Two Theorems on Generalized Diminishing Returns and their Applications to

Economic Analysis* NGO VAN LONG

Australian National University, Canberra, ACT 2600

Under certain weak assumptions such as free disposal and non-satiety, it is shown that the concavity of utility and of technology implies that the maximum value of the set of all attainable programmes is a concave function o f the initial capital stocks. For time-independent problems, this implies that along an optimal path, as a capital stock is accumulated, its shadow price falls. The usefulness of the theorems is demonstrated in a number of examples, including Kemp’s cake-eating problem and Forster’s pollution-control problem.

Z Introduction In dynamic planning models, the usual ob-

jective is to maximize a social welfare index, W, which is defined as the present value of the stream of momentary utility. Obviously W is a function of the initial stocks and the initial time. Let @ ( t ) > 0 be the discount factor. Define:

T

W(k(O), to) = Maximum 1 [P<f)/P(to)] W )

to

U [ f , v ( 0 , k(t)Idt (1)

k(to) = k“’ (2) ( i = 1,2,. . . , n) (3)

(44 ki(t) 2 0, (4b)

where: k( t ) = the capital stocks at time t [k(t)

(T is fixed, possibly infinite), where the maxi- mization is subject to:

k ( t ) = f l [ t , W l gj[ t , v(t>, k(1)l 2 0 (i = 1, 2, . . 9 m)

c R”] v(t) = the control variables [v(t) E RS]

and (4) is a set of m constraints which define the set S of admissible points: S(t) = {(v, k)lgj[t, 4 t h N t ) ] 2 0, i = 1,2, . . . , m>.

( 5 ) * I wish to thank a referee for helpful comments.

Clearly S ( t ) is a convex set i f each gj is quasi- concave.

An interesting theoretical question arises: if U is a concave function and if the set S(t) is convex, can we conclude that W(k(”, to) is a concave function of k(O)? In this paper we will show that the answer is in the affirmative, pro- vided that the functions fi satisfy certain con- ditions and that ‘free disposal’ is assumed. This is the essence of Theorem 1: a ‘generalized diminishing returns’ result.

It is well-known that a concave function is differentiable almost everywhere. When aW/akl(o) is defined, it can be shown that this partial derivative is in fact the current-value shadow price vi*(to) corresponding to the stock k1(O). If for any t j 2 to, we define:

W(k(”, r,) = Maximum [/3(t)/P(tj)]

*J

u t , 40, W l d t

V ( t ) i subject to:

and to (3) and (4), then: k(t j ) = k‘”

a W/ak,”’ = vl*(tj). (6) As is well-known, equation (6) has an economic

interpretation: the ‘shadow price’ vi(t) of the

58

DIMINISHINQ RETURNS 59 1979

capital stock k i ( t ) is equal to the marginal contri- bution of the capital stock ki to the optimal value of the objective function. Consider the special case where b(t) = exp (-pt), p > 0. In this case if T = 00 and if the functions U, fr, g j are time-independent, then W is the function of the initial stock alone. In this case, using (6) and the concavity of W, we can show that for any pair t , , f 2 (tl z t 2 ) :

n

X [v/i<tl) - v / t ( t~ . ) ] [k i ( f~) -ki(tdI 5 0 i= I

(7’)

with strict inequality if W is strictly concave. The inequality (7) is of considerable im-

portance. In optimal control problems without an interior steady state equilibrium it is usually difficult to determine the nature of the optimal path, and (7) is helpful in these cases. For example, when n = 1, (7 ) implies that the capital stock and its current value shadow price move in opposite directions. Applying this result to Kemp’s problem of ‘cake eating under uncertainty’l we are able to derive a sufficient condition for the consumption path to be monotonically decreasing. The inequality (7) can also be used to show that in Forster’s pollution-control models, consumption is mono- tonically increasing.2 These results can also be obtained using a weaker form of (7) :

5 yqt) ic i ( t ) I 0 (8)

which is obtained from (7) by taking limit. (7) and (8) constitute Theorem 2 of the present paper.

Our assumptions and notations will be made precise in Section 11, where the first theorem on generalized diminishing returns is proved and discussed. The relationship between the time path of shadow prices and that of the capital stocks is examined in Section 111. Section IV discusses applications of Theorem 1 and Theorem 2 to a variety of economic models.

i= 1

Ii Generalized Diminishing Returns Our formulation (1) to (4) is quite general.

