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Differential Equations

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Concepts of Differential Equation Consider a variable 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 the time argument. Doing so, we would write the per capita capital stock level as , rather than just writing . If we think of time as unfolding continuously, we can also think of capital as being a continuous function of time, and we can assume that the function has derivate . This derivative is the instantaneous change in per capita capital. In many presentations, is written as , but we will here use the more familiar notation . By writing the derivate as , so that we include the time argument, we are emphasizing that the value of the derivative may change with time. Because it is typically cumbersome to repeatedly write down the argument, we can also write the derivative as just , while remembering that its value may change 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) . The variable is called a state variable because it gives the state of the system at any given point in time. In our example, 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 things change.” A differential equation is one way of modeling the basic dynamic principle. The state variable indicates the way things are, while the time derivative indicates how things change. The differential equation itself is what relates the two. More precisely, a differential equation typically presents a functional relationship, showing how depends upon . When we want to emphasize the functional relationship, we can write the differential equation as , where the general functional form is the rule that tells us how the value of the time derivative is determined from the level of the state variable .

A difference equation can also be used to represent the basic dynamic principle. When a difference equation is used, time unfolds as sequential time periods, rather than unfolding continuously. We will focus on using the differential equation here.

A solution to a differential equation is a function that satisfies the differential equation for all points in time that are of interest. Some differential equations can be solved. However, most differential equations that are of interest in economics cannot be solved. Nonetheless, we can still learn about the path followed by the state variable over time by using a phase diagram technique.

Assuming the function in equation (1) is nonlinear, we can describe equation (1) as a single state variable system, and equation (1) is a nonlinear and non-autonomous differential equation. As a dynamic system, equation (1) is a single state variable system because there is only one time derivative present, which is . If the system were a two state variable system, you would see two time derivatives and two different state variables. The equation is labeled nonlinear because the functional relationship between and is not linear. Typically, nonlinear differential equations cannot be solved. However, the presence of the general functional form in equation (1) assures us that we cannot solve it. Finally, equation (1) is labeled non-autonomous because it includes variables other than state variables and time derivatives. The variables , , , and in equation (1) are exogenous variables, which are variables that describe the environment impacting the system. Usually exogenous variables are constants, which is the case in equation (1).

Classifying the variables of a differential equation system helps clarify how you the model builder believe the system works. Because our system (1) is a single equation system, we can only have one endogenous variable. The endogenous variables a differential equation system are always the time derivatives. So, for our system, is endogenous. The state variables of a system are always classified as predetermined, so is predetermined for our system. The level of the variable is predetermined at point in time because the instantaneous change in the level of we just determined by the model in the previous instant of time. Non state variables are exogenous, so , , , and are exogenous. The last value that needs to be specified for the system is the initial condition for the state variable , which is the value of the variable at the initial point in time . Thus, we can summarize the classification of variables as follows:

Classification of Variables
Endogenous (1): Exogenous (4):, , , Predetermined (1): Initial Conditions (1): It is common that differential equation models contain auxiliary equations, which are equations that endogenized (i.e., determine) the values for other variables by relating them to the state variable. Using standard techniques for analyzing a differential equation of this type, we can learn how the path for state variable will evolve over time. Once we know the path followed by the variable , we can use the following “auxiliary” equations to determine the paths followed by the other variables of interest. To illustrate, let us add the endogenous variables , , , and to our system, and assume that they are determined from the following four equations: (2) (3) (4) (5)

The variables, , and are directly related to the variable , and only related to . Thus, if we know the path followed by , we can use equations (2)-(4) to determine the paths followed by , , and . Equation (5) indicates that we need to know the path followed by , along with the value of the exogenous variable , to determine the path followed by . So we can determine the path followed by by first using equation (2) to determine the path followed by . This discussion indicates that we can classify the variables of the dynamic system (1)-(5) as follows.

