Research Methods in Nutritional Anthropology (UNU, 1989, 201 pages) 
6. Elementary mathematical models and statistical methods for nutritional anthropology 

Many foodrelated activities involve flows and exchanges. Food procurement, for instance, frequently involves the flow of food among diverse, specialized sectors of a socioeconomic system. Meat from hunters goes to gatherers, while vegetables go from gatherers to hunters. At a more basic level, energy itself can be considered the currency for transactions among the components of an ecosystem (Hannon, 1973; Johnson, 1978, pp. 7595). What is often difficult to identify, let alone describe and analyse, is the structure of the direct and indirect relations of interdependence among: (a) a set of components represented as an endogenous system; and (b) the relation of this system to exogenous environmental variables. Returning to our example we might ask: How dependent are hunters on gatherers for the food energy necessary for them to produce food energy for themselves, the gatherers, and nonproductive dependents? How dependent are gatherers on each other, hunters, and the environment? What is the nature of these interrelationships through time? Is an equilibrium point ever attained? And if so, what is it?
Even with regard to "simple" systems, these are complex questions and investigating them requires, inter alia, a precise language for representing patterned relations of interdependence among a set of elements. Many branches of mathematics (e.g. matrix algebra and graph theory) provide this language.
Here we briefly describe and present an attenuated illustration of inputoutput analysis (IOA), a mathematical model for analysing relations of interdependence in an ecomponent system. It was developed originally by econometricians (Leontief, 1966) to examine intersectorial relations in complex, national economic systems. It is also being vigorously used by ecologists to study the structure and dynamics of ecosystems (Hannon, 1973; Finn, 1976; Richey et al., 1978). An IOA can provide: (a) definitions and representations of the structure of the direct and indirect flows among the n  components of an endogenous system; (b) information on the way in which direct, exogenous inputs to, and demands on, the endogenous system ramify directly and indirectly throughout it; and (c) information on the nature of equilibrium conditions for system maintenance. IOA can take many forms: linearnonlinear, openclosed, staticdynamic. We will focus on the linear, open, static version.
Example 7
To illustrate an IOA, we will present a muchabridged description of !Kung calorie production and flows, analysed in detail by Carlson (1978), using data from Lee (1969). All values are calories x 10^{3}.
An IOA begins with a flow matrix, F. Putting the value of each element in the ith row and jth column, the f_{ij} elements denote the output of the ith row component (source of supply) to the jth column component (destination). In the open model we append another column, D, the d_{i} elements of which represent the direct demand from the ith row source to an exogenous sector. In economic production systems the matrix F normally represents inter and intraindustry flows, and D represents the nonproducing consumer sector which makes direct purchase demands on it. Thus, each row and column of F represents a finite set of einterdependent industries and the values of the f_{ij} elements are flows among them. D represents the outside demand on the system and the values of the di elements are the demands on each specific industry. The total flow, or output of the system, X, is a column vector, the x, elements of which represent the total amounts of output required from each ith row source (or industry) to meet both system and outside requirements. Thus,
(66)
For the !Kung this is:
M N
where f_{11} = 69 is the amount of meat calories provided by hunters and consumed by hunters; f_{12} = 48.02 is the amount of meat calories provided by hunters and consumed by nutgatherers; f_{21} = 126 is the amount of mongongonut calories provided by nutgatherers and consumed by hunters; and f_{22} = 87.7 is the amount of nut calories provided by nutgatherers and consumed by nutgatherers. The amount of meat calories used by nonprocuring dependents (e.g. children and old people) is d_{1}, which equals 494.04, and d_{2}=902.16 is the amount of nut calories used by nonprocuring dependents. Finally, x1, the total output of meat calories required from hunters by hunters and gatherers (f_{11} + f_{12}) and dependents (d_{1}) is (69 + 48.02) + 494.04 = 611.06. And x_{2} is interpreted similarly. Thus, reading down each column of F gives the required inputs from each ith source to a jth industry. Reading across the rows of F gives the jth destination of the outputs from each ith industry.
