Multisector Growth Models

Abstract

Multisector models are essential ingredients for general equilibrium analysis of an economy over time. They have been used extensively in the literature whenever an adequate description of the relevant issues makes it inappropriate to use aggregative models for formal analysis. A study of optimal accumulation of capital goods or optimal depletion of exhaustible resources is a key to developing a theory of economic planning. The specific results can also be viewed from a different perspective. The idea that markets and prices can be used to achieve efficiency in a decentralized manner has been central to economics. The fundamental theorems of ‘new’ welfare economics identify conditions under which competitive economies attain an efficient or Pareto optimal allocation of resources. It is natural to enquire whether in dynamic models such a connection between optimality and competitive prices can be established. One possibility is to use the basic static model and treat the same good at different points of time as different commodities. While such an approach is not entirely shorn of merit, a fundamental paper by Malinvaud (1953) suggested that when economic activity does not terminate at a known date, the outcome of a period-by-period competitive process may fail to be optimal. Indeed, the possibility (or otherwise) of designing an informationally decentralized resource allocation mechanism that leads to optimal outcomes has been the subject of considerable speculation and over the last thirty years it has become a ‘classical’ problem in dynamic models with an infinite horizon. In what follows, I shall review some recent results that throw new light on this topic.

Multisector models are essential ingredients for general equilibrium analysis of an economy over time. They have been used extensively in the literature whenever an adequate description of the relevant issues makes it inappropriate to use aggregative models for formal analysis. A study of optimal accumulation of capital goods or optimal depletion of exhaustible resources is a key to developing a theory of economic planning. The specific results can also be viewed from a different perspective. The idea that markets and prices can be used to achieve efficiency in a decentralized manner has been central to economics. The fundamental theorems of ‘new’ welfare economics identify conditions under which competitive economies attain an efficient or Pareto optimal allocation of resources. It is natural to enquire whether in dynamic models such a connection between optimality and competitive prices can be established. One possibility is to use the basic static model and treat the same good at different points of time as different commodities. While such an approach is not entirely shorn of merit, a fundamental paper by Malinvaud (1953) suggested that when economic activity does not terminate at a known date, the outcome of a period-by-period competitive process may fail to be optimal. Indeed, the possibility (or otherwise) of designing an informationally decentralized resource allocation mechanism that leads to optimal outcomes has been the subject of considerable speculation and over the last thirty years it has become a ‘classical’ problem in dynamic models with an infinite horizon. In what follows, I shall review some recent results that throw new light on this topic.

Notation

R^m denotes the m-dimensional Euclidean space; if x = (x_i) ∈ R^m we write x ≥ 0 (x is non-negative) if x_i ≥ 0 for all i; x > 0 (x is positive) if x ≥ 0 and x ≠ 0; x ≫ 0 (x is strictly positive) if x_i > 0 for all i. ${R}_{+}^m=\left\{x\in {R}^m:x\ge 0\right\}$ and ${R}_{++}^m=\left\{x\in {R}^m:x\gg 0\right\}$N is the set of all non-negative integers.

Programmes

There are m producible commodities in the economy (the term ‘commodity’ is interpreted broadly, including machines of different vintage) and a single non-producible factor of production called ‘labour’. Labour is used as an input in production but does not enter into consumption. The supply of labour in period t, denoted by L_t, is given by

$$ {L}_t={L}_0{\lambda}^t,\ \ \ \ \ {L}_0>0,\ \ \ \ \ \lambda >0;\ \ \ \ \ t\in N $$

(1)

An activity is a triplet $\left(L,X,Y\right)\in {R}_{+}\times {R}_{+}^m\times {R}_{+}^m$, where L is the quantity of labour input, X the vector of inputs of producible goods and Y the vector of outputs of producible goods. Let ${J}^{\prime}\subset {R}_{+}\times {R}_{+}^m\times {R}_{+}^m$ be the set of all technologically feasible activities. The following assumptions on J′ are made:

• (T′.1) J′ is a closed convex cone containing (0,0,0).

• (T′.2) ‘(L, X,Y) ∈ J′, L′ ≥ L, X′ ≥ X, 0 ≤ Y′ ≤ Y’ implies ‘(L′, X′, Y′) ∈ J′’.

• (T′.3) There exists $\left(\widehat{L},\widehat{X},\widehat{Y}\right)\in {J}^{\prime }$ such that $\widehat{Y}\gg \lambda \widehat{X}.$

• (T′.4) (L, X,Y) ∈ J′, L = 0 and Y ≠ 0 implies ‘Y < X’.

• (T′.5) ‘(L, X₁,Y₁) ∈ J′, (L, X₂,Y₂) ∈ J′, L > 0, X₁ ≠ X₂ and 0 < w < 1’ implies that ‘there exists Y > wY₁ + (1 − w)Y₂ such that (L, wX₁ + (1 − w) X₂, Y) ∈ J′’.

