A Summary of Economic Methodology I. The Methodology of Theoretical Economics All economic analysis begins with theory, based in part on intuitive insights that naturally spring from certain stylized facts, regularities, and other organized social phenomena. Theory can be broadly classified into two major categories. Samuelson has stated as much in his brilliant Foundations of Economic Analysis (1947). What he says is true not only for economics but for social science in general. Theory is founded either on one or both of the following (a) Constrained optimization models, or (b) Equilibrium models The first type (a) includes any models involving the maximization of utility subject to constraints, the maximization of profit (or optimization of profit and risk), the minimization of cost, dynamic programming, rational expectations, optimal turnpikes, inventory control, scheduling models, etc. The second type (b) of model includes things such as ordinary supply and demand models, Austrian type models, models of imperfect competition, IS-LM, AD-AD models, Walrasian general equilibrium models, computable general equilibrium models, contestable market models, arbitrage models, etc. Combination of the two types (a & b) are found in monopoly models, game theory models, supply and demand models based on micro-foundations, Ramsey models, overlapping generations models, etc. Virtually any model one can think of falls into one or both of these two categories. I suppose the exception might be the broad class of disequilibrium models which have become less important in recent years. It follows that if we are to consider theoretical models in which we analyze social issues, we must thoroughly understand the nature of the two categories of models above. In addition, all models from these two categories have both static and dynamic formulations. Again, Samuelson was correct to note that static and dynamic aspects of any model have a special relation. According to Samuelson, the stability conditions of dynamic models places important restrictions on the movement of variables in these models and thus affects the comparison of long run static outcomes.
Example Supply and Demand Demand comes from the maximization of utility subject to a resource constraint as well as aggregation across consumers. Supply comes from maximization of profit by the firm and aggregation across firms. Demand and supply can form a partial equilibrium for a single market. Shifting supply and demand will result in different equilibrium points. This is called comparative statics. The equilibrium of supply and demand before the shift is compared with the equilibrium after the shift. Unfortunately, we do not know from the supply and demand curves how long it takes to go from the starting point to the ending point. Also, we do not know the path the price takes over time in going from the starting price to the ending price. None of this information is included in the supply and demand model. Alfred Marshall, the famous Cambridge economist at the turn of the 19 th century and the teacher of John Maynard Keynes, attempted to give some clarity to this by breaking the dynamic transition into different periods. He called the shortest period the market period. This was when the quantity supplied was fixed and could not change. As one factor was allowed to change we moved into a short run period where the quantity supplied could be changed somewhat. Finally, in the long run period all factors of production could change and supply was free to move (and shift). This type of analysis was called period analysis and it was quite popular until the 1940s when Samuelson changed much of economics. Samuelson and others believed that differential equations (and difference equations) provided economists with the perfect tools for analyzing the dynamic behavior of economic models. It was up to the economist to provide a clear dynamic structure to the models that he or she created. If it was an equilibrium model, then a dynamic structure to the equilibrium must be added in order to allow movement of variables from one long run equilibrium to another. If it was constrained optimization models, then again the economist must provide a dynamic setting in which the phenomenon could be analyzed from start to end. Differential equations are the natural language of modeling. Example: Supply and Demand with Dynamics Suppose demand is and supply is. Let D = the quantity demanded and S = the quantity supplied, with P = price and I = income. Naturally, if we set I = I o, we can draw demand and supply curves. One way to introduce dynamics into this model is to write the differential equation as
where the > 0. Note that the only way that the price can increase in this model is for the quantity demanded to be greater than the quantity supplied. The long run equilibrium P * would be at the price where D = S. We can show this using a phase diagram.
To the left of P * the price P tends to increase, whereas to the right of P *, the price tends to decrease. There is a convergence to the long run equilibrium P *. The speed and direction of the movement to equilibrium will depend on the shape of the curve in the above diagram. The shape of the curve will be determined by our assumptions about the dynamics of the equilibrium process. This is why economists think that supply and demand models refer to long run equilibrium not short run equilibrium. According to our dynamic model, supply and demand do not have to be equal all the time. At times, quantity supplied can be larger than quantity demanded, and vice versa. Now assume that the good is a normal good and that income increases from I o to I 1. This causes the demand for the good to rise and the long run equilibrium price moves to P **. This is shown in the simple supply and demand graph below. The phase diagram on the next page shows this movement as well. The movement from one long run equilibrium P * to the new long run equilibrium is a dynamic path consisting of a (possible) jump and a smooth transition. In other words, when demand increases due to the increase in consumer income, the price in the market may first jump by an amount and then move continuously to the new higher price. The size of this jump will be determined by how the current price changes when income changes. Usually, economists ignore the possibility of a jump and assume the current price does not change at
the instant that income changes. Some more complicated models in economics, especially rational expectations models, make jumps endogenously determined so that stable paths are achieved. If we compare only the starting value P * with the ending value P **, this is called comparative statics. Clearly P * increases to P ** when income, I, increases from I o to I 1. We can write this comparative statics result as Therefore, to get the comparative statics result in dynamic models, we set equal to zero and take the partial derivative of P with respect to I. The sign of this derivative will be dependent on the stability condition for the differential equations. This is known as Samuelson s Correspondence Principle. In the supply and demand example above, setting equal to zero we get Differentiating this for P* with respect to I gives
which will be positive for a normal good if. This last inequality ensures that the differential equation is stable at its equilibrium point P * and illustrates Samuelson s Correspondence Principle. Example: The IS-LM model As we have pointed out, all models in economics are either constrained optimization or equilibrium models. The familiar IS-LM model from macroeconomics is an equilibrium model. Graphically, we draw IS-LM as follows The IS curve represents equilibrium in the goods market where aggregate expenditure just matches aggregate real GDP. The LM curve represents equilibrium in the money market, where the real demand for money is equal to the real money stock.