Next, we use the elements in F and X to construct an inputoutput matrix, A. Each aij element of A denotes the fractional amount of the output of a rowcomponent industry i, used by a columncomponent industry j, to produce a unit of j. (These aij's are often called "technological coefficients.") Or
(67) a_{ij} = f_{ij}/x_{i}
where x_{i} is the total output of industry i.
In the !Kung example
Meat Nuts
Thus, a_{11} = .113 is a ratio of meat calories used to meat calories provided and means that .113 calories of meat are used by hunters to provide a calorie of meat; and a_{21} = .206 is the ratio of nut calories used by hunters to meat calories provided by hunters in order for them to provide one calorie of meat. Column 2 is interpreted similarly. Thus, A defines the equilibrium or maintenance conditions of the foodprocurement system. For an open system to operate feasibly (i.e. meet industry and outside requirements), at least one column of A must sum to < 1. Otherwise, it will operate at a loss.
With the data in this form, we will ask two questions of the model: (a) What level of production is needed from these industries, both separately and together, to maintain the production system and satisfy the demand from the sector of nonproducing dependents? (b) What level of production would be needed from these industries, both separately and together, to maintain the production system (i.e. maintain equilibrium) and satisfy the demand of the nonproducing dependent sector if the nature of their demands changes in a specifiable way?
To answer question (1) we will represent the model as a system of linear equations.
(68) x_{1} = a_{11}x_{1} + a_{12}x_{2} + d_{1}
x_{2} = a_{21}x_{1} + a_{22}x_{2} + d_{2}
where x_{n} are the elements of X, specifying the total output required of each industry; a_{nn} are the elements of A specifying proportionally coefficients of output required of each industry from each industry; and d_{n} are elements of D specifying the outside demand on each industry. Solving for x_{n} provides the answer to question (a)  the total output required from each industry to meet industry needs and outside demand. For the !Kung
(69) x_{1} = .113x_{1} + .043x_{2} + 494.04
x_{2} = .206x_{1} + .079x_{2} + 902.16
where x1 = total required output of meat calories and x2 = total required output of nut calories. Rewriting with the outside demand on the right, and collecting terms, we have
(70) (1  .113)x_{1}  .043x_{2} = 494.04
.206x_{1} + (1  .079)x_{2} = 902.16
The solution, x_{1}, = 611.07 meat calories, and x_{2} = 1,115.75 nut calories, comes as no surprise, for we already know the total output of each industry, which is given by X. The system of equations (69) represents the structure of the system in precise terms.
The answer to question (b) reveals the full power of the model for extrapolation. Consider a change in the values of the outside demand, from 494.04 cal for meat to 600 cal and from 902.16 cal for nuts to 500 cal (perhaps attributable to changes in such factors as food preferences, trade, and resources). This change, obviously, would call for an overall decrease of 296.2 x 10^{3} calories. But what level of production would be required from each industry to meet both the revised outside demand and industry requirements? The answer to this question is not so obvious? and certainly would not be obvious in a model composed of several dozen industries.
Fortunately, the answer is easily obtained by inserting these new demand values into equations (69):
(71) x_{1} = .113x_{1} + .043x_{2} + 600
x_{2} = .206x_{1} + .079x_{2} + 500
Rewriting with the demands on the right and collecting terms as before, the solution is: x_{1} = 710.446 and x_{2} = 701.499. In other words, in order for the system of industries to meet the new combined total of 1,100 x 10^{3} calories of outside demand and remain in equilibrium (maintain interindustry flows as in A), meat calorie production (x,) would have to increase from 611.07 to 710.446 and nut calorie production (x_{2}) would have to decrease from 1,115 86 to 701.499.
We could continue to substitute edifferent D values into (69) to explore nalternative equilibrium solutions that might occur under various theoretically expected conditions. These projections, however, all depend on an assumption of stability in A, and for this reason we have used the label static for this model. More intricate, dynamic models can be developed which allow for changes in A. Closed models (without an outside demand) and models involving nonlinear equations can also be constructed. Finally, it should be noted that matrix algebra and notation provide a more compact representation of the inputoutput model, and greatly relieve the computational burden, especially when the systems have components. Matrix algebra is commonly used when electronic computers are programmed to perform calculations (Leontief, 1966).