{s}} = \left[ {\frac{\upsigma - 1}{\upsigma } + \frac{{\upbeta^{1 - \upsigma } }}{\upsigma } + \frac{{1 - \upbeta^{1 - \upsigma } }}{\upsigma }} \right]*\sum\limits_{\text{d}} {\frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{T}}_{\text{sd}} }}} . $$

(4.42)

Simplifying on the right hand side of (4.42) quickly gives (4.3).

• The fixity of the shares of variable, link and establishment costs in an industry’s total costs

Our derivation of (4.3) brings out an important feature of the Melitz model with a Pareto distribution of firm productivities: the split of an industry’s total costs between variable, link and establishment costs is fixed by the variety substitution parameter σ and the Pareto parameter α [recall that β is a function of α and σ, see (2.27)]. The split does not depend on tariff rates (T_sd), preferences (δ_sd), the number of firms (N_s), the number of firms on links (N_sd), and rather remarkably it doesn’t depend on establishment costs per firm (H_sW_s) or link set-up costs per firm (F_sdW_s). Variable costs always account for the share (σ − 1)/σ in total costs; link costs always account for the share $\upbeta^{1 - \upsigma } /\upsigma$; and establishment costs always account for the share $(1 - \upbeta^{1 - \upsigma } )/\upsigma$. As we will see in Chap. 7, the determination of the cost split via just α and σ sharply simplifies calibration in a model specified in percentage changes.

Appendix 4.2: Calibration: Establishing Relationships Between Unobservables (δ, F and H) and Base-Year Data (V, T and W)

In this appendix we derive (4.5) from the Melitz sectoral model set out in Table 4.1. The algebra is tedious but not hard. Our strategy is to derive expressions for N_sd, N_s and $\Phi_{{ \bullet {\text{sd}}}}$ in terms of V(s, d), W_s, T_sd, F_sd, H_s. σ, α (and β). These expressions lead eventually to (4.5).

• Derivation of expression for N _sd

Equation (T4.8) in Table 4.1 gives

$$ \frac{{\text{Q}_{{\hbox{min} ( {\text{s,d)}}}} }}{{\Phi_{\text{min(s,d)}} }} = {\text{F}}_{\text{sd}} *\left( {\upsigma - 1} \right). $$

(4.43)

Then via (T4.9) and (T4.10) we obtain

$$ \frac{{\text{Q}_{{ \bullet {\text{sd}}}} }}{{\Phi _{{ \bullet {\text{sd}}}} }} = {\text{F}}_{\text{sd}} *\left( {\upsigma - 1} \right)*\upbeta^{\upsigma - 1} $$

(4.44)

Equations (T4.1) and (T4.11) give

$$ {\text{V}}({\text{s}},{\text{d}}) = {\text{N}}_{\text{sd}} {\text{W}}_{\text{s}} {\text{T}}_{\text{sd}} *\frac{{\text{Q}_{{ \bullet {\text{sd}}}} }}{{\Phi_{{ \bullet {\text{sd}}}} }}*\frac{\upsigma }{\upsigma - 1} $$

(4.45)

Now using (4.44) we find that

$$ {\text{N}}_{\text{sd}} = \frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{W}}_{\text{s}} {\text{T}}_{\text{sd}} }}*\frac{{\upbeta^{1 - \upsigma } }}{{\upsigma {\text{F}}_{\text{sd}} }} $$

(4.46)

• Derivation of expression for N _s

Substituting from (4.46) and (T4.11) into (T4.5) gives

$$ 0 = \frac{1}{\upsigma }\sum\limits_{\text{d}} {\frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{T}}_{\text{sd}} }}} - \frac{{\upbeta^{1 - \upsigma } }}{\upsigma }\sum\limits_{\text{d}} {\frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{T}}_{\text{sd}} }}} - {\text{N}}_{\text{s}} {\text{H}}_{\text{s}} {\text{W}}_{\text{s}} , $$

(4.47)

leading to

$$ {\text{N}}_{\text{s}} = \left( {\frac{{1 - \upbeta^{1 - \upsigma } }}{{\upsigma \,{\text{H}}_{\text{s}} {\text{W}}_{\text{s}} }}} \right)*\sum\limits_{\text{d}} {\frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{T}}_{\text{sd}} }}} $$

(4.48)

