Back Matter
Author:
• 1 https://isni.org/isni/0000000404811396, International Monetary Fund

### APPENDIX: Algebraic Presentation of the Model

This Appendix contains an algebraic description of the model. Both the description and the model rely heavily on Keller (1980). The model contains 6 household sectors (including the foreign and public households), 13 production sectors (including the investment sector and the Armington sectors), and 25 goods and factors (including the Armington goods and sector-specific capital).

#### 1. Households sectors

The column-vector ${\mathrm{q}}_{\mathrm{H}}^{\mathrm{i}}$ contains the relative changes in the demands (supplies are measured negatively) of household i for all 25 goods. The household-specific demand functions are given by

$\begin{array}{cc}{\mathrm{q}}_{\mathrm{H}}^{\mathrm{i}}\mathrm{=}{\mathrm{N}}_{\mathrm{H}}^{\mathrm{i}}{\mathrm{P}}_{\mathrm{H}}^{\mathrm{i}}\mathrm{+}{\mathrm{n}}^{\mathrm{i}}\mathrm{\lambda }& \mathrm{\left(}\mathrm{A}\mathrm{.1}\mathrm{\right)}\end{array}$

Here, λ is a scalar representing the relative change in revenue from transaction taxes, for the public household ni is the 25-vector of income elasticities, for the nonpublic households ni is a 25-vector containing zeros, and ${\mathrm{N}}_{\mathrm{H}}^{\mathrm{i}}$ is a 25×25 matrix of uncompensated price elasticities of household i (defined according to equation (1) in Section II, subsection 3).

The symbol ${\mathrm{P}}_{\mathrm{H}}^{\mathrm{i}}$ stands for the 25-vector of after-tax prices facing household i. These prices are related to the relative changes in market prices, which are contained in the 25-vector pM, and to the changes in the 8 tax instruments, which are contained in the 8-vector t, by the 25×8 matrix ${\mathrm{T}}_{\mathrm{H}}^{\mathrm{i}}$:

$\begin{array}{cc}{\mathrm{P}}_{\mathrm{H}}^{\mathrm{i}}\mathrm{=}{\mathrm{P}}_{\mathrm{M}}\mathrm{+}{\mathrm{T}}_{\mathrm{H}}^{\mathrm{i}}\mathrm{t}& \mathrm{\left(}\mathrm{A}\mathrm{.2}\mathrm{\right)}\end{array}$

The 25-vector qH contains the relative changes in demands of the aggregate household sector, which is found by aggregating over the 8 households:

$\begin{array}{cc}{\mathrm{q}}_{\mathrm{H}}\mathrm{=}\underset{\mathrm{i}\mathrm{=}\mathrm{1}}{\overset{\mathrm{8}}{\mathrm{\Sigma }}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\stackrel{\mathrm{^}}{\mathrm{a}}}_{\mathrm{H}}^{\mathrm{i}}{\mathrm{q}}_{\mathrm{H}}& \mathrm{\left(}\mathrm{A}\mathrm{.3}\mathrm{\right)}\end{array}$

Here, ${\stackrel{\mathrm{^}}{\mathrm{a}}}_{\mathrm{H}}^{\mathrm{i}}$ stands for a diagonal 25×25 matrix with the shares of household sector i in aggregate household demands on its diagonal.

Substituting (A.2) and (A.1) into (A.3), aggregate household behavior is described by

