Appendix

Abstract

IMPACT (International Model for Policy Analysis of Agricultural Commodities and Trade) is a global agricultural sector partial equilibrium model developed in the early 1990s at the International Food Policy Research Institute in response to the lack of a long-term vision and consensus about the actions that are necessary to feed the world in the future, reduce poverty, and protect the natural resource base. IMPACT models crop and livestock commodities, including cereals, soybeans, roots and tubers, meats, milk, eggs, oilseeds, oilcakes/meals, sugar/sweeteners, and fruits and vegetables. It is specified as a set of 115 country and regional sub-­models, within each of which supply, demand, and prices for agricultural commodities are determined. The model links the various countries and regions through international trade using a series of linear and nonlinear equations to approximate the underlying production and demand functions. World agricultural commodity prices are determined annually at levels that clear international markets. Growth in crop production in each country is determined by crop and input prices, the rate of productivity growth, investment in irrigation, and water availability. Demand is a function of prices, income, and population growth. IMPACT generates projections for crop area, yield, production, demand for food, feed and other uses, prices, and trade; and for livestock numbers, yield, production, demand, prices, and trade.

Appendices

Appendix Chapter 2 Description of the IMPACT Model

IMPACT (International Model for Policy Analysis of Agricultural Commodities and Trade) is a global agricultural sector partial equilibrium model developed in the early 1990s at the International Food Policy Research Institute in response to the lack of a long-term vision and consensus about the actions that are necessary to feed the world in the future, reduce poverty, and protect the natural resource base. IMPACT models crop and livestock commodities, including cereals, soybeans, roots and tubers, meats, milk, eggs, oilseeds, oilcakes/meals, sugar/sweeteners, and fruits and vegetables. It is specified as a set of 115 country and regional sub-­models, within each of which supply, demand, and prices for agricultural commodities are determined. The model links the various countries and regions through international trade using a series of linear and nonlinear equations to approximate the underlying production and demand functions. World agricultural commodity prices are determined annually at levels that clear international markets. Growth in crop production in each country is determined by crop and input prices, the rate of productivity growth, investment in irrigation, and water availability. Demand is a function of prices, income, and population growth. IMPACT generates projections for crop area, yield, production, demand for food, feed and other uses, prices, and trade; and for livestock numbers, yield, production, demand, prices, and trade.

In order to explore the relationships among water, environment, and food production, a global modeling framework, IMPACT-WATER, has been developed that combines an extension of IMPACT with the Water Simulation Model (WSM). The WSM incorporates water availability as a stochastic variable with observable probability distributions. Water demand is estimated for irrigation and non-­irrigation uses (domestic, industrial, and livestock water demand). Irrigation water demand is projected based on irrigated area, crop evapotranspiration requirements, effective rainfall, soil and water quality (salinity leaching requirements), and basin-level irrigation-water-use efficiency. Livestock water demand is assessed based on livestock production, water price, and water consumptive use per unit of livestock production including beef, milk, pork, poultry, eggs, sheep and goats, and aquaculture fish production. Industrial water demand depends on income (GDP per capita), water use technology improvements, and water prices. Domestic water demand is estimated separately for rural and urban users, based on projections of population, income growth, and water prices. In each country or basin, income and price elasticities of demand for domestic use are synthesized based on available estimates from the literature. Committed flow is estimated as a portion of average annual runoff.

The WSM then simulates water availability for crops at a river basin scale, taking into account precipitation and runoff, water use efficiency, flow regulation through reservoir and groundwater storage, nonagricultural water demand, water supply infrastructure and withdrawal capacity, and environmental requirements at the river basin, country, and regional levels. Environmental impacts can be explored through scenario analysis of committed instream and environmental flows, salt-leaching requirements for soil salinity control, and alternative rates of groundwater pumping.

Appendix Chapter 5 A Model of Linkage

This appendix presents a simple model of linkage in a game theoretic context, drawing on the work of Limão (2005).

Suppose that two countries, Row and Column, share a river that they both pollute and also trade with one another. They can cooperate in the trade sphere by liberalizing their barriers to trade; they can cooperate in the environmental sphere by regulating pollution. They can deviate from cooperation in the trade sphere by raising trade barriers; they can deviate from cooperation in the environmental sphere by relaxing pollution regulation.