Classification of Variables
Endogenous (5): , , , , Exogenous (4):, , , Predetermined (1): Initial Conditions (1):

The core of this model is the path followed by the state variable . Once you understand the core of a model, you can use that knowledge to then understand the auxiliary aspects of the model. Notice that adding the auxiliary equations only changes the number of endogenous variables. We can think of the core of the model, which is equation (1) and the first classification of variables as a subsystem, a system upon which the auxiliary aspects of the model depend.

3. Analyzing the Steady State Finding the Steady States The steady state for the state variable is the state where the variable is not changing, or where . Setting in (1), we know SS1 . There are five variables in equation SS1, the model’s four exogenous variables and the state variable. Equation SS1 determines the steady state values of the state variable as a function of the exogenous variables. Assuming , one value of for which equation SS1 holds is . This steady state is not so interesting in that this result is simply telling us that if no capital initially exists, then none will accumulate over time, (which is because capital is assumed to be essential in the production process). If there is a non zero value for that satisfies SS1, let it be denoted by . To illustrate a situation where there is a unique value for the steady state , consider Figure 2 below. The right side of equation SS1 is linear in , with a zero intercept, as shown in the figure. The assumption implies the left side of equation SS1 also has a zero intercept. The conditions and would ensure that the production function increases at a decreasing rate as increases, as shown in the figure. If it were true that, then the only steady state would be associated with , because the two curves and would only intersect at the point in Figure 2. The Inada condition ensures that the curve initial rises above the curve, and the Inada condition ensures that the curve eventually intersects the curve. Because and for all , we know that this intersection at a capital level only occurs once, which implies is unique, as shown in Figure 2. Figure 2: The steady state for the Solow-Swan growth model

Stability of the Steady State The Inada conditions and also ensure that the steady tate associated with is stable. A steady state is locally stable if the state variable moves toward the steady state value for the variable when it is in the “neighborhood” of the steady state. Define the function so . Notice that, if , then the steady state is locally stable. This is because implies when and when . For our model given by SS1, . So, . Because, our steady state is locally stable if . Notice, in Figure 2, that the slope of the curve, at the steady state , is less than the slope of the curve. That is, . So, we find, meaning our steady state is locally stable. Examining Figure 2, we can also see that the steady state is locally unstable. In the neighborhood of , with positive, notice in Figure 2 that the slope of the curve is greater than the slope of the curve. This implies , which implies . If in the neighborhood of , then is increasing in that neighborhood, meaning it increasing away from the steady state value of zero. Figure 3 is a phase diagram. The traditional phase diagram for a single state variable differential equation is constructed by plotting the state variable along the horizontal axis and plotting the time derivative of the state variable along the vertical axis. So, for our model is plotted horizontally and is plotted vertically. The function shows how depends upon . The Inada conditions tell us that the function is positive and diverging to infinity as approaches zero, and they tell us decreases as increases, eventually becoming negative, as shown in Figure 3. The steady state value for is where equals zero. The arrows along the horizontal axis are presented to emphasize is increasing when but decreasing when . Figure 3: Phase diagram for the Solow-Swan growth model

To summarize, we know that the steady state is locally unstable and the steady state is locally stable. The Inada conditions indicate that the path taken over time by the state variable is as shown in Figure 3. If the initial capital stock level is between zero and , then increases over time and approaches . Alternatively, if is greater than , then decreases over time and approaches . Figure 4 shows these two potential time paths for by plotting as it depends upon time.

Figure 4: Two possible time paths for the state variable in the Solow-Swan growth model

4. A System of Two Differential Equations Suppose a dynamic system is represented by the following system of equations and the following classification of variables (A)

(B) Endogenous (2): , Exogenous (2): , ,
Predetermined (2): ,
Initial Conditions (2): ,

How can we characterize the paths followed by the two state variables?