$\begin{array}{cc}{\mathrm{q}}_{\mathrm{H}}\mathrm{=}{\mathrm{N}}_{\mathrm{HM}}{\mathrm{P}}_{\mathrm{M}}\mathrm{+}{\mathrm{N}}_{\mathrm{HT}}\mathrm{t}\mathrm{+}\mathrm{n\lambda }& \mathrm{\left(}\mathrm{A}\mathrm{.4}\mathrm{\right)}\end{array}$

with

$\begin{array}{cc}{\mathrm{N}}_{\mathrm{HM}}\mathrm{=}\underset{\mathrm{i}\mathrm{=}\mathrm{1}}{\overset{\mathrm{8}}{\mathrm{\Sigma }}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\stackrel{\mathrm{^}}{\mathrm{a}}}_{\mathrm{H}}^{\mathrm{i}}{\mathrm{N}}_{\mathrm{H}}^{\mathrm{i}}& \mathrm{\left(}\mathrm{A}\mathrm{.5}\mathrm{\right)}\end{array}$
$\begin{array}{cc}{\mathrm{N}}_{\mathrm{HT}}\mathrm{=}\underset{\mathrm{i}\mathrm{=}\mathrm{1}}{\overset{\mathrm{8}}{\mathrm{\Sigma }}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\stackrel{\mathrm{^}}{\mathrm{a}}}_{\mathrm{H}}^{\mathrm{i}}{\mathrm{N}}_{\mathrm{H}}^{\mathrm{i}}{\mathrm{T}}_{\mathrm{H}}^{\mathrm{i}}& \mathrm{\left(}\mathrm{A}\mathrm{.6}\mathrm{\right)}\end{array}$
$\begin{array}{cc}\mathrm{n}\mathrm{=}{\stackrel{\mathrm{^}}{\mathrm{a}}}_{\mathrm{n}}^{\mathrm{P}}{\mathrm{n}}^{\mathrm{P}}& \mathrm{\left(}\mathrm{A}\mathrm{.7}\mathrm{\right)}\end{array}$

Here, the superscript p represents the public household.

#### 2. Production sectors

The column-vector ${\mathrm{q}}_{\mathrm{F}}^{\mathrm{i}}$ contains the relative changes in the supplies (demands are measured negatively) of firm j. The firm-specific supply functions are given by

$\begin{array}{cc}{\mathrm{q}}_{\mathrm{F}}^{\mathrm{i}}\mathrm{=}{\mathrm{N}}_{\mathrm{F}}^{\mathrm{j}}{\mathrm{P}}_{\mathrm{F}}^{\mathrm{i}}\mathrm{+}\mathrm{1}{\mathrm{q}}_{\mathrm{sj}}& \mathrm{\left(}\mathrm{A}\mathrm{.8}\mathrm{\right)}\end{array}$

Here, qsj stands for the output level of firm jɩ is a 25-vector that consists of unit elements, and ${\mathrm{N}}_{\mathrm{F}}^{\mathrm{j}}$ is a 25×25 matrix of uncompensated price elasticities for firm j (defined according to equation (2) in Section II, subsection 3).

${\mathrm{P}}_{\mathrm{F}}^{\mathrm{j}}$ stands for the 25-vector of after-tax prices facing firm j. These prices are related to the tax instruments by the 25×8 matrix ${\mathrm{T}}_{\mathrm{F}}^{\mathrm{j}}$:

$\begin{array}{cc}{\mathrm{P}}_{\mathrm{F}}^{\mathrm{j}}\mathrm{=}{\mathrm{P}}_{\mathrm{M}}\mathrm{+}{\mathrm{T}}_{\mathrm{F}}^{\mathrm{j}}\mathrm{t}& \mathrm{\left(}\mathrm{A}\mathrm{.9}\mathrm{\right)}\end{array}$

The zero-profit condition for firm j is described by

$\begin{array}{cc}{\mathrm{C}}_{\mathrm{F}}^{\mathrm{j}\mathrm{\prime }}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{P}}_{\mathrm{F}}^{\mathrm{j}}\mathrm{=}\mathrm{0}& \mathrm{\left(}\mathrm{A}\mathrm{.10}\mathrm{\right)}\end{array}$

Here, ${\mathrm{c}}_{\mathrm{F}}^{\mathrm{j}}$ is the 25-vector of cost shares of firm j.

