Chapter 14: Finding the Equilibrium Solution and Exploring the Nature of the Equilibration Process

Transcription

1 Chapter 14: Finding the Equilibrium Solution and Exploring the Nature of the Equilibration Process Taking Stock: In the last chapter, we learned that equilibrium problems have an interesting dimension that is not shared by optimization problems. That new twist is the equilibration process itself. Economists are interested in the way in which equilibrium systems (such as markets) get to equilibrium; we are less interested in the process by which agents reach their optimal solutions. Other than this, optimization problems and equilibrium systems share quite a bit of common ground, especially in the steps of applying the Economic Approach: 1) Set Up the Problem 2) Find the Initial Solution For Equilibrium Models, in this step we Explore the Equilibration Process 3) Comparative Statics Last time, you also saw that: every equilibrium system consists of at least one structural equation and at least one equilibrium condition. equilibrium is a single point; while a steady-state is a repeated pattern of values there are some general ways of describing the equilibration process: oscillatory versus non-oscillatory (or direct), and convergent versus divergent. all equilibrium systems have built into them a feedback mechanism which describes how the system alters the values of the endogenous variables In this lab: The purpose of this lab is to cement your understanding of those concepts by asking you to do three things: (1) set up and solve a simple equilibrium problem, (2) explore the equilibration process under different assumptions about the nature of the feedback mechanism, and (3) generate some simple comparative statics results. Compared to the way we tackled optimization problems, we are adding a middle component "exploring the equilibration process." It is here that we will discuss the type of equilibration process (oscillatory or not, convergent or divergent) and the speed with which equilibrium is reached. 1

2 ASIDE: Let s take a moment to say a few more words about the equilibration process. You must realize that when the Economic Approach is applied to optimization problems, it is assumed that the agent optimizes instantaneously. We don't watch the consumer get better at allocating income, for example; she is at X*,Y* instantaneously. Systems, however, are different! Systems grope, stagger, and wander around. The endogenous variables are NOT immediately at their equilibrium values. If the variables get to equilibrium, the way they get there, and how long it takes them to get there are all interesting issues to be explored. Thus, after we find an equilibrium solution, we usually pause to discuss the equilibration process. In this lab, we will spend a substantial part of our time on the equilibration process. A Profit Equilibration Model Step 1: Set Up the Problem The Model in Words: This simple profit equilibration model assumes that firms are attracted to industries where economic profits are positive and exit industries where profits are negative. Of course, entry leads to a decrease in price (as supply shifts right), which results in lower profits, while exit leads to an increase in price (as supply shifts left) and higher profits. If profits are zero (remember, these are economic profits so they include opportunity costs), then firms neither enter nor exit. Equilibrium results when there is no tendency for profits to change. Thus the system is composed of firms whose sole decision is to enter or exit. Their decisions on entry and exit affect the market price, and then price influences profit which is the signal on which the firms base their decision. We offer for your close consideration the previous sentence one more time: Their decisions on entry and exit affect the market price, and then price influences profit which is the signal on which the firms base their decision. The circularity that is embedded in the system is a "feedback mechanism" and it is an essential property of an equilibrium model. The system settles down when profits are zero because firms neither enter nor exit, and this means price doesn't change and so profits remain where they were (at zero) and so firms neither enter nor exit and... 2

3 Get the picture? No? OK, we'll draw it for you: Decision: Do I enter or exit Signal: Level of Profits Result: Price Watch the ball go round and round: The industry makes some level of profits which is the SIGNAL used by individual firms on whether to enter or exit. The outcome of the individual firm s DECISIONS leads to a RESULT which determines the level of profits which is the SIGNAL used by individual firms on whether to enter or exit. The outcome of the individual firm s DECISIONS leads to a RESULT which determines the level of profits which is the SIGNAL used by individual firms on whether to enter or exit. The outcome of the individual firm s DECISIONS leads to a RESULT which determines the level of profits... Let s stay in Step 1: Set Up the Problem, but turn to a mathematical presentation. The Model in Equations: The profit equilibration model is very simple. It has ONE structural equation and ONE equilibrium condition. ALL equilibrium models have structural equations and equilibrium conditions. Structural Equation: Profits, as you well know, are determined by the following function: π=tr-tc=pq-c(q) where P is market price, Q is firm output, and C(Q) is the cost of producing Q units of output. In order to simplify things, let's assume Q and C(Q) are given and exogenous. We are interested in modelling the firm's entry or exit decision, not its output decision, so we will assume away the question of how much to produce. Suppose that Q=100 and C(Q)=1000; then the profit equation gives us the following relation between price and profits: π=p