Suppose for simplicity that we can represent cooperation in either trade or environmental policies as a symmetric Prisoner’s Dilemma. Let 0 represent the payoff to each government from cooperation in each sphere. Suppose also that the effect of each policy on each government’s payoff is independent of the policy adopted in the other sphere.

First, suppose that there is no linkage between trade and environmental policies, so that we can treat the game in each sphere as independent of the game in the other sphere. Let the following matrix represent the effects on the payoffs to the governments of Row and Column, respectively, for each game:

Effect on payoff (Row, Column)	Cooperate	Deviate
Cooperate	0, 0	−1, d i
Deviate	d i , −1	−g i , −g i

where i = T for the Prisoner’s Dilemma in the trade sphere and i = E for the Prisoner’s Dilemma in the environmental sphere, 0 < d _i , and 0 <g _i <1. The dominant strategy in this one-shot game is for each player to deviate from cooperation, but the resulting Nash equilibrium yields (−g _i , −g _i ) as its payoffs, which is Pareto inferior to the (0, 0) payoffs from cooperation.

Suppose, however, this game is repeated over an infinite time horizon in a supergame, and each government observes the other government’s actions at the end of each period. The Nash equilibrium in the one-shot game, with each player deviating from cooperation, repeated infinitely is an equilibrium of the supergame. This “grim strategy,” however, is not the only equilibrium of the supergame, which allows one player to retaliate in the future against the other player for deviating from cooperation in the current period. We know from the Folk Theorem in game theory that the parties can sustain cooperation through the threat of future retaliation as long as both the rate of time preference and the probability that the supergame ends in any stage of the supergame are sufficiently small.

With no linkage between trade and environmental policies, we can treat the supergame in each sphere as independent of the other supergame. For each sphere, suppose that each player adopts the following strategy:

• 1. Cooperate in the first stage.

• 2. In stage t, cooperate as long as both players have cooperated in all t-1 preceding stages; otherwise, deviate.

This strategy is an example of a “trigger strategy,” because a single deviation from cooperation triggers a permanent end to cooperation. If one player ever deviates, it gains d _i during the period of deviation but destroys cooperation in all subsequent periods, as both players revert to the “grim strategy” of deviation forever, which is itself an equilibrium. The grim strategy costs each player g _i in each period they play the Nash equilibrium of the stage game.

Suppose for simplicity that both governments share the same discount factor. Let D ^* be the discount factor, and let p be the probability that the supergame continues for one more period, where 0 <D ^* <1 and 0 < p < 1. Then each government discounts future payoffs at the rate D = pD ^*, where 0 < D <1.

The proposed trigger strategies will sustain cooperation in each period as equilibrium of each supergame as long as the following incentive constraint is satisfied in each period for each i:

$${d_i}\ >= {g_i}/({{ 1}} - D)$$

which we can also express as follows:

$$D > = d_i / (d_i + g_i)$$

Thus, as long as the discount factor D is sufficiently close to 1, we can attain the cooperative payoffs (0, 0) in each period as equilibrium. Let D _i represent the smallest D that satisfies the incentive constraint, that is:

$$D_i = d_i / (d_i + g_i )$$

Now suppose we allow linkage between trade and environmental policies. Under this regime, deviation from cooperation in either sphere may trigger retaliation in either or both spheres. That is, each government chooses to cooperate in both spheres as long as the history of play for both players in both spheres has been cooperation in each past period. Otherwise, the government reverts to the grim strategy in either or both spheres.

As the players can create the greatest incentives for cooperation by threatening the most costly retaliation, the governments would create the greatest scope for cooperation (i.e., sustain cooperation in both spheres for the lowest D) by retaliating for any deviation by reverting to grim strategies in both spheres. If the parties adopt this trigger strategy, then any party choosing to deviate from cooperation would choose to deviate in both spheres rather than just one. In this case, we only need to consider cooperation, deviation, or grim strategies in both spheres simultaneously. If the parties adopt this proposed trigger strategy, then they can sustain cooperation in both spheres as long as the following incentive constraint is satisfied in each period:

$${d_T} + {d_E}\ <= D \left( {{g_T} + {g_E}} \right)/({{\rm 1 }} - D^*),$$

which we can also express as follows:

$$D > = \left( {{d_T} + {d_E}} \right)/\left( {{d_T} + {d_E} + {g_T} + {g_E}} \right)$$

Thus, as long as the discount factor D is sufficiently close to 1, we can again attain the cooperative payoffs (0, 0) in each period as equilibrium.

