Guang-Zhen Sun∗
Monash University and National Taiwan University.
Abstract: This short article offers economically intuitive proofs of the Euler equation and the maximum principle based on one of the best known results in economics, namely that the marginal utility of one extra dollar spent on each consumption goods is the same for all the consumption goods as required by budget-constrained utility maximization.
Keywords: The Euler equation, marginal utility, the maximum principle.
MSC 2000: 49K10, 91B02.
1.Introduction
That the marginal utility of one extra dollar (MUD) spent on each consumption goods is the same for all the consumption goods as required by budget-constrained utility maximization, or, the same thing put in economics jargon, the marginal rate of substitution between any two goods equals their price ratio, is unquestionably one of the best known results to students in economics. For convenience, we will refer to this result as the MUD principle below.1 On the other hand, in the inter-temporal decision context, the Euler equation proves to be a far more powerful tool, from which one can readily obtain the MUD principle in its inter-temporal version wherein the same goods (service) at different time is formally viewed as different goods defined by the date and hence MUD remains the same across time. This short article aims to show that one can indeed reverse the reasoning, making use of the MUD principle to derive the Euler equation. (The standard proof of Euler equation using the calculus of variations is found in almost any textbook in mathematical economics, e.g., [3], pp.377-9). Furthermore, by similar argument, the maximum principle can also be established. As such, the principle of MUD that underlies the well known condition for constrained utility maximization,
∗ Correspondence; Department of Economics, Monash University, Clayton Vic. 3800, Australia. E-mail:
1 This result is justly attributed in the history of economics literature to Hermann Heinrich Gossen (1810-1858), a brilliant predecessor of the Marginalism Revolution in the history of economics, and is therefore refereed to as “Gossen’s Second Law” (e.g., [4], p.220 and [5], pp.551-2). Nonetheless, the term
“Gossen’s Second Law” does not appear to be well known among contemporary economists presumably due to widely held reluctance in study of the history of economics. Yet its content is of course known to any economics student.
2
may derive, or at least better understand, other results which may otherwise take sophisticated calculations to obtain.
It may be noted that, as shown below, the reasoning to establish the Euler equation and the maximum principle is straightforward once appropriately famed in economics logic. Somehow surprisingly, however, no such proof is available in the literature (as far as I know). For a discrete-time version of the Ramsey growth model, Heal ([1], pp.272-6) proposed an economically intuitive argument, namely hypothetically postponing an infinitesimal amount of consumption from one year to another along the optimal consumption path does not increase the maximand, to derive the necessary condition of the optimal consumption-saving plan. The same idea was then elaborated by Jones ([2], pp. 224-8) in terms of social welfare to characterize the Keynes-Ramsey rule as a necessary condition of a simple Ramsey growth model. In his textbook of macroeconomics, Romer on a few occasions made masterful use of economic reasoning to obtain the first order conditions for some well specified models, e.g., the Ramsey growth model with a constant relative risk aversion (CRRA) utility function ([6], pp.49-56). However, to intuitively derive the Euler equation for models of well-defined economic issues with specified functional forms is one thing, and to prove the Euler equation in its general form is quite another.
2. Economic Intuition
Consider∈
∫
= T
T t t C t x
dt t t x t x F Max V
0 0
1 ( ( ), ( ), )
) , ( )
( & where
F
∈C
2s.t.
x
(t
0)=x
0 and x(
T) =
xT (1) wherex
(t) is piecewise continuously differentiable and T could be infinity.Recall that for the simplest budget-constrained utility maximization problem wherein the prices are the same for all the n consumption goods
X ,...,
1X
n (n≥ 2
), with I being the income budget,Max
u
(x
1,L,x
n) s.t.x
1+L+x
n =I
(2) the principle of MUD requires that,∂
u
/∂x
i =∂u
/∂x
j,∀i
,j
∈{1,Ln
} (3)∫
That is, V can be treated as the maximized value of the integral of F, as a function of )
turns out, problem (4) is essentially no more than a continuous version of problem (2).
