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White Chalak Luit does not require a unique fixed point, the potential response functions required forcausal discourse are not defined.The importance of game theory generally in describing the outcomes of interactionsof goal-seeking agents and the fact that the unique fixed point requirement prohibits thePCM from supporting causal discourse in important cases strongly motivates formulatinga causal framework that drops this requirement. As we discuss below, the SSframework does not require a unique fixed point, and it applies readily to games generally.Moreover, recognizing and enforcing the distinction between y i (i’s best responsestrategy) and z i (an arbitrary setting of i’s strategy) turns out to be an important componentto eliminating this requirement.Another noteworthy aspect of this example is that a is a fixed list of elements thatdefine the game. Although elements of a may differ across players, they do not vary fora given player. This distinction should be kept in mind when referring to the elementsof a as background “variables."Example 3.2 (Optimization in Consumer Demand) The neoclassical theory ofconsumer demand posits that consumers determine their optimal goods consumption bymaximizing utility subject to a budget constraint (see, e.g., Varian, H., 2009). Supposefor simplicity that there are just two goods, say beer and pizza. Then a typical consumersolves the problemmaxz 1 ,z 2U(z 1 ,z 2 ) s.t. m = z 1 + pz 2 ,where z 1 and z 2 represent quantities consumed of goods 1 (beer) and 2 (pizza) respectivelyand U is the utility function that embodies the consumer’s preferences for thetwo goods. For simplicity, let the price of a beer be $1, and let p represent the price ofpizza; m represents funds available for expenditure, “income" for short 1 . The budgetconstraint m = z 1 + pz 2 ensures that total expenditure on beer and pizza does not exceedincome (no borrowing) and also that total expenditure on beer and pizza is not less thanm. (As long as utility is increasing in consumption of the goods, it is never optimal toexpend less than the funds available.)Solving the consumer’s demand problem leads to the optimal consumer demandsfor beer and pizza, y 1 and y 2 . It is easy to show that these can be represented asy 1 = r a 1 (p,m;a) and y 2 = r a 2 (p,m;a),where r1 a and ra 2are known as the consumer’s market demand functions for beer andpizza. The “a" superscript stands for “agent," corresponding to notation formally introducedbelow. The attributes a include the consumer’s utility function U (preferences)and the admissible values for z 1 , z 2 , p, and m, e.g., R + := [0,∞).Now consider how this problem maps to the PCM. First, we see that a and (p,m)correspond to the background variables u, as these are not determined within the system.Next, we see that y := (y 1 ,y 2 ) corresponds to PCM endogenous variables v. Finally,1. Since a beer costs a dollar, it is the “numeraire," implying that income is measured in units of beer.This is a convenient convention ensuring that we only need to keep track of the price ratio betweenpizza and beer, p, rather than their two separate prices.6
Linking Granger Causality and the Pearl Causal Model with Settable Systemswe see that the consumer demand functions ri a correspond to the PCM structural functionsf i . Also, because the demand for beer, y 1 , does not enter the demand functionfor pizza, r2 a , and vice versa, there is a unique fixed point for this system of equations.Thus, the PCM supports causal discourse in this system.Nevertheless, this system is one where, in the PCM, the causal discourse naturalto economists is unavailable. Specifically, economists find it natural to refer to “priceeffects" and “income effects" on demand, implicitly or explicitly viewing price p andincome m as causal drivers of demand. For example, the pizza demand price effectis (∂/∂p)r2 a (p,m;a). This represents how much optimal pizza consumption (demand)will change as a result of a small (marginal) increase in the price of pizza. Similarly,the pizza demand income effect is (∂/∂m)r2 a (p,m;a), representing how much optimalpizza consumption will change as a result of a small increase in income. But in thePCM, causal discourse is reserved only for endogenous variables y 1 and y 2 . The factthat background variables p and m do not have causal status prohibits speaking abouttheir effects.Observe that the “endogenous" status of y and “exogenous" status of p and m isdetermined in SS by utility maximization, the “governing principle" here. In contrast,there is no formal mechanism in the PCM that permits making these distinctions. Althoughcausal discourse in the PCM can be rescued for such systems by “endogenizing"p and m, that is, by positing additional structure that explains the genesis of p and min terms of further background variables, this is unduly cumbersome. It is much morenatural simply to permit p and m to have causal status from the outset, so that price andincome effects are immediately meaningful, without having to specify their determiningprocesses. The SS framework embodies this direct approach. Those familiar withtheories of price and income determination will appreciate the considerable complicationsavoided in this way. The same simplifications occur with respect to the primitivevariables appearing in any responses determined by optimizing behavior.Also noteworthy here is the important distinction between a, which represents fixedattributes of the system, and p and m, which are true variables that can each take a rangeof different possible values. As WC (p.1774) note, restricting the role of attributesby “lumping together" attributes and structurally exogenous variables as backgroundobjects without causal status creates difficulties for causal discourse in the PCM:[this] misses the opportunity to make an important distinction betweeninvariant aspects of the system units on the one hand and counterfactualvariation admissible for the system unit values on the other. Among otherthings, assigning attributes to u interferes with assigning natural causalroles to structurally exogenous variables.By distinguishing between attributes and structurally exogenous variables, settable systemspermit causal status for variables determined outside a given system, such as whenprice and income drive consumer demand.Example 3.3 (Learning in Structural Vector Autoregressions) Structural vectorautoregressions (VARs) are widely used to analyze time-series data. For example,7
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