4 In other words, the firm produces 100 units no matter the price, and the costs are $1000. We are not focusing on the firm as a profit-maximizer here, but simply as an agent which decides whether or not to enter or exit. The profit equation is then pretty simple: If P=2, then π= If P=27, then π=1700. Notice that once we assign numbers for the exogenous variables Q and C(Q), we have a concrete problem on our hands. The endogenous variable is P, the price of what the firm produces. P will be determined by the working of the system. Remember, if π>0, P will fall because positive profits induce entry and supply shifts right; for π<0, the opposite occurs exit leads to decrease in supply. Equilibrium Condition: We can find the equilibrium solution by finding the point where profits are zero; for there, price has no tendency to change because there will be neither entry nor exit when profits are zero. Prices won't change so profits won't change, so no one will enter or exit and so prices won't change... and so on. We find ourselves in a position of rest. The equilibrium condition is therefore: π=0 Endogenous and Exogenous Variables: As for the variables, P is an endogenous variable that is determined by the forces within the system; while Q and C(Q) are exogenously given in this model. By considering the Structural Equations, Equilibrium Conditions, and Endogenous and Exogenous Variables, we have completed the first step, Set Up the Problem. Now, all we have to do is find the equilibrium solution! Step 2: Finding the Equilibrium Solution Solving equilibrium models requires finding the values of the endogenous variables that meet the equilibrium condition(s). As we saw in our work with optimization problems, there are usually several different ways to solve a problem. Knowing more than one way is useful just in case the one way you know doesn t work, because doing the problem a second way enables you to check your work, and because different methods help us better understand what we re doing. 4

5 A Pencil and Paper (or by hand ) Solution: To solve an equilibrium problem with pencil and paper, you simply follow a recipe (just like we did with optimization). STEP (1): Write the structural equation(s) and equilibrium condition(s) For this problem, we have: and Structural equation: π=p Equilibrium Condition: π=0 P is the endogenous variable. It will continue to change value as long as π 0 due to the continuing entry and exit. STEP (2): Force the structural equations to obey the equilibrium conditions. We do this by writing: 0=P e Notice that we attached an "e" subscript to P when we made the structural equation equal the equilibrium condition. This is just like putting an asterisk (*) on the endogenous variable when we set the first order conditions equal to zero. As before, we have, strictly speaking, found the answer. We rewrite the equation, however, for ease of human understanding, as a reduced form expression. STEP (3): Solve for the equilibrium value of the endogenous variable; that is, rearrange the equation in Step 2 so that you have P e by itself on the left-hand side and only exogenous variables on the right. This is called a reduced form: it tells you the equilibrium value of the endogenous variable for any set of values for the exogenous variables. P e =10 (We trust the algebra is not too daunting on this one!) "That's it?" you ask? Yes, that is it. This system is in equilibrium when Price=10 because profits are zero there and so there is no entry or exit and so there is no change in price. Once price settles down and stops moving, the system is in equilibrium. That s remarkably easy, you say. Yes, this example is really pretty simplistic. It gets more complicated later, but the basic idea remains the same! 5