The 25-vector qp contains the relative changes in supplies of the aggregate production sector, which is found by aggregating over the 13 firms:

$\begin{array}{cc}{\mathrm{q}}_{\mathrm{F}}\mathrm{=}\underset{\mathrm{j}\mathrm{=}\mathrm{1}}{\overset{\mathrm{13}}{\mathrm{\Sigma }}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\stackrel{\mathrm{^}}{\mathrm{a}}}_{\mathrm{F}}^{\mathrm{j}}{\mathrm{q}}_{\mathrm{F}}& \mathrm{\left(}\mathrm{A}\mathrm{.11}\mathrm{\right)}\end{array}$

Here, ${\stackrel{\mathrm{^}}{\mathrm{a}}}_{\mathrm{F}}^{\mathrm{j}}$ represents a diagonal 25×25 matrix with the shares of firm j in aggregate firm supplies on its diagonal.

Using (A.8), (A.9), (A.10), and (A.11), aggregate firm behavior is described by

$\begin{array}{cc}{\mathrm{q}}_{\mathrm{F}}\mathrm{=}{\mathrm{N}}_{\mathrm{FM}}{\mathrm{P}}_{\mathrm{M}}\mathrm{+}{\mathrm{N}}_{\mathrm{FT}}\mathrm{t}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{+}{\mathrm{A}}_{\mathrm{F}}{\mathrm{q}}_{\mathrm{S}}& \mathrm{\left(}\mathrm{A}\mathrm{.12}\mathrm{\right)}\end{array}$
$\begin{array}{cc}\mathrm{U}\mathrm{=}{\mathrm{C}}_{\mathrm{FM}}^{\mathrm{\prime }}{\mathrm{P}}_{\mathrm{M}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{+}{\mathrm{C}}_{\mathrm{FT}}^{\mathrm{\prime }}\mathrm{t}& \mathrm{\left(}\mathrm{A}\mathrm{.13}\mathrm{\right)}\end{array}$

with

$\begin{array}{cc}{\mathrm{N}}_{\mathrm{FM}}\mathrm{=}\underset{\mathrm{j}\mathrm{=}\mathrm{1}}{\overset{\mathrm{13}}{\mathrm{\Sigma }}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\stackrel{\mathrm{^}}{\mathrm{a}}}_{\mathrm{F}}^{\mathrm{j}}{\mathrm{N}}_{\mathrm{F}}^{\mathrm{j}}& \mathrm{\left(}\mathrm{A}\mathrm{.14}\mathrm{\right)}\end{array}$
$\begin{array}{cc}{\mathrm{N}}_{\mathrm{FT}}\mathrm{=}\underset{\mathrm{j}\mathrm{=}\mathrm{1}}{\overset{\mathrm{13}}{\mathrm{\Sigma }}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\stackrel{\mathrm{^}}{\mathrm{a}}}_{\mathrm{F}}^{\mathrm{j}}{\mathrm{N}}_{\mathrm{F}}^{\mathrm{j}}{\mathrm{T}}_{\mathrm{F}}^{\mathrm{j}}& \mathrm{\left(}\mathrm{A}\mathrm{.15}\mathrm{\right)}\end{array}$

and

$\begin{array}{cc}\mathrm{j}\mathrm{-}\mathrm{th}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{column}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{of}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{C}}_{\mathrm{FM}}\mathrm{=}{\mathrm{c}}_{\mathrm{F}}^{\mathrm{j}}& \mathrm{\left(}\mathrm{A}\mathrm{.16}\mathrm{\right)}\end{array}$
$\begin{array}{cc}{\mathrm{C}}_{\mathrm{FT}}\mathrm{=}\mathrm{\left(}{\mathrm{T}}_{\mathrm{F}}^{\mathrm{j}}\mathrm{\right)}\prime {\mathrm{c}}_{\mathrm{F}}^{\mathrm{j}}& \left(\begin{array}{c}\mathrm{A}\mathrm{.17}\end{array}\right)\end{array}$
$\begin{array}{cc}\mathrm{j}\mathrm{-}\mathrm{th}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{column}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{of}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{A}}_{\mathrm{F}}\mathrm{=}{\mathrm{a}}_{\mathrm{F}}^{\mathrm{j}}& \mathrm{\left(}\mathrm{A}\mathrm{.18}\mathrm{\right)}\end{array}$

