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# Differential equations

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### Differential equations

1. 1. Differential Equations by tarungehlots Concepts of Differential Equation Consider a variable k that might denote the per capita capital stock level in an economy.When convenient, we can recognize that the level of capital depends upon time by addition thetime argument. Doing so, we would write the per capita capital stock level as k (t ) , rather thanjust writing k . If we think of time as unfolding continuously, we can also think of capital asbeing a continuous function of time, and we can assume that the function has derivate dk/ dt .This derivative is the instantaneous change in per capita capital. In many presentations, dk/ dt is written as k , but we will here use the more familiar notation dk / dt k (t ) . By writing thederivate as k (t ) , so that we include the time argument, we are emphasizing that the value of thederivative may change with time. Because it is typically cumbersome to repeatedly write downthe argument, we can also write the derivative as just k , while remembering that its value maychange with time. A differential equation is an equation that relates the time derivative of a variable to its level.An example is the equation(1) k sf (k ) gk nk k.The variable k is called a state variable because it gives the state of the system at any givenpoint in time. In our example, k gives the state or level of per capita capital stock.2. A Dynamic System The basic dynamic principle is the idea that “the way things are determines the way thingschange.” A differential equation is one way of modeling the basic dynamic principle. The statevariable k indicates the way things are, while the time derivative k indicates how things change.The differential equation itself is what relates the two. More precisely, a differential equationtypically presents a functional relationship, showing how k depends upon k . When we want toemphasize the functional relationship, we can write the differential equation as k (k ) , wherethe general functional form (k ) is the rule that tells us how the value of the time derivative k is determined from the level of the state variable k . 1
2. 2. A difference equation can also be used to represent the basic dynamic principle. When adifference equation is used, time unfolds as sequential time periods, rather than unfoldingcontinuously. We will focus on using the differential equation here. A solution to a differential equation is a function k (t ) that satisfies the differential equationfor all points in time t that are of interest. Some differential equations can be solved. However,most differential equations that are of interest in economics cannot be solved. Nonetheless, wecan still learn about the path followed by the state variable over time by using a phase diagramtechnique. Assuming the function f (k ) in equation (1) is nonlinear, we can describe equation (1) asa single state variable system, and equation (1) is a nonlinear and non-autonomous differentialequation. As a dynamic system, equation (1) is a single state variable system because there isonly one time derivative present, which is k . If the system were a two state variable system,you would see two time derivatives and two different state variables. The equation is labelednonlinear because the functional relationship between k and k is not linear. Typically,nonlinear differential equations cannot be solved. However, the presence of the generalfunctional form f (k ) in equation (1) assures us that we cannot solve it. Finally, equation (1) islabeled non-autonomous because it includes variables other than state variables and timederivatives. The variables s , g , n , and in equation (1) are exogenous variables, which arevariables that describe the environment impacting the system. Usually exogenous variables areconstants, which is the case in equation (1). Classifying the variables of a differential equation system helps clarify how you themodel builder believe the system works. Because our system (1) is a single equation system, wecan only have one endogenous variable. The endogenous variables a differential equationsystem are always the time derivatives. So, for our system, k is endogenous. The statevariables of a system are always classified as predetermined, so k is predetermined for oursystem. The level of the variable k is predetermined at point in time t because theinstantaneous change k in the level of k we just determined by the model in the previousinstant of time. Non state variables are exogenous, so s , g , n , and are exogenous. The lastvalue that needs to be specified for the system is the initial condition k0 for the state variable k ,which is the value of the variable k at the initial point in time t 0 . Thus, we can summarizethe classification of variables as follows:Classification of VariablesEndogenous (1): k Exogenous (4): s , g , n ,Predetermined (1): kInitial Conditions (1): k0 It is common that differential equation models contain auxiliary equations, which areequations that endogenized (i.e., determine) the values for other variables by relating them to thestate variable. Using standard techniques for analyzing a differential equation of this type, we 2
3. 3. can learn how the path for state variable k will evolve over time. Once we know the pathfollowed by the variable k , we can use the following “auxiliary” equations to determine thepaths followed by the other variables of interest. To illustrate, let us add the endogenous variables y , r , w , and c to our system, and assumethat they are determined from the following four equations:(2) y f (k )(3) r f (k )(4) w f (k ) f (k )k(5) c 1 syThe variables y , r , and w are directly related to the variable k , and only related to k . Thus, ifwe know the path followed by k , we can use equations (2)-(4) to determine the paths followedby y , r , and w . Equation (5) indicates that we need to know the path followed by y , alongwith the value of the exogenous variable s , to determine the path followed by c . So we candetermine the path followed by c by first using equation (2) to determine the path followed by y . This discussion indicates that we can classify the variables of the dynamic system (1)-(5) asfollows.Classification of VariablesEndogenous (5): k , y , r , w , cExogenous (4): s , g , n ,Predetermined (1): kInitial Conditions (1): k0 The core of this model is the path followed by the state variable k . Once you understand thecore of a model, you can use that knowledge to then understand the auxiliary aspects of themodel. Notice that adding the auxiliary equations only changes the number of endogenousvariables. We can think of the core of the model, which is equation (1) and the firstclassification of variables as a subsystem, a system upon which the auxiliary aspects of themodel depend.3. Analyzing the Steady StateFinding the Steady StatesThe steady state for the state variable k is the state where the variable is not changing, or wherek 0 . Setting k 0 in (1), we knowSS1 sf (k ) g n k. 3