ECO2114 2024/2025 Chapter 3 Equilibrium Analysis PDF

Department of Economic Theory & Analysis, Makererere University Semester I Academic Year 2024/2025 ECO 2114 Mathematical Economics Issued By: Bruno L. Yawe Issue Date: 17th August 2024 CHAPTER 3 EQUILIBRIUM ANALYSIS IN ECONOMICS THE MEANING OF...

Department of Economic Theory & Analysis, Makererere University Semester I Academic Year 2024/2025 ECO 2114 Mathematical Economics Issued By: Bruno L. Yawe Issue Date: 17th August 2024 CHAPTER 3 EQUILIBRIUM ANALYSIS IN ECONOMICS THE MEANING OF EQUILIBRIUM Just like any economic term, equilibrium, can be defined in various ways. According to one definition, an equilibrium is “a collection of selected interrelated variables, so adjusted to one another that no inherent tendency to change prevails in the model which they constitute.” Several words in this definition deserve special attention. (a) The term “selected” underscores the fact that there are variables which, by the analyst’s choice, have not been included in the model. Therefore, the equilibrium under discussion can have relevance only in the context of the particular set of variables chosen, and if the model is enlarged to include additional variables, the equilibrium state pertaining to the smaller model will cease to apply. (b) The word “interrelated” suggests that, in order for the equilibrium to obtain, all variables in the model must simultaneously be in a state of rest. The state of rest of each variable must be compatible with that of every other variable; otherwise some variable(s) will be changing, thereby also causing the others to change in a chain reaction and no equilibrium can be said to exist. 1 (c) The word ‘inherent” implies that, in defining an equilibrium, the state of rest involved is based only on the balancing of internal forces of the model, while the external factors are assumed fixed. Operationally, this means that parameters and exogenous variables are treated as constants. When the external factors actually change, there may result a new equilibrium defined on the basis of the new parameter values. When defining the new equilibrium, the new parameter values are again assumed to persist and stay unchanged. In essence, an equilibrium for a specified model is a situation that is characterized by a lack of tendency to change. It is for this reason that the analysis of equilibrium, more specifically, the study of what the equilibrium state is like, is referred to as STATICS. The fact that an equilibrium implies no tendency to change may tempt someone to conclude that an equilibrium necessarily constitutes a desirable or ideal state of affairs, on the ground that only in the ideal state would there be a lack of motivation for change. Such a conclusion is not warranted. Even though a certain equilibrium may represent a desirable state and something to be striven for (e.g. a profit-maximizing situation from the firm’s viewpoint), another equilibrium position may be quite undesirable and thus something to be avoided, for instance an underemployment equilibrium level of national income. The only warranted interpretation is that an equilibrium is a situation which, if attained, would tend to perpetuate itself, barring any changes in the external forces. The desirable variety of equilibrium namely as goal equilibrium will be treated later as optimization problems. In this section, the discussion will 2 be confined to the nongoal type of equilibrium, resulting not from any conscious aiming at a particular objective, but from an impersonal or suprapersonal process of interaction and adjustment of economic forces. Examples of this are the equilibrium attained by a market under given demand and supply conditions and the equilibrium of national income under given conditions of consumption and investment patterns. PARTIAL MARKET EQUILIBRIUM – A LINEAR MODEL In a static-equilibrium model, the standard problem is that of finding the set of values of endogenous variables which will satisfy the equilibrium condition of the model. This is because once we have identified those values; we have in effect identified the equilibrium state. Let us illustrate this with a so-called “partial-equilibrium market model,” i.e. a model of price determination in an isolated market. Constructing the Model Since only one commodity is being considered, it is necessary to include only three variables in the model, namely the quantity demanded of the commodity ( Qd ); the quantity supplied of the commodity ( Q s ); and its price (P). The quantity is measured, say, in (xxx) kilograms or pounds per week while the price can be measured in (xxxx) shillings, US dollars euros, etc. Having chosen the variables, we next make certain assumptions regarding the working of the market. 