${\stackrel{\mathrm{^}}{\mathrm{a}}}_{\mathrm{F}}^{\mathrm{j}}$, is the 25-vector with shares of firm j in aggregate firm supplies.

In order to arrive at a closed-form solution for the aggregate firm sector, the 13-vector qs, which contains the output levels of the firms, is eliminated from (A.12) to arrive at 25–13 = 12 independent equations in qp Together with (A.13) these equations yield 25 independent equations for the aggregate production sector.

#### 3. Equilibrium

Combining the equations for the aggregate household and production sectors with the equilibrium condition

$\begin{array}{cc}{\mathrm{q}}_{\mathrm{H}}\mathrm{=}{\mathrm{q}}_{\mathrm{F}}& \mathrm{\left(}\mathrm{A}\mathrm{.19}\mathrm{\right)}\end{array}$

and fixing the world price of the foreign good, the price-vector pM and total tax revenue λ can be solved for.

## References

• Armington, P.,A Theory of Demand for Products Distinguished by Place of Production,Staff Papers, International Monetary Fund (Washington, D.C.), Vol. 16 (July 1969), pp. 15978.

• Crossref
• Export Citation
• Beveridge, W.A., and M.R. Kelly,fiscal Content of Financial Programs Supported by Stand-By Arrangements in the Upper Credit Tranches, 1969-78,Staff Papers, International Monetary Fund (Washington, D.C.), Vol. 27 (June 1980), pp. 20549.

• Crossref
• Export Citation
• Bovenberg, A.L.,The General Equilibrium Approach: Relevant for Public Policy?Paper presented to the 41st Congress of the International Institute of Public Finance, Madrid, Spain (August 1985).

• Export Citation
• Bovenberg, A.L., and W.J. Keller,Nonlinearities in Applied General Equilibrium Models,Economics Letters, Vol. 14 (February 1984), pp. 5359.

• Crossref
• Export Citation
• Cooper, R., and K. McLaren,The Orani-Macro Interface: An Illustrative Exposition,The Economic Records, Vol. 59 (June 1983), pp. 16679.

• Crossref
• Export Citation
• Cornielje, O.J.C., and W.J. Keller,Exploring Malinvaud-type Disequilibrium Models Using Virtual Taxes,Discussion Paper, Free University Amsterdam (Amsterdam, 1984).

• Export Citation
• Dervis, K., J. de Melo, and S. Robinson, General Equilibrium Models for Development Policy (New York.: Cambridge University Press, 1982).

• Export Citation
• Devarajan, S., and H. Sierra,Growth Without Adjustment: Thailand, 1973-1982,Research Paper, World Bank (Washington, D.C., 1985).

• Export Citation
• Ebrill, L.P.,The Effects of Taxation on Labor Supply, Savings, and Investment in Developing Countries: A Survey of the Empirical Literature,International Monetary Fund (Washington, D.C.), DM/84/23 (1984).

• Export Citation
• Gandhi, V.P.,Vertical Equity of General Sales Taxation in Developing Countries,International Monetary Fund (Washington, D.C.), DM/79/52 (1979).

• Export Citation
• Goldstein, M., and M.S. Khan,The Supply and Demand for Exports: A Simultaneous Approach,Review of Economics and Statistics, Vol. 60 (1978), pp. 27586.

• Crossref
• Export Citation
• Johansen, L., A Multi-Sectoral Study of Economic Growth (Amsterdam: North-Holland, 1960).