How do we interpret the equilibrium point where IS and LM intersect? Is this a long run equilibrium or does this represent a short run equilibrium. Is it something which the economy is ideally trying to achieve or does it hold at every moment? Is this equilibrium something that will eventually happen if nothing in the economy changes? Is the IS and LM intersecting now? Will they intersect always? Do we ever really arrive at the intersection point? How long does it take to reach the intersection point if we are not there now days, weeks, months, years? Finally, and most importantly, if we are not at the intersection point and are out of equilibrium, what determines the point we are at and where we go? What determines the economy s current position and dynamic movement if it is not equilibrium? These are just a few of the questions we have about IS-LM, but the same questions are involved in every model in economics, including the supply and demand model above. Equilibrium and constrained optimization are powerful and useful concepts in economics, but if we have not achieved equilibrium or a point of extrema, how do we get there? This is the subject of economic dynamics. Comparing long run equilibrium points under differing market conditions is called comparative statics. Comparing the dynamic paths to long run equilibrium under differing market conditions is called comparative dynamics. Example: IS-LM with dynamics added Most modern economic models and analysis introduce dynamics by means of differential equations. In the 1960s the IS-LM model was dynamized by the following With this dynamic setup, the only way that output could increase is if aggregate expenditure (C+I+G+NX) exceeded current output Y. And, the only way that interest rates could rise is for the demand for real balances to exceed the real money stock. Note that the model assumed that the price level was exogenous, but this too could be endogenized by including a labor market and an Okun s Law relation. The basic model also ignored the exchange rate, but this could also be included by adding a balance of payments equation, suitably dynamized. During the 1980s and 1990s a great deal of attention was placed on introducing endogenous expectations into these kinds of models. Later, the IS-LM model was pretty much abandoned in favor of micro-based dynamic models like the Ramsey model and OLG model.
Note that the IS-LM model does not include any relations that point to utility maximization. There is no explicit formulation of profit maximization either. It is a loose aggregation of certain general features of the macroeconomy. It is a typical equilibrium model (type b above) and does not represent a constrained optimization outcome. Moreover, the dynamics represent a very crude type of Keynesian analysis. Nevertheless, many older macroeconomic texts discussed the dynamics of the macroeconomy using this type of system. Hundreds of academic papers were written in the 1960s and 1970s using this formulation of the dynamics of the economy. Assuming there is an increase in government spending, the IS curve would shifts to the right and a new equilibrium would emerge. How would he economy adjust. The typical analysis in the past followed the differential equations above and the dynamic path would be as in the following diagram. The dynamic path shown in the graph above is not the only path possible. Other types of paths from one long run equilibrium E to another long run equilibrium E can occur. The
dynamic path is determined by the differential equations which are added to the IS-LM model. All theoretical economic analyses are interested in determining (i) the movement in long run equilibrium points called comparative statics, and (ii) the dynamic paths under different market conditions called comparative dynamics. In the graph above we know that long run income and interest rates are driven up when government spending increases. This is the comparative static result. We also assert that income and interest rates will cycle to the new long run equilibrium given the particular model and dynamics. To determine if this model is useful, we need to now look at data to see if there is any empirical confirmation of the results given in the theoretical model. Rarely do we get empirical results that perfectly confirm what we expect in the theoretical model. This leads us to consider new theoretical models with results that more nearly conform to the data. One question which we have not asked is the following. Suppose that we are on a dynamic path leading from one equilibrium to another. Now suppose that the conditions associated with the new equilibrium change once again. How does the system behave? Clearly, this is not like the above supply and demand or IS-LM graphs. We are not moving from one long run equilibrium to another long run equilibrium. We are moving from a point of disequilibrium to a point of long run equilibrium. This is not comparative statics. For the case of the IS=LM model above it is possible that the interest rate moves down to long run equilibrium if we are on the upper part of the path when the change occurs. It is possible that the interest rate moves up if we are on the lower segment of the path. This is shown in the diagram below. Suppose the economy is at long run equilibrium E and government spending increases. This moves IS to the right and a new long run equilibrium is established at E. The economy moves along the dynamic path from A to B and so on. However, if the economy is at A and government spending again increases, then the interest rate will change from A to E which is an increase. If the economy is at B when the IS shifts, then the interest rate will decrease from B to E. The effect is uncertain depending on where the economy is when the change occurs. This is why we define comparative statics as being from one long run equilibrium to another long run equilibrium. Note that the change of income and interest rate from E to E is clear, and the change from E to E is also clear. The same cannot be said about the change from A to E versus the change from B to E. This illustrates an important point in the use of our theoretical tools such as the IS and LM or the demand and supply curves earlier. Our analysis always focuses on movements from one long run equilibrium to the other long run equilibrium. The initial conditions which determine the explicit solution to the differential equations is almost never specified in economic models. Typically, economists simply assume that the economy is already at its long run equilibrium. Moreover, changes in the model s parameters