• Derivation of expression for $\varvec{\varPhi}_{{ \bullet \varvec{sd}}}$

Rearranging (T4.7) gives

$$ \Phi _{\text{min(s,d)}} = \left( {\frac{{{\text{N}}_{\text{sd}} }}{{{\text{N}}_{\text{s}} }}} \right)^{ - 1/\upalpha } $$

(4.49)

Now substitute in from (4.46) and (4.48) to obtain

$$ \Phi _{\text{min(s,d)}} = \left( {\frac{{\frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{T}}_{\text{sd}} }}*\frac{1}{{{\text{F}}_{\text{sd}} }}}}{{\frac{{\left( {\upbeta^{\upsigma - 1} - 1} \right)}}{{{\text{H}}_{\text{s}} }}*\sum\nolimits_{\text{j}} {\frac{{{\text{V}}({\text{s}},{\text{j}})}}{{{\text{T}}_{\text{sj}} }}} }}} \right)^{ - 1/\upalpha } . $$

(4.50)

Then (T4.9) gives

$$ \Phi_{{ \bullet {\text{sd}}}} = \upbeta *\left( {\frac{{\frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{T}}_{\text{sd}} }}*\frac{1}{{{\text{F}}_{\text{sd}} }}}}{{\frac{{\left( {\upbeta^{\upsigma - 1} - 1} \right)}}{{{\text{H}}_{\text{s}} }}*\sum\nolimits_{\text{j}} {\frac{{{\text{V}}({\text{s}},{\text{j}})}}{{{\text{T}}_{\text{sj}} }}} }}} \right)^{ - 1/\upalpha } . $$

(4.51)

• Using the expressions for N _sd and $\varvec{\varPhi}_{{ \bullet \varvec{sd}}}$ to derive ( 4.5 )

Equation (T4.3) implies that

$$ {\text{P}}_{{ \bullet {\text{sd}}}} {\text{Q}}_{{ \bullet {\text{sd}}}} {\text{N}}_{\text{sd}} = \updelta_{\text{sd}}^{\upsigma } {\text{D}}_{\text{d}} \left( {\frac{{{\text{P}}_{\text{d}} }}{{{\text{P}}_{{ \bullet {\text{sd}}}} }}} \right)^{\upsigma - 1} *{\text{N}}_{\text{sd}}\quad . $$

(4.52)

Then using (T4.11) and (T4.2) we obtain

$$ {\text{V}}({\text{s}},{\text{d}}) = \left( {{\text{D}}_{\text{d}} } \right)*\frac{{\updelta_{\text{sd}}^{\upsigma } {\text{P}}_{{ \bullet {\text{sd}}}}^{1 - \upsigma } *{\text{N}}_{\text{sd}} }}{{\sum\limits_{\text{j}} {\updelta_{\text{jd}}^{\upsigma } {\text{P}}_{{ \bullet {\text{jd}}}}^{1 - \upsigma } *{\text{N}}_{\text{jd}} } }}\quad . $$

(4.53)

Dividing through by V(d, d) leads to

$$ \frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{V}}({\text{d}},{\text{d}})}} = \frac{{\updelta_{\text{sd}}^{\upsigma } {\text{P}}_{{ \bullet {\text{sd}}}}^{1 - \upsigma } *{\text{N}}_{\text{sd}} }}{{\updelta_{\text{dd}}^{\upsigma } \text{P}_{{ \bullet {\text{dd}}}}^{1 - \upsigma } *{\text{N}}_{\text{dd}} }} $$

(4.54)

which, via (T4.1), can be written as

$$ \frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{V}}({\text{d}},{\text{d}})}} = \frac{{\updelta_{\text{sd}}^{\upsigma }\Phi_{{ \bullet {\text{sd}}}}^{\upsigma - 1} *{\text{N}}_{\text{sd}} *\left( {{\text{W}}_{\text{s}} {\text{T}}_{\text{sd}} } \right)^{1 - \upsigma } }}{{\updelta_{\text{dd}}^{\upsigma }\Phi_{{ \bullet {\text{dd}}}}^{\upsigma - 1} *{\text{N}}_{\text{dd}} *\left( {{\text{W}}_{\text{d}} {\text{T}}_{\text{dd}} } \right)^{1 - \upsigma } }}\quad . $$

(4.55)