While k ( t ) is referred to as the vector of ‘capi-

2 See Fors.ter (1976). Forster’s proof that con- sumption is increasing along the optimal path in- volves integral comparison. Our proof is much simpler, bcing a straightforward application of Theorem 2.

See Kemp (1976), Chapter 23.

tal stocks’, in fact it is the vector of stock variables. Accumulated past consumption, or accumulated past profits, or the stock of poltu- tion (as in Forster, 1976), the stock of a natural resource, or the expected rate of infla- tion, etc., are admitted in our formulation.

The assumption that U is concave in ~ ( t ) and k( t ) is a standard assumption in economics and often it reflects diminishing returns.3 The concavity of the g,s can he justified similarly. What assumption can we place on the transition functions f,s? This question seems to require careful considerations.

If ki ( t ) is a productive stock, it is usual to assume the corresponding fi[v( t ) , k( t ) ] is con- cave. For example, if n = 1 and k is the capital stock per head, we may write:

k = F ( k ) - ~ ( t ) - 6k f ( v , k ) (9) where F ( k ) is the concave per capita produc- tion function, v ( t ) is per capita consumption and 6 is the rate of physical depreciation of k. Notice that in many instances a non-concave transition function can be transformed into a concave one by suitable choice of the control variables. Thus, if s ( t ) is the saving ratio, then:

k = s F ( k ) - bk (10) is obviously not concave in (s, k), but if we adopt (9) instead of ( lo ) , then we have a concave transition equation.

If k , ( t ) is a ‘bad’ stock, such as the stock of pollution, it is usual to assume that the correspoiiding transition equation is convex. For example, let k , ( t ) = P ( t ) be the stock of pollution, v,(t) be the output of the con- sumption good and v2(t) be the expenditure on cleaning up the environment. We may then write:

k , ( t ) = P ( r ) = z [ v , ( t ) ] - h[v,(t)] - a P ( t ) ( 1 1 )

where z is a non-negative, convex and in- creasing function, and k is a non-negative concave and increasing function. T h e con- cavity of h reflects diminishing returns in the cleaning-up activity and the convexity of z reflects the increasing cost ( in terms of pollu- tion) of output. Notice that (11) is convex in both v, and v2 because --ti is convex.

3 See Arrow and Kurz (1970), p. 43.

60 THE ECONOMIC RECORD MARCH

We now come to an assumption that will play a useful part in the proofs of the 'general- ized diminishing returns' theorem, the assump- tion of 'free disposal'. Consider equation (9). It implies that the part of output that is not consumed is invested. While in most cases of interest this is true, it will be technically convenient to allow a free disposal activity y ( t ) 2 0 and write:

k = F(k) - v( t ) - y( t ) - 6k (12)

Notice that y ( t ) is a control variable that does appear in the utility function or any of the functions gj. In this sense, the disposal is 'free'. In most cases of interest it is easy to prove that y ( t ) = 0 along an optimal programme. In fact the variable y ( t ) is introduced mainly for technical convenience.

Similarly, for a 'bad' stock such as pollution, we will assume a free disposal activiy y d ( t ) 2 0 and rewrite ( I 1 ) as:

k, = P = z[vl(t)] - k[vZ( t ) ] + yi(t) - aP(t) (13)

In this case free disposal means that we can add to the stock of pollution. Again, this as- sumption is made mainly for technical reasons, and should not cause alarm, because in most cases of interest it is easy to show that y c ( t ) = 0 in the optimal programme.

We can now formally state the free disposal assumption: Assumption A . I (Free Disposal). If f J t , v( t ) , k ( t ) ] is concave in (v, k ) but not linear we assume that:

(1 5) with y,(t) 2 0. Iff,[t, v(t), k(t)] is convex but not linear, we assume that

with y,(t) 2 0. Iff, is linear in (v, k), it is ob- viously concave and convex. In this case we write:

= f ( v , k ) - Y ( 0 .

3 fi(v, k ) + yi(t)* (14)

ki = fip, v O > , k(0l - Yl (0

(16) k __ , - h[t, v w , W l + Y,(O

i, =f#, v (0 , k(t)l. Let N be the set of indices { 1,2, . . . , n}, and let

{i I i E N and fi convex but not linear}

Nl E {i I i E N and fi concave but not linear}

N, (17a)

(17b) (174 N3 = { i I i E N and fi is linear}.