One can interpret the assumptions on J′ as follows: (T′.1) means that the technology exhibits constant returns to scale, that inaction is possible and that the production process is continuous (limits of feasible input–output combinations are always feasible). (T′.2) formalizes the idea of free disposal: if (L, X, Y) is feasible, any non-negative output vector smaller than Y is feasible from any input vector larger than (L, X). (T′.3) means that the technology is sufficiently productive: given the natural growth rate λ there is some activity that leads to an increase in per capita stocks. (T′.4) stresses the essential role of labour in the production process: if an activity uses no labour at all, then its net production is non-positive. Finally, (T′.5) is a restrictive strict convexity assumption on the technology.

Given an initial stock ${Y}_0\in {R}_{++}^m$ a production programme from Y₀ is a pair of sequences (X, Y) ≡ (X_t,Y_t)_t∈N such that

$$ \left({X}_t,{Y}_{t+1}\right)\in {J}^{\prime },\ \ \ \ \ {X}_t\le {Y}_t\ \ \ \ \ \mathrm{for}\ \ \ \mathrm{all}\ \ \ \ \ t\in N $$

(2)

A production programme generates a consumption programme C = (C)_t)_t∈N defined by C = Y_t − X_t

It is convenient to use per-capita variables.

Define:

$$ {x}_t={X}_t/{L}_t,\ \ \ \ \ {y}_t={Y}_t/{L}_t,\ \ \ \ \ {c}_t={C}_t/{L}_t\ \ \ \ \ \mathrm{for}\ \ \ \ \ t\in N $$

(3)

Using the assumption (T′.1), we can rewrite the feasibility conditions (2) as:

$$ \left(1,{x}_t,\lambda {y}_{t+1}\right)\in {J}^{\prime },\ \ \ \ \ {x}_t\le {y}_t\ \ \ \ \ \mathrm{for}\ \ \ \mathrm{all}\ \ \ t\in N $$

(4)

Define the set

$$ J{R}_{+}^m\times {R}_{+}^m\ \ \ \ \ \mathrm{as}\ \ \ \ \ J=\left\{\left(x,y\right):\left(1,x,\lambda y\right)\in {J}^{\prime}\right\} $$

Then (4) is equivalent to

$$ {}^{`}x\left({x}_t,{y}_{t+1}\right)\in J,\ \ \ \ \ {x}_t\le {y}_t\ \ \ \ \ \mathrm{for}\ \ \ \mathrm{all}\ \ \ \ \ t\in {N}^{\prime} $$

Also, one gets

$$ {c}_t={y}_t-{x}_t\ \ \ \ \ \mathrm{for}\ \ \ \mathrm{all}\ \ \ t\in N. $$

For brevity,

$$ \left(\mathbf{x},\mathbf{y},\mathbf{c}\right)={\left({x}_t,{y}_t,{c}_t\right)}_{t\in N} $$

is called a programme from the initial stock y₀. A programme is a complete specification of inputs x_t, outputs y_t and consumptions c_t (measured in per capita terms) in all periods. The relevant constraints are indicated in (4). The set of all programmes from y₀ is denoted by F(y₀).

Evaluation of Programmes

In order to evaluate the welfare-implications of alternative programmes, one introduces an appropriate criterion. With respect to any such criterion, the question of existence of a ‘best’ or a ‘maximal’ programme ought to be settled first. In infinite horizon models, subtle consistency problems may arise owing to the fact that an evaluation criterion need not be representable by a real valued function which is continuous in the same topology as that in which F(y₀) is compact. Consider, first, the notion of intertemporal efficiency. A programme (x, y, c) in F(y₀) is intertemporally efficient if there does not exist another programme (X′,Y′,C′) such that c_t − c_t ≥ 0 for all t and ${c}_t^{\prime }-{c}_t>0$ for some t. It is easy to see that F(y₀) contains an infinite number of efficient programmes. A basic question in the literature has been the relation between programmes that are efficient and those that meet the criterion of intertemporal profit maximization at discounted prices. Formally, a programme $\left(\overline{\mathbf{x}},\overline{\mathbf{y}},\overline{\mathbf{c}}\right)$ in F(y0) is said to satisfy the condition of intertemporal profit maximization if there exists a (non-zero) sequence $\overline{\mathbf{p}}={\left(\overline{p}\right)}_{t\in N}$ of price vectors (in R^m) such that for all t ∈ N one has:

$$ {\overline{p}}_{t+1}{\overline{y}}_{t+1}-{\overline{p}}_t{\overline{x}}_t\ge {\overline{p}}_{t+1}y-{\overline{p}}_tx\ \ \ \ \ \mathrm{for}\ \ \ \left(x,y\right)\in J $$

(5)

Two well-known results (due to Malinvaud 1953) clarify the relationship between efficiency and intertemporal profit maximization.