Now we use (4.46) and (4.51) to eliminate N_sd, N_dd, $\Phi_{{ \bullet {\text{sd}}}}$ and $\Phi_{{ \bullet {\text{dd}}}}$ for all s and d:

$$ \frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{V}}({\text{d}},{\text{d}})}} = \frac{{\updelta_{\text{sd}}^{\upsigma } \upbeta^{\upsigma - 1} *\left( {\frac{{\frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{T}}_{\text{sd}} }}*\frac{1}{{{\text{F}}_{\text{sd}} }}}}{{\frac{{\left( {\upbeta^{\upsigma - 1} - 1} \right)}}{{{\text{H}}_{\text{s}} }}*\sum\nolimits_{\text{j}} {\frac{{{\text{V}}({\text{s}},{\text{j}})}}{{{\text{T}}_{\text{sj}} }}} }}} \right)^{{\frac{(1 - \upsigma )}{\upalpha }}} *\left( {\frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{W}}_{\text{s}} {\text{T}}_{\text{sd}} }}*\frac{{\upbeta^{1 - \upsigma } }}{{\upsigma {\text{F}}_{\text{sd}} }}} \right)*\left( {{\text{W}}_{\text{s}} {\text{T}}_{\text{sd}} } \right)^{1 - \upsigma } }}{{\updelta_{\text{dd}}^{\upsigma } \upbeta^{\upsigma - 1} *\left( {\frac{{\frac{{{\text{V}}({\text{d}},{\text{d}})}}{{{\text{T}}_{\text{dd}} }}*\frac{1}{{{\text{F}}_{\text{dd}} }}}}{{\frac{{\left( {\upbeta^{\upsigma - 1} - 1} \right)}}{{{\text{H}}_{\text{d}} }}*\sum\nolimits_{\text{j}} {\frac{{{\text{V}}({\text{d}},{\text{j}})}}{{{\text{T}}_{\text{dj}} }}} }}} \right)^{{\frac{(1 - \upsigma )}{\upalpha }}} *\left( {\frac{{{\text{V}}({\text{d}},{\text{d}})}}{{{\text{W}}_{\text{d}} {\text{T}}_{\text{dd}} }}*\frac{{\upbeta^{1 - \upsigma } }}{{\upsigma {\text{F}}_{\text{dd}} }}} \right)*\left( {{\text{W}}_{\text{d}} {\text{T}}_{\text{dd}} } \right)^{1 - \upsigma } }} $$

(4.56)

By simplifying (4.56) we arrive at:

$$ {\text{F}}_{\text{sd}} = {\text{F}}_{\text{dd}} *\left( {\frac{{{\text{J}}({\text{s}},{\text{d}})}}{{{\text{J}}({\text{d}},{\text{d}})}}} \right)^{{\frac{(1 - \upsigma )}{(1 + \upalpha - \upsigma )}}} *\left( {\frac{{{\text{W}}_{\text{s}} {\text{T}}_{\text{sd}} }}{{{\text{W}}_{\text{d}} {\text{T}}_{\text{dd}} }}} \right)^{{\frac{ - \upsigma *\upalpha }{(1 + \upalpha - \upsigma )}}} *\left( {\frac{{\updelta_{\text{sd}}^{{}} }}{{\updelta_{\text{dd}}^{{}} }}} \right)^{{\frac{\upsigma *\upalpha }{(1 + \upalpha - \upsigma )}}} *\left( {\frac{{{\text{H}}_{\text{s}} }}{{{\text{H}}_{\text{d}} }}} \right)^{{\frac{(1 - \upsigma )}{(1 + \upalpha - \upsigma )}}} \quad $$

(4.57)

where J(s, d) is, as defined in (4.6), the share of region s’s sales accounted for d. Finally, we substitute base-year values (marked with bars) for J’s, W’s and T’s into (4.57) to obtain (4.5).

Appendix 4.3: A Percentage Change Version of the Melitz Model: Derivation of Table 4.2 from Table 4.1

As explained in Dixon et al. (1992) there are three useful rules for converting from a levels presentation of equations to a percentage-change presentation.

• Multiplicative rule

$$ {\text{Z}} = {\text{X}}*{\text{Y}}\;{\text{becomes}}\;{\text{z}} = {\text{x}} + {\text{y}} $$

(4.58)

where the lowercase symbols represent percentage changes in the uppercase symbols. (4.58) can be derived by total differentiation:

$$ \Delta {\text{Z}} = {\text{X}}*\Delta {\text{Y}} + \Delta {\text{X}}*{\text{Y}}\quad . $$

(4.59)

Dividing on the left-hand side by Z and on the right-hand side by X * Y and multiplying both sides by 100 gives (4.58).