Obviously,

4 c N i c N (17d) where 4 is the null set. Assumption A.2 Each f r is either concave or con- vex (or possibly both). That is, we assume

Assumption A.3 The set S defined by (5 ) is convex. Notice that S is convex if each gi is quasi-concave, Assumption A.4 The free disposal activities are not used (i.e., y(t) = 0) along optimal pro- grammes. Theorem 1 (Generalized diminishing returns)

Under assumptions A.l to A.4, the welfare index is concave (strictly concave) if U is concave (strictly concave) in (v, k). Proof

Let [vA(t), kA(t)] and [vB(t), kB(t)] be optimal programmes when the initial capital stocks are koA and koB respectively. For &to) = ukoA + (1 - a)koB, a E (0, l), consider the feasible pro- gramme defined by :

vc(t) = avA(t) + (1 - a)vB(t) (18) k"(t) = akA(t) + (1 - a)kB(t) (19)

N1 U Nz U N3 = N.

with y t defined by k,"(t) =fr[t, W), k'(t)l - Y , W

= aLiA(t) + (1 -a) i i B ( t ) , ie N~ (20) &Yt) =f i [ t , fO>, k"(t)l + ~lc(t)

= kiA(t) + (1 - a)iciB(t), i E N, (21) y t ( t ) = 0, i E N3. (22) The concavity (convexity) of f r , i E ZVi ( i e Nz) implies that yie(t) 2 0, all i. The convexity of S implies that (vc, k') E S.

Now from the concavity of U ~ [ t , v"(t), kc(t)] 2 aU[t, vA(t), kA(t)]

Integrating (23) yields: + (1 - a)v[t, vB(t), kB(t>l. (23)

J[p(t)/p(t,,)] u[t, vC(t)ldt 2 a W(kOA, to) to + (1 - a)W(koB, to) . (24

The proof is complete by noting that: W[akoR + (1 - a)koB, to] 2 Left hand side of

(24). Q.E.D.

Theorem 1 has a discrete-time counterpart.

Maximize C P(t) U[t, v(t), k(t)]

For any fixed T I co, consider the problem: T

t = t a subject to:

M t ) 3 ki(t + 1) - k,(t) = f , [ t , ~ ( 0 , k(01 k(t0) = k'0'

g,[c v(0, k(0l 2 0 k,(t) 2 0.

1979 DIMINISHING RETURNS 61

Assumptions A.l to A.4 will be retained, with Ak,(t) replacing ki(t). Obviously we have: Theorem 1‘ (Generalized diminishing returns in discrete time) Let

W[k(O), to] = Maximum C P(t) U[t, v(t), k(t)]

where T 5 co and where the maximization is subject to (1’) to (4’). Under A.1, A.2, A.3 and A.4 (with k,(t) replaced by Akt(r)), the welfare index W is concave (strictly concave) if U is concave (strictly concave) in (v , k).

T

f= fo

ZZZ The Time Paths of Shadow Prices and Capital Stocks

With the help of Theorem 1 we will be able to establish a basic relationship between the time path of shadow prices and that of the capital stocks.

Let

~ [ k ‘ o ) , to] = max e-dl--io) U [ f , v(t), k(t)]dt (25)

to

with T 5 00, and subject to k(t,) = k(O), (26)

g,P, v (0 , k(0l 2 0 (27) k, =f;-[t, v ~ , k(t)] - r,(t>, for i E N~ (28) ic., = htt, v(t>, k(t)l + Y,W, for] E N~ (29) k, = fi[tl m, Wl forjeN, (30) and the non-negativity of yi(t).

In what follows it will be assumed that the functions U, g, andfi belong to theclass C(,), and that the constraint qualifications4 are satisfied. Under these assumptions, if [v*(t), y*(t), k*(t)l is an optimal path, then there exists a vector of shadow prices’ v/*(t) associated with k(t) such that :

yi,*(t) = w,* - a w k , (31) k,(O = w a w , (32)

where L is the Lagrangean function associated with problem (25).

4See, for example, Hadley and Kemp (1971), p. 283, and Long and Vousden (1976).

5 There is also a constant multiplier $,, 2 0 associated with the integrand. We adopt the stan- dard assumption that q,, is positive, and hence we set $,, = 1. For sets of sufficient conditions en- suring q0 > 0, see Long and Vousden (1976).

