A di¤erential equation system consists of two or more di¤erential equations which are mutually related to each other. Such a system can be written as
_x (t) = Ax (t) + b;
where the vector x (t) is given by x = (x1; x2; x3; :::; xn)0; A is an n n matrix with elements aij and b is a vector b = (b1; b2; b3; :::; bn)0: Note that elements of A and b can be functions of time but not functions of x:
With constant coe¢ cients, such a system can be solved in various ways, e.g. by deter-mining so-called Eigenvalues and Eigenvectors. These systems either result from economic models directly or are the outcome of a linearization of some non-linear system around a steady state. This latter approach plays an important role for local stability analyses (compared to the global analyses we undertook above with phase diagrams). These local stability analyses can be performed for systems of almost arbitrary dimension and are therefore more general and (for the local surrounding of a steady state) more informative than phase diagram analyses.
Please see the references in “further reading” on many textbooks that treat these issues.
4.6 Further reading and exercises
There are many textbooks that treat di¤erential equations and di¤erential equation sys-tems. Any library search tool will provide many hits. This chapter owes insights to Gandolfo (1996) on phase diagram analysis and di¤erential equations and inter alia -to Brock and Malliaris (1989), Braun (1975) and Chiang (1984) on di¤erential equations.
See also Gandolfo (1996) on di¤erential equation systems. The predator-prey model is treated in various biology textbooks. It can also be found on many sites on the Internet.
The Leibniz rule was taken from Fichtenholz (1997) and can be found in many other textbooks on di¤erentiation and integration.
The AK speci…cation of a technology was made popular by Rebelo (1991).
Exercises chapter 4
Applied Intertemporal Optimization Using phase diagrams
1. Phase diagram I
Consider the following di¤erential equation system,
_x1 = f (x1; x2) ; _x2 = g (x1; x2) : Assume
fx1(x1; x2) < 0; gx2(x1; x2) < 0; dx2 dx1 f(x
1;x2)=0
< 0; dx2 dx1 g(x
1;x2)=0
> 0:
(a) Plot a phase diagram for the positive quadrant.
(b) What type of …xpoint can be identi…ed with this setup?
2. Phase diagram II
(a) Plot two phase diagrams for
_x = xy a; _y = y b; a > 0: (4.6.1) by varying the parameter b.
(b) What type of …xpoints do you …nd?
(c) Solve this system analytically. Note that y is linear and can easily be solved.
Once this solution is plugged into the di¤erential equation for x; this becomes a linear di¤erential equation as well.
3. Phase diagram III
(a) Plot paths through points marked by a dot “ ” in the …gure below.
(b) What type of …xpoints are A and B?
y
O x
&x=0
&y=0
A
B
4. Local stability analysis
Study local stability properties of the …xpoint of the di¤erential equation system (4.6.1).
5. Phase diagram and …xpoint
Grossman and Helpman (1991) present a growth model with an increasing number of varieties. The reduced form of this economy can be described by a two-dimensional di¤erential equation system,
_n (t) = L
a v (t); _v (t) = v (t) 1 n (t) ;
where 0 < < 1 and a > 0. Variables v (t) and n (t) denote the value of the representative …rm and the number of …rms, respectively. The positive constants and L denote the time preference rate and …x labour supply.
(a) Draw a phase diagram (for positive n (t) and v (t)) and determine the …xpoint.
(b) What type of …xpoint do you …nd?
6. Solving linear di¤erential equations Solve _y (t) + y (t) = ; y (s) = 17 for
(a) t > s;
(b) t < s:
(c) What is the forward and what is the backward solution? How do they relate to each other?
7. Comparing forward and backward solutions Remember that Rz2
z1 f (z) dz = Rz1
z2 f (z) dz for any well-de…ned z1; z2 and f (z) : Replace T by t0 in (4.3.8) and show that the solution is identical to the one in (4.3.7). Explain why this must be the case.
8. Derivatives of integrals
(e) Show that the integration by parts formulaRb
a _xydt = [xy]ba Rb
ax _ydt holds.
9. Intertemporal and dynamic budget constraints
Consider the intertemporal budget constraint which equates the discounted expen-diture stream to asset holdings plus a discounted income stream,
Z 1 (a) Show that solving the dynamic budget constraint yields the intertemporal
bud-get constraint if and only if limT !1A (T ) exph RT
t r ( ) d i
= 0:
(b) Show that di¤erentiating the intertemporal budget constraint yields the dy-namic budget constraint.