6 Microsoft Excel Solution Strategies: Let s turn to the computer and use Excel to help us present alternative ways of solving this simple profit equilibration problem. Getting Started: Open the Excel spreadsheet called C14Lab.xls. Begin at the Introduction and return to this handout when you are prompted to do so. Return to here... EXPLORING THE EQUILIBRATION PROCESS Having found an initial equilibrium solution via pencil and paper, table and graph, and Excel s Solver, we now turn, not to Comparative Statics, but to an exploration of the equilibration process. The outline looks something like this: A Profit Equilibration Model Step 1: Set Up the Problem Step 2: Finding an Equilibrium Solution EXPLORING THE EQUILIBRATION PROCESS YOU ARE HERE Step 3: Comparative Statics Exploring the equilibration process basically reduces to an analysis of how the system might reach equilibrium. What happens, for example, if we start out with P 10? Can we get to equilibrium? If so, how and how fast will we get there? These questions depend upon the equilibration process. As we saw with calculus in the context of optimization problems, mathematics was an aid and often a shortcut in answering questions. In a moment, many of you will be introduced for the first time to a difference equation. As before, we will use the power of mathematical reasoning and presentation to make our work easier. You will be learning some new ideas and terminology. We will proceed slowly and carefully. 6

7 The equilibration process tells us how the endogenous variables in the system respond to the forces in the model. Let's suppose that P at any point in time is determined according to the following difference equation: P(t+1) = P(t) + P(t+1) where P(t+1) = theta π(t) Let s figure out what this says in plain English. P stands for price, and we've added a label that tells you what period that price belongs to. P(t) stands for price at time t; and P(t+1) stands for price in the NEXT period after t,time period t+1. The label t or t+1 allows us to make general statements regarding time. For any value of t, t+1 is the NEXT period and t+8 is eight periods later. The difference equation above tells you how P(t) and P(t+1) are related; or in other words, how P changes over time. You read the difference equation like this: The price in time period t+1 is equal to the price in time period t (or what the price was the period previous to t+1) plus the change in the price from time period t to t+1 where the change in the price from time period t to t+1 is equal to theta times the level of profits in time period t Basically, the equation above says that the change ( ) in price in the next period is some proportion (theta) of the level profits in the current period. Theta is some exogenous variable whose value depends on the system you're looking at. Different values of theta will generate different kinds of equilibration. That's the story on a general level. Now, let's slowly and carefully try to figure out more precisely what's going on by walking through some numbers. Suppose the initial price is P=18. Then, our structural equation tells us that π=800 (since 18 x = 800). Entry occurs (since π>0) and price falls (since supply shifts to the right). By how much does price fall? This is crucial to a specific description of the equilibration process. Suppose that, because of the speed with which firms can enter the industry, P falls by 1.5% of profits so that the value of theta is According to our description of the equilibration process, the change in P in the next period will be P(t+1) = x 800 = - 12, so that the price next period will be 6 since that s equal to Let's see what happens the next period. With P=6, the price next period will change by x = 6 so that next period s price will be = 12. "-400?" you ask? Sure, at P=6, π = because 6 x = and that negative $400 of profit leads to exit and an increase in price of 6 to 12. 7