• Keller, W.J., Tax Incidence: A General Equilibrium Approach (Amsterdam: North-Holland, 1980).

• Shoven, J., and J. Whalley,Applied General Equilibrium Models of Taxation and International Trade: Introduction and Survey,Journal of Economic Literature, Vol. 22 (September 1984), pp. 100751.

• Export Citation
• Tanzi, V.,Quantitative Characteristics of the Tax Systems of Developing Countries,International Monetary Fund (Washington, D.C.), DM/83/79 (1983).

• Export Citation

The author would like to thank Shanta Devarajan and Hecktor Sierra for providing data, Wouter Keller for providing computational assistance, and Vito Tanzi, Ved Gandhi, Sheetal Chand, Partho Shome, Thanos Catsambas, Somchai Richupan, Wouter Keller, Shanta Devarajan, Sweder van Wijnbergen, Phil Young, and Sara Chernick for their comments and suggestions.

For the tax content of stand-by arrangements, see Beveridge and Kelly (1980).

This study used the computer software “The Keller Model, Free University Amsterdam, for IBM-PC,” which was kindly provided by W.J. Keller.

In order to compute the effects of large changes in policy, global nonlinear methods require global information on the various structural relations. The method applied in this paper only requires local information. Linear models, however, can account for non-linearities by adopting a procedure of iterative linearization. See Bovenberg and Keller (1984).

Although the static model does not compute changes in stocks, the simulated changes in prices provide some information on the dynamic effects of taxation. To illustrate, profitability in each production sector provides an indication for the effects on investment decisions in each sector and the interindustry allocation of capital (see Section III).

For a similar model of the Australian economy--the Orani model--the reference period has been estimated to be about one and a half to two years. See Cooper and McLaren (1983).

This paper defines services as a good.

All foreign goods can be aggregated into a single foreign good because Thailand takes the prices of foreign goods as exogenously given.

Under certain conditions (see Keller (1980)), this utility function can be derived from the preferences for public goods of private households.

For a more detailed discussion of the systems of indirect taxation in developing countries, see Gandhi (1977) and Tanzi (1983).

The elasticities refer to the reference period, which is about three years. See subsection 2 above (p. 4).

See Keller (1980) for a derivation of (1) as well as for a more detailed discussion on nested CES utility functions.

Empirical studies on savings behavior in developing countries have not resolved whether an increase in interest rates will raise the savings rate. See, for example, Ebrill (1984).

Given the technical nature of this section, the reader who is mainly interested in the policy implications of the simulation results can skip this section and continue with Section IV, which summarizes the major lessons and policy implications from this study.

These changes are presented in the first columns of Tables 2-4.

This comparison amounts to the difference between the second and first columns of Tables 2-4.

This comparison amounts to the difference between the third and second columns of Tables 2-4.

This comparison amounts to the difference between the fourth and third columns of Tables 2-4.

This comparison amounts to the difference between the fifth and the second columns of Tables 2-4.

These relative effects can be interpreted as the reverse effects of a tax package which reduces cascading and rate differentiation. This tax package consists of a cut in the rates of excises and the business tax compensated for by the introduction of an income-based VAT such that public consumption is unaffected. The comparison in this subsection amounts to the difference between the sixth and first columns in Tables 2-4.

Table 1, Part A contains these differential tax rates.

This comparison amounts to the difference between the seventh and first columns of Tables 2-4.

Dervis, de Melo, and Robinson (1982), for example, show in a general equilibrium study on Turkey that overvalued exchange rates and import rationing induce relatively large efficiency losses, particularly when accompanied by rent seeking. These results suggest that in many developing countries the efficiency losses from implicit taxes dominate the efficiency costs from explicit taxes. For a more general way to model “implicit” taxes in a general equilibrium framework, see Cornielje and Keller (1984).

Indirect Taxation in Developing Countries: A General Equilibrium Approach
Author: Ary Lars Bovenberg