Consequently, the marginal change in V caused by an infinitesimal increase in
x
&(t) at any two pointst
1,t
2∈(t
0,T
) should be the same at optimum.For the sake of illustration, we may view the decision horizon of problem (4) as a period from year
t to year T. To simplify notation, the solution to problem (4) is
0 still denoted asx
&(t) in the rest of this section. As the length of each “year” approaches zero,F x t x t t V
( & . Hypothetically increasing
x
&(t) by an infinitesimal amount at any year but nowhere else leads to the same change in V, by the MUD principle. For the particular year t*
, letx
&(t) increase by a small number δ . Thus,As a consequence, the change in V directly caused by a hypothetically infinitesimal change in
x
&(t) at t=
t*
is4
It follows from (6) and (8) that the change in V caused by an infinitesimal change in ) of which w.r.t. time t
*
consequently equals zero. Hence the Euler equation,) 0
3. Proof of the Euler Equation
We now present a rigorous treatment of the above economic intuition by considering a specific manner in which
x
&(t) is increased infinitesimally around t=
t*
. As above, unnecessary — concern effectively applies to the logic of Equations (2) and (3), as any small increase inx
i (i
∈{1,L,n
}) alone violates the budget constraint.∫ ⋅ ∂
The principle of MUD requires the above must be the same for any value of t* .
Differentiation of RHS of (11) w.r.t. t*
thus equals zero, i.e.,
applies to multi-dimension Euler equations as well, for which the analysis is essentially
the same, yet with a bit more cumbersome notations.
4. Proof of the Maximum Principle
Consider the control problem with a fixed value of the state variable at the terminal-point, The solution to problem (12) is denoted as u
ˆ t ( )
and the corresponding state values over ]t
∈[t
0,T
as xˆ t ( )
. Clearly, for any arbitrary differentiable functionπ
(t),6
neatness. A careful reader might be concerned with the possibility that
u
(t) may also depend on the values of xˆ t ( ' )
andx&
ˆ t( ') for some t' ≠
t , even on the path applies to such a general case.Problem (12) can thus be equivalently formulated as
)
The solution to (15) is still signified as
x
(t) hereafter to simplify notation. Similar argument to that in the preceding subsection yields,dV
(δ
;x
t*)/d δ
which by the MUD principle must be the same for any t , and of which differentiation wrt t therefore equals zero. Hence,
)
Acknowledgements.
The author is grateful to the two referees for helpful comments and the National Science Council of the Republic of China (Taiwan) for support during his visiting professorship at Department of Economics of National Taiwan University.References
[1] G. M. HEAL, The Theory of Economic Planning, North-Holland, Amsterdam, 1973.
[2] H. G. JONES, An Introduction to Modern Theories of Economic Growth, McGrew-Hill, New York, 1976.
[3] K. LANCASTER, Mathematical Economics. Dover, New York, 1987.
[4] H. LANDRETH and D. C. COLANDER, History of Economic Thought, 3rd edition. Houghton Mifflin Company, Boston, 1994.
[5] J. NIEHANS, “Gossen”, in The New Palgrave: A Dictionary of Economics, edited by John Eatwell, Murray Milgate and Peter Newman, Macmillan, London. 1987, pp. 550-554.
[6] D. ROMER, Advanced Macroeconomics, Third edition. McGraw-Hill, New York, 2006.
G-Z Sun
I made a research trip during March 25-Apr 5, 2007 back to my Monash University, Melbourne, Australia, to discuss with my Monash colleagues, especially Professor Yew-Kwang Ng, one leading theorist in increasing returns to specialization, on some key conceptual issues
confronting my NSC research project as well as possible extension thereof. It proved fruitful. In addition, one unexpected outcome generated from this trip, a pleasant surprise to myself indeed, is that I conceived a new research agenda that I reckon will significantly extend this NSC project, and I plan to carry it out in the near future. The flight tickets cost NT 27772.00, which was later on reimbursed from the NSC grant and which was also the total cost of this research trip.