• Power rule

$$ {\text{Z}} = {\text{X}}^{\upalpha } \;{\text{becomes}}\;{\text{z}} = \upalpha *{\text{x}}\quad . $$

(4.60)

To derive (4.60) we start from

$$ \Delta {\text{Z}} = \frac{{{{\partial}} {\text{X}}^{\upalpha } }}{{{{\partial}} {\text{X}}}}*\Delta {\text{X}} = \upalpha *{\text{X}}^{\upalpha - 1} *\Delta {\text{X}}\quad . $$

(4.61)

Dividing on the left-hand side by Z and on the right-hand side by ${\text{X}}^{\upalpha }$ and multiplying both sides by 100 gives (4.60).

• Addition rule

$$ {\text{Z}} = {\text{X}} + {\text{Y}}\;{\text{becomes}}\;{\text{either}}\;{\text{Z}}*{\text{z}} = {\text{X}}*{\text{x}} + {\text{Y}}*{\text{y}}\;{\text{or}}\;{\text{z}} = {\text{S}}_{\text{x}} *{\text{x}} + {\text{S}}_{\text{y}} *{\text{y}} $$

(4.62)

where S_x and S_y are the shares of X and Y in Z.

To derive (4.62) we start from

$$ \Delta {\text{Z}} = \Delta {\text{X}} + \Delta {\text{Y}} $$

(4.63)

which can be written as

$$ {\text{Z}}\frac{{\Delta {\text{Z}}}}{\text{Z}} = {\text{X}}\frac{{\Delta {\text{X}}}}{\text{X}} + {\text{Y}}\frac{{\Delta {\text{Y}}}}{\text{Y}} $$

(4.64)

quickly leading to (4.62).

The conversions from (T4.1), (T4.3), (T4.4), (T4.7), (T4.8), (T4.9), (T4.10) and (T4.11) in Table 4.1 to the percentage change versions in Table 4.2 are achieved by straight-forward applications of the multiplicative and power rules.

• Derivation of (T4.2P)

Applying all three rules to (T4.2) we obtain

$$ \left( {1 - \upsigma } \right)*{\text{p}}_{\text{d}} = \sum\limits_{\text{s}} {\frac{{{\text{N}}_{\text{sd}} *\updelta_{\text{sd}}^{\upsigma } *\text{P}_{{ \bullet {\text{sd}}}}^{1 - \upsigma } }}{{\sum\limits_{\text{j}} {{\text{N}}_{\text{jd}} *\updelta_{\text{jd}}^{\upsigma } *{\text{P}}_{{ \bullet {\text{jd}}}}^{1 - \upsigma } } }}*} \left( {{\text{n}}_{\text{sd}} + \upsigma \widehat{\updelta }_{\text{sd}}^{{}} + (1 - \upsigma )*{\text{p}}_{{ \bullet {\text{sd}}}}^{{}} } \right)\quad . $$

(4.65)

Then working with (T4.3) and (T4.11) we find that

$$ \frac{{{\text{N}}_{\text{sd}} *\updelta_{\text{sd}}^{\upsigma } *{\text{P}}_{{ \bullet {\text{sd}}}}^{1 - \upsigma } }}{{\sum\limits_{\text{j}} {{\text{N}}_{\text{jd}} *\updelta_{\text{jd}}^{\upsigma } *{\text{P}}_{{ \bullet {\text{jd}}}}^{1 - \upsigma } } }} = \frac{{{\text{V}}({\text{s}},{\text{d}})}}{{\sum\limits_{\text{j}} {{\text{V}}({\text{j}},{\text{d}})} }}\quad . $$

(4.66)

Substituting from (4.66) into (4.65) gives (T4.2P).