Let D _{T + E} represent the smallest D* that satisfies the incentive constraint, that is:

$${D_{T + E}} = \left( {{d_T} + {d_E}} \right)/\left( {{d_T} + {d_E} + {g_T} + {g_E}} \right)$$

Using the definition of D _T and D _E we can express D _T+E as a weighted average of D _T and D _E :

$${D_{T + E}} = {{\rm }}[{D_T}\left( {{d_T} + {g_T}} \right) + {D_E}\left( {{d_E} + {g_E}} \right)]/\left( {{d_T} + {d_E} + {g_T} + {g_E}} \right).$$

Therefore, if D _T < D _E , then D _T < D _T+E < D _E . That is, if cooperation is easier to sustain in the trade supergame than in the environmental supergame, in the sense that the parties can sustain cooperation in trade policies without linkage for lower D than they can in environmental policies, then linkage will promote greater environmental cooperation, in the sense that the parties can sustain cooperation in environmental policies for lower D with linkage than they can without linkage. We can interpret this result as implying greater environmental cooperation under linkage if we imagine applying linkage to a population of cases with D as a random variable. If cooperation is easier to sustain in the trade sphere than in the environmental sphere, in the sense that D _T < D _E , then linkage would yield environmental cooperation in a larger fraction of those cases than the absence of linkage would yield.¹

Appendix Chapter 6 Description of the H-O Theorem Applied to Water

To prove the H-O theorem when applied to water as an input in the agricultural process, we start with the equilibrium identity expressing a country’s net water exports as the difference between water absorbed in production and water absorbed in consumption:

$$ {A_i}{T_i} = {A_i}{Q_i} - {A_i}{C_i} $$

((1))

where Ai = 1 x N is the vector of water requirements or the total amount of water needed to produce one unit of agricultural output in each of N countries.

T _i = N x 1 is the vector of net trade agricultural flows of country i;

Q _i = N x 1 is the vector of country i’s final agricultural outputs;

C _i = N x 1 is the vector of country i’s final agricultural consumption.

Full employment implies A _i Q _i = E _i , where E _i is the country i’s water supplies. Thus, the vector of water embodied in net trade is:

$$ {A_i}{T_i} = {E_i} - {A_i}{C_i} $$

((2))

This identity is transformed into a testable hypothesis by making one or more of the following assumptions (Bowen et al. 1987):

(A1) Assumption 1: All individuals face the same commodity prices;

(A2) Assumption 2: Individuals have identical and homothetic tastes; and

(A3) Assumption 3: All countries have the same factor input matrix, A_i=A.

The postulate of identical input matrices (A3) is replaced by the assumption of factor price equalization and internationally identical technologies.

Assumptions (A1) and (A2) imply that the consumption vector of country i is proportional to the world agricultural vector (Q _w ), C _i =s _i Q _w , where s _i is country i’s consumption share (Bowen et al. 1987). The consumption share can be derived by premultiplying the net trade identity (T _i =Q _i −s _i Q _w ) by the vector of common goods prices:

$$ {s_i} = \frac{{\left( {{Y_i} - {B_i}} \right)}}{{{Y_w}}} $$

((3))

where Y _i is the agricultural product and B _i is the agricultural trade balance of country i. If agricultural trade is balanced, then s _i equals country i’s share of world agricultural product (Y _w ).

If, in addition, the water input vector is identical, we can write

A _i C _i =s _i AQ _w =s _i E _w , where E _w =Σ _i E _i is the world water supply. Then, (2) can be written as:

$$ A{T_i} = {E_i} - {E_w}\frac{{\left( {{Y_i} - {B_i}} \right)}}{{{Y_w}}} $$

((4))

Equation (4) specifies an exact relationship between water content and water endowment. This relationship can be tested by measuring the net export vector T _i , the water input vector A, and the excess water supplies E _i -s _i E _w , and computing the extent to which these data violate the equality given by (4). Such analysis requires some sensible way of measuring the distances between two vectors: the vector equal to the water contents of trade for each country, and the vector equal to the excess water supplies for each country. The empirical analysis used data on estimates of virtual water flows for N=126 countries.