The following Lemma can now be stated: Lemma 1

Let:

W(k(O), to) E

then, for j = 1, 2, . . . , n,

For a proof of Lemma 1, see Hadley and Kemp (1971), pp 117 ff. The classical argu- ment of Hadley and Kemp can be easily ex- tended to allow for constraints of various kinds.6

Now, if the programme [v*(t), y3’(t), k*(t)], t E (to, T ) maximizes (25), then for any tl > I,, the remaining portion of that programme must maximize:

e-P(f--fo) U[t , v*(r) , k*(r)]dt (33) -io f

t,vj*(ro) = a w/ak,(O). (34)

T ,. j e--p(t-tJ) U [ t , v ( 9 , k(t)]dt

f j

(35)

and if w*(t) is the shadow price path for (25 ) then y*(t), t 2 ti is the shadow price path for (35). Thus if we define:

W(k(J), ti) = Maximum e + ’ ( f - t J ) U[ t , v(t), { v ( t ) , Y ( O > i k(0ldt (3 6)

tJ

subject to:

and (26) to (30), then k(t,) = k(J) (37)

a W/ak,(J)l,,tJ,k(J)=k*(t,) = wi*(t,). (38)

If T = co and if the function U, g, and fi are time-independent i.e., U = U[v(t) , k(r)], gj = g,[v(t), kM1, f, =h[v(t), W l , so that the only place where t appears explicitly is in the discount factor, then the welfare index W is the function of the initial stock alone. In this case W(k(’), to) in (25) and W ( / C ( ~ ) , t j ) in (36) are replaced by W(k(”) and W(k(’)) respectively. The concavity of W(k) is of great use here in deriving Theorem 2 below.

Consider the path [v*(t), y*(t), k*( f ) ] which maximizes (25). For any t i , t2 such that t , # r,, tl 2 to, tz 2 to, we have, from the concavity of W(k) :

6 Alternatively, a proof can be based on Hestenes’ theorem 11.1 (Hsstenes, 1966). For more heuristic proofs, see Arrow and Kurz (1970), pp. 34-5, or Intriligator (1971), Chapter 14.

62 THE ECONOMIC RECORD MARCH

5 [aw/akl*(t,)ltki*(t2) - kt*(tl)l i - 1

2 W[k*(t2)] - W[k*(t,)] (39)

2 [a ~/ak,*(t ,)I[kl*(tl> - kl*(t2)1 i-1

2 W[k*(t,)] - W[k*(t2)]. (40) Adding (39) to (a), we have:

5 [a W&*(t,) - a Wlak,*(t,>lk*(t2) i = l

- kt,*(tl)] 2 0. (41) Using (41) and (38) we obtain the following theorem : Theorem 2 If T = co and if the functions U, g,, fi are time-independent, then under assumptions A.l, A.2 and A.3 of Section ZZ, the optimal shadow price and quantity paths of problem (25) satisfy:

if U is concave in (v, k). Strict inequality holds if U is strictly concave in (v, k ) . Corollary Under the hypotheses of Theorem 2,

C [yl*(t,) - ytT(t2)l[ki*(tl) - ki*(t2)1 5 0 (42)

i2, Gi*(t)i i*(t) 5 0. (43) i= 1

Inequality (43) is obtained from (44) by dividing both sides by (tz - ti)' and taking limit t2 4 t l .

ZV Applications The class of economic problems where our

theorems can be usefully applied is quite wide. We will consider some simple models which illustrate possible applications.

Optimal consumption of an unknown reserve It is well-known that with a concave utility

function, the optimal way to eat a cake of a known size must satisfy the condition that when- ever C(t) > 0, C(t) is given by:

where p > 0 is the rate of time preference. When the size of the cake is unknown, then the planned consumption is not necessarily a monotone decreasing function of time. This result is due to Kemp (1976), Chapter 23. Kemp's formulation is :

C = p U ' / U " < 0 (44)

Maximize H[Q(t)]U[C(t)]e-Ptdt (45) 0 s"

subject to: Q ( f , = C(t) Q(0) = 0, C(t) 2 0.

where Q(t) is planned accumulated consumption

and n(Q) is the probability that the initial size of the cake exceeds Q, with n(0) = 1 and n'(Q) < 0.

L = n(Q)U(C) + YC f 1C. (46) If U'(0) = "3, then, for all Q such that n(Q) > 0, we have :

The Lagrangean is:

n(Q)U'= -y. (47) Using Theorem 2 we can derive a sufficient con-

dition for planned consumption to be decreasing over time. The sufficient condition is that rI(Q)U(C) be strictly concave in ( Q , C). For example, let n ( Q ) = 0 for all Q 2 > 0, and:

n ( Q ) = (e - QY/G 0 < cc < 1 , Q 5 5 U(C) = A 0 a + P < 1.