10. A budget constraint with many assets
Consider an economy with two assets whose prices are vi(t). A household owns ni(t) assets of each type such that total wealth at time t of the household is given by a (t) = v1(t) n1(t) + v2(t) n2(t) : Each asset pays a ‡ow of dividends i(t) : Let the household earn wage income w (t) and spend p (t) c (t) on consumption per unit of time. Show that the household’s budget constraint is given by
_a (t) = r (t) a (t) + w (t) p (t) c (t)
where the interest rates are de…ned by
r (t) (t) r1(t) + (1 (t)) r2(t) ; ri(t) i(t) + _vi(t) vi(t)
and (t) v1(t) n1(t) =a (t) is de…ned as the share of wealth held in asset 1:
11. Optimal saving
Let optimal saving and consumption behaviour (see ch. 5, e.g. eq. (5.1.6)) be de-scribed by the two-dimensional system
_c = gc; _a = ra + w c;
where g is the growth rate of consumption, given e.g. by g = r or g = (r ) = : Solve this system for time paths of consumption c and wealth a:
12. ODE systems
Study transitional dynamics in a two-country world.
(a) Compute time paths for the number ni(t) of …rms in country i: The laws of motion are given by (Grossman and Helpman, 1991; Wälde, 1996)
_ni = nA+ nB Li ni (L + ) ; i = A; B; L = LA+ LB; ; > 0:
Hint: Eigenvalues are g = (1 ) L > 0 and = (L + ).
(b) Plot the time path of nA. Choose appropriate initial conditions.
Finite and in…nite horizon models
One widely used approach to solve deterministic intertemporal optimization problems in continuous time consists of using the so-called Hamiltonian function. Given a certain maximization problem, this function can be adapted - just like a recipe - to yield a straightforward result. The …rst section will provide an introductory example with a
…nite horizon. It shows how easy it can sometimes be to solve a maximization problem.
It is useful to understand, however, where the Hamiltonian comes from. A list of examples can never be complete, so it helps to be able to derive the appropriate optimality conditions in general. This will be done in the subsequent section. Section 5.4 then discusses what boundary conditions for maximization problems look like and how they can be motivated. The in…nite planning horizon problem is then presented and solved in section5.3which includes a section on transversality and boundedness conditions. Various examples follow in section 5.5. Section 5.7 …nally shows how to work with present-value Hamiltonians and how they relate to current-value Hamiltonians (which are the ones used in all previous sections).
5.1 Intertemporal utility maximization - an intro-ductory example
5.1.1 The setup
Consider an individual that wants to maximize a utility function similar to the one en-countered already in (4.4.8),
U (t) = Z T
t
e [ t]ln c ( ) d : (5.1.1)
The planning period starts in t and stops in T t: The instantaneous utility function is logarithmic and given by ln c ( ) : The time preference rate is : The budget constraint of this individual equates changes in wealth, _a ( ) ; to current savings, i.e. the di¤erence
107
between capital and labour income, r ( ) a ( )+w ( ) ; and consumption expenditure c ( ), _a ( ) = r ( ) a ( ) + w ( ) c ( ) : (5.1.2) The maximization task consists of maximizing U (t) subject to this constraint by choosing a path of control variables, here consumption and denoted by fc ( )g :
5.1.2 Solving by optimal control
This maximization problem can be solved by using the present-value or the current-value Hamiltonian. We will work with the current-value Hamiltonian here and in what follows.
Section 5.7 presents the present-value Hamiltonian and shows how it di¤ers from the current-value Hamiltonian. The current-value Hamiltonian reads
H = ln c ( ) + ( ) [r ( ) a ( ) + w ( ) c ( )] ; (5.1.3) where ( ) is a multiplier of the constraint. It is called the costate variable as it corre-sponds to the state variable a ( ) : In maximization problems with more than one state variable, there is one costate variable for each state variable. The costate variable could also be called Hamilton multiplier - similar to the Lagrange multiplier. We show further below that ( ) is the shadow price of wealth. The meaning of the terms state, costate and control variables is the same as in discrete time setups.
Omitting time arguments, optimality conditions are
@H
@c = 1
c = 0; (5.1.4)
_ = @H
@a = r : (5.1.5)
The …rst-order condition in (5.1.4) is a usual optimality condition: the derivative of the Hamiltonian (5.1.3) with respect to the consumption level c must be zero. The second optimality condition - at this stage - just comes “out of the blue”. Its origin will be discussed in a second. Applying logs to the …rst …rst-order condition, ln c = ln ; and computing derivatives with respect to time yields _c=c = _ = : Inserting into (5.1.5) gives the Euler equation
_c
c = r, _c
c = r : (5.1.6)
As this type of consumption problem was …rst solved by Ramsey in 1928 with some support by Keynes, a consumption rule of this type is often called Keynes-Ramsey rule.
This rule is one of the best-known and most widely used in Economics. It says that consumption increases when the interest rate is higher than the time preference rate. One reason is that a higher interest rate implies - at unchanged consumption levels - a quicker increase in wealth. This is visible directly from the budget constraint (5.1.2). A quicker increase in wealth allows for a quicker increase in consumption. The second reason is that
a higher interest rate can lead to a change in the consumption level (as opposed to its growth rate). This channel will be analyzed in detail towards the end of ch. 5.6.1.
Equations (5.1.2) and (5.1.6) form a two-dimensional di¤erential equation system in a and c: This system can be solved given two boundary conditions. How these conditions can be found will be treated in ch. 5.4.