5. Using a Phase Diagram to Characterize the Two Dimensional Differential Equation System

We now characterize the path followed by and diagrammatically in a “phase diagram.” The first step in constructing the phase diagram is to plot the “nullclines” associated with the steady state conditions and . From (B), we can see that implies . The usual restrictions on the production function imply there is a unique value such that . (Notice that we need the Inada conditions and to conclude this. The restrictions and alone are not enough.) When , the assumption implies, so which implies the right side of (B) is negative. Because and are typically assumed for the utility function, we know must be positive for the left side of (B) to be negative. Thus, when , . Using analogous reasoning, when , . This is shown by the arrows in Figure 1.

We obtain the nullcline by setting in condition (A), which implies
. Because , we know when . The Inada conditions and allow us to draw conclusions about the shape of this nullcline consumption level . The assumption ensures that when is near zero, which implies when is near zero. Moreover, ensures increases faster than as increases from a number near zero, which implies is increasing as increases from zero. The assumption ensures that, when is large enough, increases faster than as increases. This implies that, as increases, must reach a peak and then decline thereafter. This nullcline separates the region where is increasing from where it is decreasing. Condition (A) indicates that, when is greater than its nullcline value for a given level , then . Conversely, when is less than its nullcline value for a given level , then . This is shown by the arrows in Figure 1.

Figure 1: Dynamics implied by the Euler Equation and Capital Accumulation Constraint for the Neoclassical Model of Optimal Growth

Starting at any point in the of Figure 1, the arrows tell us the direction of movement. This allows us to trace the path from any starting point . The starting point for is given to us in the problem. The starting point for is a choice. Without additional restrictions, we cannot determine which value of will occur. However, proceeding with some analysis can provide an indication of what those restrictions might be.

Figure 2 is drawn under the assumption that . (After understanding this analysis, it should be apparent how to present the case for .) Path 1 is associated with a relatively high initial consumption choice . Because is in the upper left quadrant of the phase diagram, the direction of movement must be up and to the left ( increasing and decreasing). . Path 2 is still associated with a relatively high initial consumption choice , but one where falls in the lower left quadrant. Initially, the direction of movement must be up and to the right ( increasing and increasing). However, the nullcline is eventually reached, and the direction of movement of is reversed. Thus, the paths of type 2 eventually become like those of type 1. For all paths of type 1 and type 2, must eventually occur. To rule out such paths as equilibrium paths, we need only add the reasonable condition to the problem.

Figure 2: Phase Diagram for the Neoclassical Model of Optimal Growth


Path 3 is associated with a relatively low initial consumption choice . Because is in the lower left quadrant of the phase diagram, the direction of movement must initially be up and to the right ( increasing and increasing). However, on this path, the nullcline is reached, and the direction of movement of is reversed. Eventually, such a path must involve a negative consumption level. One way to rule out such a path is to recognize that, at any given point in time , when the capital-consumption combination is , the consumer can discontinuously increase or decrease the consumption level, for such an action is equivalent to the choosing when . On path 3, by choosing to jump paths by discontinuously increase , (by choosing to reduce the saving level ), the consumer can move to a new path that will provide a higher consumption level in every period, for all time, which is clearly preferable. Thus, it is not optimal to choose a consumption level that generates a path of type 3.

Having ruled out paths of types 1 and 2 because they are not feasible, and paths of type 3 because they are clearly not optimal, we can conclude that the unique optimal choice for , given the initial capital level is the level for that generates path 4, or the level shown in Figure 2. On this path, consumption and capital each increase en route to the steady state , never crossing a nullcline.

It is worth noting that, if the consumer believes that the economy would end at a particular point in time , rather than assuming the economy will never end, then it would be rational to choose a consumption path such that . That is, the consumer should choose the consumption path where capital is just used up when the economy terminates. (The Euler equation and capital accumulation equation would be the same; only the “boundary condition” condition would change to .) This implies, rather than choosing path 4 in Figure 2, the consumer would choose a path either of Type 1 or Type 2, so that vertical axis where is eventually reached. Which path is chosen would depend upon the length of time T. The optimal initial consumption level becomes lower as the length of time T becomes longer.…...

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