3.1 Test of Qualitative Hypothesis

The traditional implication of the H-O theory is that water abundance determines which agricultural commodities are exported and which are imported, in other words, the sign of net exports. A typical element of (4) can be written as:

$$ \frac{{\left( {{{{F_i}^A} \mathord{\left/{\vphantom {{{F_i}^A} {{E_w}}}} \right.} {{E_w}}}} \right)}}{{\left( {{{{Y_i}} \mathord{\left/{\vphantom {{{Y_i}} {{Y_w}}}} \right.} {{Y_w}}}} \right)}} = \left[ {\frac{{\left( {{{{E_i}} \mathord{\left/{\vphantom {{{E_i}} {{E_w}}}} \right.} {{E_w}}}} \right)}}{{\left( {{{{Y_i}} \mathord{\left/{\vphantom {{{Y_i}} {{Y_w}}}} \right.} {{Y_w}}}} \right)}}} \right] - 1 $$

((5))

where F _i is an element of the water content vector ${F_i} = A{T_i}$ and ${F_i}^A = \left( {{F_i} - {{{E_w}{B_i}} \mathord{\left/{\vphantom {{{E_w}{B_i}} {{Y_w}}}} \right.} {{Y_w}}}} \right)$ is the water content if agricultural trade is balanced. The quantity on the right-hand side of (5) is a measure of the relative abundance of water. If this equation is correct, the water content of trade can be used as an indirect measure of water abundance and the sign of the net trade in factor service, corrected for the trade imbalance, will reveal the abundance of water compared with other resources, on the average. This sign proposition is tested by computing the proportion of sign matches between corresponding elements in each row of the vector of adjusted water contents and the vector of water abundance ratios. In addition, Fisher’s Exact Test (one-tail) is used to test the hypothesis of independence between the sign of the water contents and of the excess water shares against the alternative of positive association.

Equation (5) also implies that trade reveals the relative abundance of water resources when two countries are considered at a time. If the adjusted net exports by country i of water exceeds the adjusted net exports by country i’s of water, $\left( {{{{F_i}^A} \mathord{\left/{\vphantom {{{F_i}^A} {{E_w}}}} \right.} {{E_w}}}} \right){{\rm /}}\left( {{{{Y_i}} \mathord{\left/{\vphantom {{{Y_i}} {{Y_w}}}} \right.} {{Y_w}}}} \right)$ $> \left( {{{{F_{i^\prime}}^A} \mathord{\left/{\vphantom {{{F_i^{\prime}}^A} {{E_w}}}} \right.} {{E_w}}}} \right){{\rm /}}\left. {{{Y_{i^\prime}} \mathord{\left/{\vphantom {{{Y_i}} {{Y_w}}}} \right.} {{Y_w}}}}\right)$ if, and only if country i is more abundant in water than country i’s, $\left( {{{{E_i}} \mathord{\left/{\vphantom {{{E_i}} {{E_w}}}} \right.} {{E_w}}}} \right){{\rm /}}\left( {{{{Y_i}} \mathord{\left/{\vphantom {{{Y_i}} {{Y_w}}}} \right.} {{Y_w}}}} \right)$ $> \left( {{E_{i^{\prime}}} \mathord{\left/\vphantom {{{E_{i^{\prime}}} {{E_w}}}} \right.} {{E_w}}} \right){{\rm /}}\left( {{{Y_{i^\prime}} \mathord{\left/\vphantom {{{Y_{i^\prime}} {{Y_w}}}} \right.} {{Y_w}}}} \right)$. More generally, for each country, the ranking of adjusted net water exports$\left( {{{{F_i}^A} \mathord{\left/{\vphantom {{{F_i}^A} {{E_i}}}} \right.} {{E_i}}}} \right)$, should match the ranking of water by their abundance. This rank proposition is tested for each country by computing the Kendall rank correlation between corresponding rows of the vector of adjusted factor content and the vector of factor abundance ratios.

Table 6.1 summarizes the water content data by listing for each country the ratio of the adjusted net trade in water to the national endowment of water in 2002: $(F_i^A /E_i )$. According to these data, for example, Mexico imports 0.46 percent of the services of its water stock.

Formal tests of the commodity of the adjusted net factor export data $(F_i^A /E_w )/(Y_i /Y_w )$ with the factor abundance data $[(F_i^A /E_w )/(Y_i /Y_w )] - 1$ are reported in Table 6.2.

Appendix Chapter 9 Tables

Table A.9.1 List of PRINWASS case studies

Table A.9.2 International investments flows involving PSP in water and sanitation by region (1990-2005) (Author’s elaboration based on World Bank 2006b)