Derivation of (T4.5P)

Applying the addition and multiplication rules to (T4.5) we obtain

$$ \begin{aligned} 0 & = \frac{1}{\upsigma }*\sum\limits_{\text{d}} {{\text{N}}_{\text{sd}} \frac{{{\text{P}}_{{ \bullet {\text{sd}}}} }}{{{\text{T}}_{\text{sd}} }}*{\text{Q}}_{{ \bullet {\text{sd}}}} *\left( {{\text{n}}_{\text{sd}} + {\text{p}}_{{ \bullet {\text{sd}}}} + {\text{q}}_{{ \bullet {\text{sd}}}} - {\text{t}}_{\text{sd}} } \right)} \\ & \quad - \sum\limits_{\text{d}} {{\text{N}}_{\text{sd}} {\text{F}}_{\text{sd}} {\text{W}}_{\text{s}} *\left( {{\text{n}}_{\text{sd}} + {\text{f}}_{\text{sd}} + {\text{w}}_{\text{s}} } \right)} - {\text{N}}_{\text{s}} {\text{H}}_{\text{s}} {\text{W}}_{\text{s}} *\left( {{\text{n}}_{\text{s}} + {\text{h}}_{\text{s}} + {\text{w}}_{\text{s}} } \right) .\\ \end{aligned}\quad $$

(4.67)

(T4.11) gives us

$$ \frac{{{\text{N}}_{\text{sd}} {\text{P}}_{{ \bullet {\text{sd}}}} {\text{Q}}_{{ \bullet {\text{sd}}}} }}{{{\text{T}}_{\text{sd}} }} = \frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{T}}_{\text{sd}} }}\quad . $$

(4.68)

From (T4.8), (T4.9), (T4.10), (T4.1) and (T4.11) we have

$$ {\text{N}}_{\text{sd}} {\text{F}}_{\text{sd}} {\text{W}}_{\text{s}} = \frac{{\upbeta^{1 - \upsigma } }}{\upsigma }*\frac{{{\text{P}}_{{ \bullet {\text{sd}}}} {\text{Q}}_{{ \bullet {\text{sd}}}} {\text{N}}_{\text{sd}} }}{{{\text{T}}_{\text{sd}} }} = \frac{{\upbeta^{1 - \upsigma } }}{\upsigma }*\frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{T}}_{\text{sd}} }}\quad . $$

(4.69)

(T4.5) implies that

$$ \begin{aligned} {\text{N}}_{\text{s}} {\text{H}}_{\text{s}} {\text{W}}_{\text{s}} & = \frac{1}{\upsigma }*\sum\limits_{\text{d}} {{\text{N}}_{\text{sd}} \frac{{{\text{P}}_{{ \bullet {\text{sd}}}} }}{{{\text{T}}_{\text{sd}} }}*{\text{Q}}_{{ \bullet {\text{sd}}}} - \sum\limits_{\text{d}} {{\text{N}}_{\text{sd}} {\text{F}}_{\text{sd}} {\text{W}}_{\text{s}} } } \\ & \quad = \frac{1}{\upsigma }*\sum\limits_{\text{d}} {\frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{T}}_{\text{sd}} }} - \frac{{\upbeta^{1 - \upsigma } }}{\upsigma }*\sum\limits_{\text{d}} {\frac{{{\text{V}}({\text{s}},{\text{d}})}}{{{\text{T}}_{\text{sd}} }}} }. \\ \end{aligned}\quad $$

(4.70)

Substituting from (4.68), (4.69) and (4.70) into (4.67) leads to (T4.5P).

• Derivation of (T4.6P)

Applying the addition and multiplication rules to (T4.6) we obtain

$$ \begin{aligned} {\text{W}}_{\text{s}} {\text{L}}_{\text{s}} *\left( {{\text{w}}_{\text{s}} + \ell_{\text{s}} } \right) & = \sum\limits_{\text{d}} {\frac{{{\text{W}}_{\text{s}} {\text{N}}_{\text{sd}} {\text{Q}}_{{ \bullet {\text{sd}}}} }}{{\Phi _{{ \bullet {\text{sd}}}} }}} *\left( {{\text{w}}_{\text{s}} + {\text{n}}_{\text{sd}} + {\text{q}}_{{ \bullet {\text{sd}}}} - \upphi_{{ \bullet {\text{sd}}}} } \right) \\ & \quad + \sum\limits_{\text{d}} {{\text{W}}_{\text{s}} {\text{N}}_{\text{sd}} } {\text{F}}_{\text{sd}} *\left( {{\text{w}}_{\text{s}} + {\text{n}}_{\text{sd}} + {\text{f}}_{\text{sd}} } \right) + {\text{W}}_{\text{s}} {\text{N}}_{\text{s}} {\text{H}}_{\text{s}} *\left( {{\text{w}}_{\text{s}} + {\text{n}}_{\text{s}} + {\text{h}}_{\text{s}} } \right) \\ \end{aligned}\quad . $$

(4.71)

By using (4.3), (T4.1), (4.69), (4.70) and (T4.11) we arrive at (T4.6P).