8 Now, with the price = 12, the price next period will be 9 and after that it will be And so on, until the price "settles down", i.e., reaches an equilibrium solution. Figuring out the path of the endogenous variable price over time is one way of describing the equilibration process. We will do this in a moment. More on theta: The key to the equilibration process is theta. Notice that we made theta < 0. Why? Because price falls when profits are greater than zero; and P rises when π less than 0. If the economic argument that price and profits are inversely related holds true, theta will be negative. The size and sign of the exogenous variable theta describes the speed and type of the equilibration process. More on Excel: If you are thinking that there s no way you are going to chug through repeated calculations of the sort found on the previous page as we slowly worked from P = 18 to 6 to 12 to 9 to 10.5, don t worry, YOU WON T HAVE TO! This type of mindless calculation is what computers are exceptionally good at!!! Return to Excel now and go to the sheet called Eq Process. Examine the cells and the three graphs that we've created. Click on the cells (like A7, B4, C10, and D8 to name a few) to see how the formulas are set up. It s exactly what we were doing on the previous page. NOTE: It s probably worth spending a little time figuring out how these cells are set up because you will be setting up cells like this pretty soon. DO NOT just click on a cell and say, Yes. A Fine. DO think about the cell formulas and what it is doing. Try to figure out how each cell formula relates to the difference equation that describes the equilibration process. After you have explored the cells a bit, explore how the equilibration process changes when theta changes. To do this, click on the cell called theta (D1) and change its value a few times to and , for example. Notice how the table of values and the three graphs change. You are now ready to answer the key question regarding the equilibration process: How does the equilibration process vary as theta varies? Your task is to use the three graphs and the cell values of Price and Profit in the Eq Process sheet to observe the different types of equilibration processes. Depending on the type of equilibration process, we can divide the possible values of theta from negative infinity to positive infinity into seven distinct regions (where a region may be a range or a single point). Your assignment is to carefully keep track of what you see as theta varies, then answer the question above by describing the different types of equilibration processes as theta varies from one extreme (negative infinity) to another (positive infinity). 8

9 We provide structure for your investigations into the key question, How does the equilibration process vary as theta varies? by having you fill in the table on the very last page of this document. As you can see (by taking a quick look at the last page), we have provided SEVEN distinct categories of theta that correspond to the SEVEN distinct types of equilibration in the system. By trying different values of theta, you should be able to determine the SEVEN distinct types of equilibration. Draw rough sketches of each case under the heading Picture of Equilibration Process over Time. You can also experiment with different values of the initial Price (in cell B3). At a given theta, change this value from 18 to 1 or 28. Try 10 and think about what s happening there. To get you started we have completed one case for you. If > theta > then you will observe the case of oscillatory convergence. A representative graph of this case of price over time is: Price=ƒ(t) We ve already recorded this case on the provided answer sheet and provided a Type of Equilibration heading for the other six cases. Your job is to find the value or values of theta that yield those equilibration patterns. Be advised that the answers are not self-evident. This is a research question, and you will need to apply systematically the method of trial and error in order to answer the question. We suggest using some scratch paper to record your ideas while you are playing around with values of theta, saving the answer sheet to present your final results. You will get the best and quickest results if you are careful and systematic in your approach. When you are finished with your exploration of the equilibration process, return here in order to continue with Comparative Statics. 9

10 Summing Up Thus Far... We have done several things so far with this equilibrium system: set up the problem by specifying the structural equation and equilibrium condition solved for the equilibrium solution using algebra, table/graph, and Solver explored the nature of the equilibrium process as theta changes Ahead... Our final step is to explore one more aspect of equilibrium systems: comparative statics. We remind you that in the Economic Approach, comparative statics is the fundamental vehicle for predictions and explanations of observed changes. Although equilibrium systems are different from optimization problems in that we are concerned with the nature of the equilibration process, they are similar in that we often interpret observed values as equilibrium values. Changes in observed values, then, may be interpreted as caused by exogenous shocks as the system moves from one equilibrium solution to another. We track the values of the equilibrium endogenous variables for a given exogenous variable, ceteris paribus. Do NOT confuse the study of the equilibration process with comparative statics. They are two separate issues. Keep clear exactly where you are in the problem: A Profit Equilibration Model Step 1: Set Up the Problem Step 2: Finding an Equilibrium Solution Exploring the Equilibration Process Step 3: Comparative Statics YOU ARE HERE Step 3): Comparative Statics The final step in the application of the economic approach to equilibrium models is comparative statics. Here we are interested in how equilibrium values of the system respond to shocks. We do not ask how or if we get to equilibrium (these questions are explored in step 2); we merely compare initial equilibrium values to new equilibrium values. As before, comparative statics can be done by Actual Comparison or Reduced Form (differential calculus) analysis. The former relies on discrete changes of finite length; while the latter considers infinitesimally small changes and, therefore, uses the shortcuts or rules of differential calculus. 10