do not cause a re-initialization of the economy. Changes in these parameters are usually assumed to have no effect on initial or re-initialization conditions and thus there is no jump in the endogenous variables. In fact, there is no reason why that the variables could not jump with a change in the parameters. Economists do not allow such jumps because their methodology calls for comparisons of two long run equilibrium points. That is, two sets of parameter values produce two sets of differential equations. If we begin at the first set of equation s long run equilibrium, and the parameters change, the system will evolve to a new long run equilibrium. The path of this transition is the dynamic path of the economy. No jump is assumed since we are merely comparing what happens under two sets of different dynamic equations. It is interesting to note that the comparative static movement from E to E and from E to E is directly related to the stability conditions on the differential equation system we have written above for the IS-LM model. This is what is called the Correspondence Principle. In dynamic models where there are many variables and graphical methods are impossible, the
comparative statics are often signed using the Correspondence Principle. Sometimes the dynamics of the economic model are already part of the model and do not need to be appended. This is the case with the Solow growth model. Example: The Solow-Swan Neoclassical Growth Model Let be an aggregate production function of the economy where real GDP, (full) employment of labor, and physical capital stock. We typically assume constant returns to scale and thus the production function is linear homogeneous in and. Assume that the growth rate of labor is L L n, a constant. Further assume that aggregate private real consumption is a function of output and can be written as gross aggregate real investment follow the usual identity I K K C Y. Finally, let, where = the physical depreciation rate on capital. These are the basic features of the Solow growth model. The fact that the production function is linear homogeneous allows us to write where. The national income equilibrium can be written as. Substituting the consumption function in for C, the investment identity in for, and dividing by, we get Using the homogeneity of the production function, we can write this as Which is a differential equation in sometimes lead, through simplification, to a dynamic model.. This shows that the structure of the model can The Solow model is usually illustrated using a phase diagram. With and assuming certain conditions on the production function are satisfied, a unique and stable equilibrium emerges at k =. Given an initial point, usually with no discussion how it is
determined, the economy moves to the equilibrium point k *. Changes in the parameters can affect the long run steady state value of k. Comparative statics involves looking at these changes in k * in response to changes in the parameters. Comparative dynamics looks at the changes in the transitional time path of k as the parameters change. Example: Simplest example of comparative statics and dynamics Let s suppose that we want to create the simplest example illustrating economic methodology with respect to theory. Assume that we have a very simple economic model involving in the end only one variable x. Like most economic models we assume that if x is in long run equilibrium and there are no changes in the parameters in the model, then x will remain forever in long run equilibrium. We can assume that the dynamic equation describing the motion of x over time is given by where the and are parameters and are assumed to be constant. If the values of and change, then they do so from one set of constant values to another set of constant values. To get the long run steady state value of x we set equal to zero and solve for. Therefore, the steady state of is equal to
How suppose we want to compute the comparative statics. This means that we differentiate with respect to the parameters and. Thus, and The sign of the first derivative will be positive if b > 0. The sign of the second derivative will be positive if a > 0. Let s draw a phase diagram of the model. This diagram assumes that a > 0 and b < 0. Therefore, the long run equilibrium point,, exists and is stable. Consider the effect of an increase in a and an increase in b (or a decreases in b ), respectively. This is shown in the two phase diagrams below. In both cases ( b decreases and a increases) we have that x** the steady state value increases. This agrees with our comparative static derivatives above and shows that the stability condition allows us to sign the comparative statics.
Finally, it is possible to show the exact dynamic path of x for any given initial condition. We do this by solving the differential equation explicitly. Let. The differential equation can now be written as and thus the solution to this differential equation is. Substituting in to this explicit solution gives. If the initial condition is x = x o when t = 0, we can
write the exact explicit solution as. Note that we do not specify how that is determined. Since b < 0 it is clear that as t grows larger, x converges on the long run value x * = -a/b. If x o is below x * then x will converge to x * from below. If it is above x *, x will converge to x * from above.