Robust Comparative Statics  Susan Athey, Paul Milgrom, and John Roberts

Robust Comparative Statics Susan Athey, Paul Milgrom, and John Roberts DRAFT ONLY - DO NOT CITE OR CIRCULATE Comments Welcome This Draft: October, 1998 OUTLINE Chapter 0: Introduction. Part I: Univariate Optimization Problems Chapter 1: An Introduction to the Theory. Chapter 2: Single Variable Problems with Separable Contexts. Chapter 3: Single Variable Problems with non-Separable Contexts. Part II: Multivariate Optimization Problems. Chapter 4: Multivariate Comparative Statics on Product Sets. Chapter 5: Elementary Lattice Theory and Comparative Statics on Lattices. Part III: Stochastic Optimization Problems. Chapter 6: Supermodularity and monotonicity. Chapter 7: Log-Supermodularity. Chapter 8: Single Crossing Properties. Part IV: Equilibrium Models. Chapter 9: One dimensional models. Chapter 10: Multidimensional models. Chapter 11: Conclusion. 2 0 Introduction This monograph describes a new approach to comparative statics analysis that has developed rapidly in the past several years. Comparative statics – the study of how the solutions of an economic model change as the model parameters and specification are changed – is important because (1) most of the testable predictions of economic theory are comparative statics predictions and (2) many economic equilibrium analyses are built from comparative statics analyses of the model’s components. The results of comparative statics analyses thus form the basis for much of our understanding of the behavior of the economy. Comparative statics predictions can be qualitative or quantitative, but the focus of the new methods is on the former. Typical examples of qualitative predictions include assertions that increasing employment taxes or a minimum wage will increase unemployment, or that relaxing trade restrictions will cause industrial pollution to fall in developed countries and rise in less developed ones. Other examples include statements about how interactions among decisions in an economic system will multiply or attenuate certain effects, such as the prediction that a plant closing will reduce income in a community by more than the lost wages of the laid-off workers, or that a reduction in oil supplies will increase oil prices by more in the short-run than in the long-run. Notice that each of these predictions is about how a change in some condition will affect other variables in the system without regard to all the other conditions that define the economic environment. That is, the economist who makes these general predictions intends for them to be robust – independent of the specification of the fine details of the environment. The objective of this monograph is to develop and present the logic of robust comparative statics. Alternative Methods of Comparative Statics Analysis Traditionally, there have been three main mathematical approaches to comparative statics analyses of optimization problems. The first relies on building models that can be solved explicitly, and then deriving the functional relationship between the exogenous parameters and the endogenous variables. The disadvantage of this “explicit solutions” approach is that it usually entails making severe assumptions for tractability and leaves open the question of how crucially the qualitative conclusions depend on those restrictions. This approach gives us few hints about whether its conclusions are robust. The second, “implicit functions” approach is based on calculus, although its main features have been extended in the area of operations research known as “convex programming.” In optimization problems it involves assuming that the first-order conditions characterize the solution to the problem and that these define the optimizing choices as a differentiable function of the varying parameter. A formula giving the derivative of this implicit function in terms of the first and second derivatives of the objective function can be calculated, and one then seeks to find conditions that determine the sign of the derivative. This approach has wider applicability than 3 the explicit solutions approach, and in fact the Implicit Function Theorem from calculus actually gives sufficient conditions for the solution function to exist locally. Still, in order to apply this approach, the objective function must satisfy a number of conditions that are economically restrictive and that in most cases are actually unrelated to the comparative statics conclusions. For example, the objective function must be suitably differentiable and the choice variables must be continuous, which bars applications to discrete choice problems and to some other problems as well. In addition, it is common to assume that the objective is strictly concave because this guarantees that the optimum of the problem is unique for each value of the parameter. The implicit functions approach has a number of advantages. One is that when it applies to a particular problem, it gives necessary and sufficient conditions for the comparative statics conclusion to hold in that problem. A second advantage is that it can be applied before particular functional forms are assumed, giving some insight into the generality of the conclusions. Further, it is not limited to optimization models; it can be adapted for use on comparative statics for equilibrium analysis as well. Finally, these methods are systematic in the sense that the way to apply them across different problems is standardized and easily taught. The third approach to comparative statics analysis is sometimes referred to as the “revealed preference” approach, where results are derived directly from the inequalities that define a global optimum. This approach has been used with great success in price theory to obtain very general results. However, because it is unsystematic and relies on the analyst’s ingenuity in crafting an argument that works in the particular problem at hand, this approach tends to be used only when the assumptions required for the application of the implicit function approach are not satisfied. Besides these three general approaches, there is a fourth approach using dual functions (the “indirect utility” and “indirect profit” functions). This approach generates very elegant demonstrations of the comparative statics properties of the response of a competitive firm to changing market prices or of a consumer to changing prices and income. This approach is quite powerful when the relevant parameters are prices, and dual functions may help empirical researchers to draw inferences about production functions from data about prices and profits. However, dual functions are not even defined unless the relevant parameters are prices, so the scope of application of the duality approach is inherently limited. Because we are interested in more widely applicable methods, we will not study the duality approach in this monograph. Robust Comparative Statics and Critical Sufficient Conditions The defining characteristic of the new approach is that it systematically seeks the best possible conditions for robust comparative statics conclusions. Here, “robust” means that the qualitative conclusions should remain unchanged when the model is altered or generalized in particular ways that the modeler has in mind. “Best” means that the condition should be the weakest condition to yield the desired kinds of robustness. We call these best conditions for robust conclusions “critical sufficient conditions.” Critical sufficient conditions are defined by splitting the economic model into a part that is isolated to be the focus of analysis and a second part about which we profess to make only very 4 “weak” assumptions, or no assumptions at all. For example, suppose that the government considers new regulations which alter the cost of training and benefits for low skill workers. Our analysis of the problem might focus on the portion of costs affected by the new regulations, while imposing only weak restrictions on other factor prices and the production possibility set. These latter restrictions combined with the way the two parts of the model fit together, in this case as components of the firm’s profits, define what we call the context of the analysis. A critical sufficient condition is the weakest condition on the isolated part of the model that imply the desired conclusion for all fully specified models in the context. For example, the critical sufficient condition for the new regulation to reduce employment among such workers could be that the regulation increases the incremental cost of employing them. We emphasize robustness because it is our view that even our best models are always incomplete and approximate. We must therefore be concerned that the conclusions we draw from our models are valid predictions about a more complex reality. A conclusion that does not hold robustly may be a highly misleading indication about how the real economy would actually respond to the sort of changes being investigated. This leads us to ask: how robust is the particular comparative statics prediction? That is, over what range of models does the prediction hold? Also, once we have specified a range of models we believe is most relevant, what is the weakest assumption that supports a robust conclusion? The new methods provide a general framework and a collection of theorems to answer such questions. One surprising finding of our analysis that contributes greatly to the appeal of this approach is that the extra price to be paid to obtain robust predictions is often quite small. Price theory provides familiar examples to illustrate

Load more