Appendix 4.4: The Irrelevance of the Absolute Values of the Initial H_ss for Calibration and Estimation

For given values of the parameters σ and α, denote an initial set of variable values that satisfy the model in Table 4.1 by a superscript I. Then increase all of the H values by 1% while holding constant the observable variables V, W, T and D. A new set of variable values, denoted by the superscript N, that satisfy the equations in Table 4.1 is given by (4.72)–(4.88):

$$ {\text{H}}_{\text{s}}^{\text{N}} = {\text{H}}_{\text{s}}^{\text{I}} *(1.01)\quad {\text{for}}\;{\text{all}}\;{\text{s}} $$

(4.72)

$$ \updelta_{\text{sd}}^{\text{N}} = \updelta_{\text{sd}}^{\text{I}} \quad {\text{for}}\;{\text{all}}\;{\text{s}}\;{\text{and}}\;{\text{d}} $$

(4.73)

$$ \text{F}_{\text{sd}}^{\text{N}} = \text{F}_{\text{sd}}^{\text{I}} *(1.01)^{{{{(\upsigma - 1)} \mathord{\left/ {\vphantom {{(\upsigma - 1)} {(\upsigma - 1 - \upalpha )}}} \right. \kern-0pt} {(\upsigma - 1 - \upalpha )}}}} \quad {\text{for}}\;{\text{all}}\;{\text{s}}\;{\text{and}}\;{\text{d}} $$

(4.74)

$$ {\text{V}}^{\text{N}} ({\text{s}},{\text{d}}) = {\text{V}}^{\text{I}} ({\text{s}},{\text{d}})\quad {\text{for}}\;{\text{all}}\;{\text{s}}\;{\text{and}}\;{\text{d}} $$

(4.75)

$$ {\text{L}}_{\text{s}}^{\text{N}} = {\text{L}}_{\text{s}}^{\text{I}} \quad {\text{for}}\;{\text{all}}\;{\text{s}} $$

(4.76)

$$ {\text{W}}_{\text{s}}^{\text{N}} = {\text{W}}_{\text{s}}^{\text{I}} \quad {\text{for}}\;{\text{all}}\;{\text{s}} $$

(4.77)

$$ {\text{D}}_{\text{d}}^{\text{N}} = {\text{D}}_{\text{d}}^{\text{I}} \quad {\text{for}}\;{\text{all}}\;{\text{d}} $$

(4.78)

$$ {\text{T}}_{\text{sd}}^{\text{N}} = {\text{T}}_{\text{sd}}^{\text{I}} \quad {\text{for}}\;{\text{all}}\;{\text{s}}\;{\text{and}}\;{\text{d}} $$

(4.79)

$$ {\text{N}}_{\text{s}}^{\text{N}} = {\text{N}}_{\text{s}}^{\text{I}} *(1.01)^{ - 1} \quad {\text{for}}\;{\text{all}}\;{\text{s}} $$

(4.80)

$$ \Phi_{\hbox{min} (s,d)}^{\text{N}} =\Phi_{{\hbox{min} (\text{s},\text{d})}}^{\text{I}} *(1.01)^{{{1 \mathord{\left/ {\vphantom {1 {(\upsigma - 1 - \upalpha )}}} \right. \kern-0pt} {(\upsigma - 1 - \upalpha )}}}} \quad {\text{for}}\;{\text{all}}\;{\text{s}}\;{\text{and}}\;{\text{d}} $$

(4.81)

$$ \Phi_{{ \bullet \text{sd}}}^{\text{N}} =\Phi_{{ \bullet \text{sd}}}^{\text{I}} *(1.01)^{{{1 \mathord{\left/ {\vphantom {1 {(\upsigma - 1 - \upalpha )}}} \right. \kern-0pt} {(\upsigma - 1 - \upalpha )}}}} \quad {\text{for}}\;{\text{all}}\;{\text{s}}\;{\text{and}}\;{\text{d}} $$