11 Comparative Statics via the Method of Actual Comparison: For the profit equilibration model, suppose that the cost, C(Q), of each firm's output is not $1000, but $1100. This is the shock. Our task is to find the new equilibrium and then compare the initial equilibrium (P e =10) to the new equilibrium. We know that we can find the new equilibrium for this new problem with C(Q) = 1100 either by pencil and paper or by using the computer (table/graph or Solver). You should be able to show via any and all of these 3 ways that the new P e =11. You should be able to explain why the C(Q) elasticity of equilibrium price is 1. Notice that the phrase "C(Q) elasticity of equilibrium price is following the established pattern of "exogenous variable elasticity of equilibrium endogenous variable." In this case, the percentage change in C(Q) is 10% (from 1000 to 1100) and the percentage change in the equilibrium price is also 10% (from 10 to 11). Thus, the % P e /% C(Q) = 10%/10% = Comparative Statics via the Method of the Reduced Form: Having (Step 1) Set Up the Problem (i.e., identified exogenous and endogenous variables and written expressions for the structural equations and equilibrium conditions), we can (Step 2) Find the Equilibrium Solution by solving for the values of the endogenous variables where the equilibrium conditions hold. In our profit equilibrium model, that gave us 0=P e and, obviously, P e =10. Since we want to explore the Comparative Statics properties of the model, we do the problem again with "letters" instead of numbers for the exogenous variables. This will enable us to examine how P e varies with a shock. Thus, we have 0=P e Q-C(Q) where we know that Q and C(Q) are exogenous. Solving for P e, we get P e =C(Q)/Q. The above answer is a REDUCED FORM expression for P e. Put in any values for (i.e., evaluate P e at given values of) Q and C(Q) (like 100 and 1000) and you'll get the equilibrium price for those values of the exogenous variables. For Q=100 and C(Q)=1000, Pe=10. 11

12 Having found our reduced form expression of Pe as a function of exogenous variables, we can very quickly explore how P e varies with changes in those exogenous variables by using the derivative. For example, That tells us two things: dp e dc( Q ) = 1 Q 1) The sign is positive at Q=100 which means as C(Q) increases, so does P e. 2) The derivative expression does not contain C(Q) so that means P e is linear in C(Q). Increases in C(Q) will lead to increases in Pe at a constant rate. Of course, you knew that, since P e =C(Q)/Q when graphed with P e on the y axis and C(Q) on the x axis is a straight line with slope 1/Q. NOTE: The similarity between what we did above and the Method of the Reduced Form in optimization problems is EXACT down to using the derivative of the reduced form to explore how the equilibrium value responds to an exogenous shock! We can find these same properties of how Pe responds to shocks in C(Q) by using the Method of Actual Comparison. Let's do that now. Return to your Solver sheet. Using Solver and the Comparative Statics Wizard, create a a graph of how the equilibrium price is related to C(Q). You might try C(Q) = 1000, 1100, 1200, and Calculate the C(Q) elasticity of P e from C(Q) = 1000 to 1100, 1100 to 1200, and 1200 to In a text box in your Comparative Statics sheet, explain why your C(Q) elasticity of Pe results are a bit unusual and if you think it's a coincidence or some general principle that is at work here. Thus, your completed tasks for this lab include: In the C14Lab.xls workbook: A Solver sheet that shows how Solver can be used to find the equilibrium solution. A Comparative Statics sheet with a graph of Pe=ƒ(C(Q) Q=100), calculated elasticities, and a text box with an explanation. In a separate paper and pencil assignment (but using the C14Lab.xls file to do the work): A filled in C14Lab Answers sheet at the end of this document. 12

13 NAME C14Lab Answers Value of Theta (θ) Type of Equilibration Picture of Equilibration Process over Time Oscillatory Divergence Uniform Oscillation > θ > θ = Oscillatory Convergence Price Time Instantaneous Convergence Non-Oscillatory (Direct) Convergence No Movement Non-Oscillatory (Direct) Divergence 13