3 We must specify an equilibrium condition – something indispensable in an equilibrium model. The standard assumption is that equilibrium obtains in the market if and only if the excess demand is zero ( Qd − Qs = 0 ), that is, if and only if the market is cleared. But this immediately raises the question of how Qd and Q s are determined. To answer this, we assume that Qd is a decreasing linear function of P (xxxx as P increases, Qd decreases and v-v). Conversely, Q s is postulated to be an increasing linear function of P (xxxx as P rises, so does Q s and v-v), with the condition that no quantity is supplied unless the price exceeds a particular positive level. In all, the model will contain one equilibrium condition plus two behavioral equations which govern the demand and supply sides of the market, respectively. Mathematically, the model can be written as follows: Qd = Q s Qd =  −  P [ ,   0] (1) Qs = − + P [ ,   0] Four parameters, α, β, θ, and π, appear in the two linear functions, and all of them are specified to be positive. When the demand function is graphed as in the next figure, its vertical intercept is at α and its slope is –β, which is negative, as required. The supply function also has the required type of slope, π being positive, but its vertical intercept is negative at –θ. Why did we want to specify such a negative vertical intercept? The answer is that, in doing so, we force the supply curve to have a positive horizontal intercept at P1; thereby satisfying the condition 4 stated earlier that supply will not be forthcoming unless the price is positive and sufficiently high. Figure 1 Partial Market Equilibrium Qd , Qs Qd =  −  P (demand) Qs = − + P (sup ply) (P , Q ) Q = Qd = Qs 0 P1 P P − Note that contrary to the usual practice, quantity rather than price has been plotted on the vertical axis in figure 1. This is in line with the mathematical convention of placing the dependent variable on the vertical axis. Having constructed the model, the next step is to solve it, i.e. obtain the solution values of the three endogenous variables, namely (xxx) Qd , Q s , and P. The solution values to be denoted Qd , Qs , and P are those values that satisfy the three equations in (1) simultaneously, i.e. they are the values which, when substituted into the three equations, make the equations a set of true statements. In the context of an equilibrium model, those values may also be referred to as the equilibrium values 5 of the said variables. Because Qd = Qs , they can be replaced by a single symbol Q. Therefore, an equilibrium solution of the model may simply be represented by an ordered pair ( P , Q ). In case the solution is not unique, several ordered pairs may each satisfy the system of simultaneous equations. There will then be a solution set with more than one element in it. However, the multiple equilibrium situation cannot arise in a linear model. Solution by Elimination of Variables One way of finding a solution to an equation system is by successive elimination of variables and equations through substitution. In (1), the model contains three equations in three variables. 1(a ) Qd = Qs 1(b) Qd =  −  P [ ,   0] (1) 1(c) Qs = − + P [ ,   0] We know that in equilibrium, Qd = Q s. Therefore, by substituting equations 1(b) and 1(c), into 1(a) we have  −  P = − + P  +  = P + P  +  = P ( +  ) (2)  + P =  + Note that P is expressed in terms of the parameters, which represent the given data for the model. This is typical of all solution values. Thus, P is a determinate value as it ought to be. Also note that P is positive as price should be. This is because all the four parameters are positive by model specification. 6 To obtain Q we substitute P into either equation 1(b) or 1(c). By substituting P into 1(b), we obtain  +   Q = −   +    −  Q = −  + (3)  [ +  ] −  −  Q =  +  +  −  −   −  Q = =  +  + Note that Q is an expression in terms of parameters only. Since the denominator (π+ β) is positive, the positivity of Q requires that the numerator (απ – βθ) be positive as well. Hence to be economically meaningful, the present model should contain an additional restriction that απ > βθ. DO IT YOURSELF # 1. Try substituting P into equation 1(c). Is your result different from equation (3)? DO IT YOURSELF #2. Review the meaning of the restrictions in a graphical approach, Chiang (1984) pg 39. DO IT YOURSELF #3 Exercise 3.2 Chiang (1984) page 40. DO IT YOURSELF#4 Partial Market Equilibrium - A Nonlinear Model section 3.3 Chiang (1984) pp: 40-45. DO IT YOURSELF #5 Exercise 3.3 pg 45. DO IT YOURSELF #6 Make own note and attempt All Exercises for 3.4 GENERAL MARKET EQUILIBRIUM DO IT YOURSELF #7 Make own note and attempt All Exercises for 3.5 Equilibrium in National-Income Analysis 7