(4.82)

$$ \text{Q}_{{ \bullet \text{sd}}}^{\text{N}} = \text{Q}_{{ \bullet \text{sd}}}^{\text{I}} *(1.01)^{{{\upsigma \mathord{\left/ {\vphantom {\upsigma {(\upsigma - 1 - \upalpha )}}} \right. \kern-0pt} {(\upsigma - 1 - \upalpha )}}}} \quad {\text{for}}\;{\text{all}}\;{\text{s}}\;{\text{and}}\;{\text{d}} $$

(4.83)

$$ \text{Q}_{{\hbox{min} (\text{s},\text{d})}}^{\text{N}} = \text{Q}_{{\hbox{min} (\text{s},\text{d})}}^{\text{I}} *(1.01)^{{{\upsigma \mathord{\left/ {\vphantom {\upsigma {(\upsigma - 1 - \upalpha )}}} \right. \kern-0pt} {(\upsigma - 1 - \upalpha )}}}} \quad {\text{for}}\;{\text{all}}\;{\text{s}}\;{\text{and}}\;{\text{d}} $$

(4.84)

$$ \text{N}_{\text{sd}}^{\text{N}} = \text{N}_{\text{sd}}^{\text{I}} *(1.01)^{{{{(1 - \upsigma )} \mathord{\left/ {\vphantom {{(1 - \upsigma )} {(\upsigma - 1 - \upalpha )}}} \right. \kern-0pt} {(\upsigma - 1 - \upalpha )}}}} \quad {\text{for}}\;{\text{all}}\;{\text{s}}\;{\text{and}}\;{\text{d}} $$

(4.85)

$$ \text{P}_{{ \bullet \text{sd}}}^{\text{N}} = \text{P}_{{ \bullet \text{sd}}}^{\text{I}} *(1.01)^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {(\upsigma - 1 - \upalpha )}}} \right. \kern-0pt} {(\upsigma - 1 - \upalpha )}}}} \quad {\text{for}}\;{\text{all}}\;{\text{s}}\;{\text{and}}\;{\text{d}} $$

(4.86)

$$ \text{Q}_{\text{sd}}^{\text{N}} = \text{Q}_{\text{sd}}^{\text{I}} \quad {\text{for}}\;{\text{all}}\;{\text{s}}\;{\text{and}}\;{\text{d}} $$

(4.87)

$$ \text{P}_{\text{d}}^{\text{N}} = \text{P}_{\text{d}}^{\text{I}} \quad {\text{for}}\;{\text{all}}\;{\text{d}} $$

(4.88)

The validity of this new solution can be established by substituting into left and right hand sides of (T4.1)–(T4.11).

From (4.75), (4.77), (4.78) and (4.79) we see that the proportionate change in the H_ss has no effect on observables V, W, D and T. The F_jrs move uniformly, see (4.74). Thus if the ${\text{F}}_{\text{jr}}^{\text{I}} {\text{s}}$ satisfied Balistreri et al.’s preferred structure, then the ${\text{F}}_{\text{jr}}^{\text{N}} {\text{s}}$ satisfy this structure with proportionate changes in Out_s and In_d.

With regard to Balistreri et al.’s estimating method described by (4.21)–(4.24), we now see that rescaling ${\overline{\text{H}}}_{\text{j}}$ for all j cannot affect the estimates of α and θ. With rescaled H’s the optimal values for ${\text{V}}^{\text{e}} ({\text{s}},{\text{d}})$ can be achieved with the same values for α and θ, the same values for the variables with bars in (4.22), and with rescaled F_jrs.

Readers can check that (4.72)–(4.88) is not the only way to reset the values of the variables in Table 4.1 so that the observables, V, W, D and T, remain unchanged in response to a 1% change in the H_ss. For example, instead of (4.74), the F_sds could increase by 1%. Then we could write down another solution for Table 4.1 with the original values of V, W, D and T. In this new solution, there would be no change in $\Phi _{{ \bullet {\text{sd}}}}^{{}}$ rather than no change in ${\text{Q}}_{\text{sd}}^{{}}$. ${\text{P}}_{{ \bullet {\text{sd}}}}^{{}}$ would not change, N_sd and N_s would decrease by 1%, ${\text{Q}}_{{ \bullet {\text{sd}}}}^{{}}$ would increase by 1%, etc. We suspect that what this means for the Balistreri et al. estimating method is that a normalization must be imposed on Out_s in (4.24). However, on reading the article it wasn’t clear to us how this was done.