A > 0,o < p < 1

The strict concavity of the integrand implies that W(Q) is concave and hence, using the corollary:

But whenever C(t) > 0, Q(r) > 0 and hence @(t) 5 0. From (47):

hence: i. < 0.

Optimal consumption and pollution control In a recent paper, Forster (1976) showed that

under certain conditions it is optimal to drive the stock of pollution to zero. In his model, among paths that lead to a completely clean environ- ment, there are paths along which consumption at first increases and then decreases. Can these paths be optimal? Using a series of inequalities, Forster showed that the answer is in the negative. This answer can be obtained much more simply with the use of Theorem 2.

Q( l ) i ( t ) 5. 0.

U'(C)d(Q)Q f n(Q)U"(C)C = - I$

In Forster's model, the society wishes to :

Maximize e-PtU(C, P)dt j: 0

subject to:

where P is the stock of pollution, 4" is the fixed output, to be allocated between consumption, C, and pollution control, 4" - C. The function g is convex (consumption adds to pollution at an increasing rate) while h is concave (diminishing returns in pollution control). U(C, P) is concave, with Up < 0, Up, < 0, U0 5 0. Let y be the shadow price of P, it is easily seen that:

P = g(c) - h(40 - C ) - aP

acpy > o aqaP 5 o

1979 DIMINISHING RETURNS 63

whenever C > 0 (which is always the case in Forster's model).

Now: c = (ac/aP)E; + (acpv) 9

and along paths which drive pollution to zero, P < 0, and, from our (43), @ 2 0. Hence C 2 0 along the optimal path which leads to a pollution-free environment.

The deterniination of the optimal initial size of the firm

Consider now the discrete time formulation (1')-(4'). In that formulation, k(to) is given. In many cases, k(to) is not inherited, but is itself a n object of choice, at a cost. From Theorem l', the marginal present value of k(to) is downward sloping, because:

M P V [ ~ ( ~ , ) ] = a w/ak(o) (48) and, form the concavity of Wwith respect to k"):

a 2 w/(ak(oy 5 0. (49) If we postulate that k(O) can be purchased and

the marginal cost function MC(k(O)) is non- decreasing, then the optimal initial size of the firm, k(O)*, is uniquely determined. Comparative static exercises can then be performed. For example if U [ t , v(t), k( t ) ] takes the form of:

(50) where F is gross profit and II is the profit tax rate, then an increase in II will shift the MPV(k'O') curve downward proportionately reducing I%(')*.

u [ t , v(t), 4 t ) l = (1 - JW[t, ~ ( 0 , k(0I

Other applications In finite horizon planning models, the term-

inal capital stock is usually left free or perhaps partially free, subject to some arbitrary terminal constraint. It is more appealing to introduce a bequest function, W [ k ( T ) ] , and in this case Theorem 1 provides a useful characterization of thc bcquest function. The transversality con- dition at the terminal time, T, is then:

$(T) = d w [ k ( T ) ] / d k ( T ) (51) where JI( T ) is the terminal value of the shadow price. Hence the locus of points ( k , J I ) that satisfies the transversality condition (51) is upward-sloping, because, from (51 ) :

d$(T)/dk(T) = W"[k(T)] < 0. (52) Another application of Theorem 2 is that, in

one-state-variable problems, the saddle branches leading to stationary states have non-positive slopes, provided that the assumptions of Theorem 2 hold. Under these conditions, the capital stock and its shadow price move in opposite directions, which is a very intuitive result.

REFERENCES Arrow, K. J. and M. Kurz (1970), Public Invest-

menf, the Rate of Return, and Optimal Fiscal Policy (John Hopkins, Baltimore).

Forster, B. A. (1976), 'Optimal Consumption in a Polluted Environment', Essay 2, in J. D. Pitohford and S. J . Turnovsky (eds), Appli- cations of Control Theory to Economic Analy- sis (North-Holland, Amsterdam).

Hadley, G . and M. C. Kemp (1971), Variational Methods in Economics (North-Holland, Am- sterdam).

Hestenes, M. R. (1966), Calculus of Variations and Optimal Control Theory (John Wiley, New York).

Intriligator, M. D. (1971), Mathematical Optimi- zation and Economic Theory (Prentice Hall, Englewood Cliffs).

Kernp, M. C. (1976), Three Topics in the Theory of International Trade (North-Holland, Am- sterdam).

Long, N. V. and N. Vousden (1976), 'Optimal Control Theorems', Essay 1, in J. D. Pitch- ford and S. J. Turnovsky (eds), Applications of Control Theory to Economic Analysis (North-Holland, Amsterdam).