• R.1. If$\left(\overline{\mathbf{x}},\overline{\mathbf{y}},\overline{\mathbf{c}}\right)$ in F(y₀) satisfies the condition of intertemporal profit maximization at prices $\overline{\mathbf{p}}=\left({\overline{p}}_t\right)$ with ${\overline{p}}_t\gg 0$ and if

$$ \underset{t\to \infty }{\lim }{\overline{p}}_t{\overline{x}}_t=0 $$

(6)

then $\left(\overline{\mathbf{x}},\overline{\mathbf{y}},\overline{\mathbf{c}}\right)$ is efficient. It should be emphasized that the ‘transversality’ condition (6) suggests that a profit-maximizing programme (satisfying (5)) may fail to be efficient due to an overaccumulation of capital inputs. This point has been further explored in Majumdar (1974), Mitra (1976) and Majumdar and Mitra (1976).

Before stating the next result we introduce the notion of non-tightness. A pair (x, y) ∈ J, is non-tight if there exists (u, v) ≫ 0 such that (x + u, y + v) ∈ J A programme $\left(\overline{\mathbf{x}},\overline{\mathbf{y}},\overline{\mathbf{c}}\right)$ is non-tight if (x_t, y_t+1) is non-tight for all t. The non-tightness condition requires that an increase of all inputs of producible good leads to an increase in all outputs; roughly speaking, the marginal productivities are all strictly positive.

• R.2. If$\left(\overline{\mathbf{x}},\overline{\mathbf{y}},\overline{\mathbf{c}}\right)\in F\left({y}_0\right)$ is efficient and non-tight, there exists a non-zero sequence $\overline{\mathbf{p}}={\left({\overline{p}}_t\right)}_{t\in N}$ with ${\overline{p}}_t\ge 0$ for all t such that $\left(\overline{\mathbf{x}},\overline{\mathbf{y}},\overline{\mathbf{c}}\right)$ satisfies the condition of intertemporal profit maximization at $\overline{\mathbf{p}}=\left({\overline{p}}_t\right)$.

The results R1–R2 can be viewed as an extension of pricing theory characterizing productive efficiency in a static model developed by T.C. Koopmans, who also noted that the condition (6) casts doubts on the feasibility of designing an informationally decentralized resource allocation mechanism that will guarantee efficient outcomes since a verification of (6) cannot be made on the basis of a finite number of observations of ${\overline{p}}_t$ and ${\overline{x}}_t$ (Koopmans 1958, pp. 111–27). A formal treatment of this problem has not been available until very recently (see Hurwicz and Majumdar 1984).

A difficulty with the notion of efficiency is that there is an embarrassingly rich class of efficient programme, with widely diverging consumption assignments to different periods. A more precise study of the optimal ‘trade-offs’ in consumption between two periods can be made if one introduces a one period utility or felicity function. Maximization of a discounted sum of one period utilities generated by consumptions c = (c_t) has been a well-studied evaluation criterion. However, I shall focus on programmes that are optimal according to Weizsacker’s ‘overtaking’ criterion which avoids discounting. Suppose that consumptions in any period generate utility according to a function $u:{R}_{+}^m\to R$ which satisfies the following properties:

• (U.1) u is continuous on ${R}_{+}^m$

• (U.2) u is strictly increasing on ${R}_{++}^m$

• (U.3) u is strictly concave on ${R}_{+}^m$

The continuity property (U.1) means that small changes in consumption levels lead to small changes in utility-levels. The condition (U.2) means that all commodities are desirable; finally, the condition (U.3) formalizes the idea of diminishing marginal utility. It should be stressed that (U.2) and (U.3) do restrict the scope of the model. A programme (x*,y*,c*) in F(y₀) is optimal if

$$ \underset{T\to \infty }{\lim}\sup \sum_{t=0}^T\left[u\left({c}_t\right)-u\left({c}_t^{\ast}\right)\right]\le 0 $$

(7)

for all programmes (x, y, c) in F(y₀). Convexity of F(y₀) and (U.3) imply that there can be at most one optimal programme. The question of existence has been the subject of extended discussion, and, indeed, a well-known method of proof also establishes an important long-run characteristic (‘turnpike’ property) of a broad class of programmes. A programme $\left(\overline{\mathbf{x}},\overline{\mathbf{y}},\overline{\mathbf{c}}\right)$ in F(y₀) is competitive if there exists a sequence $\overline{p}={\left({\overline{p}}_t\right)}_{t\in N}$ such that:

$$ u\left({\overline{c}}_t\right)-{\overline{p}}_t{\overline{c}}_t\ge u(c)-{\overline{p}}_tc\ \ \ \ \ \mathrm{for}\ \ \ \mathrm{all}\ \ \ c\in {R}_{+}^m; $$

(8)

$$ {\overline{p}}_{t+1}{\overline{y}}_{t+1}-{\overline{p}}_t{\overline{x}}_t\ge {p}_{t+1}y-{p}_tx\ \ \ \ \ \mathrm{for}\ \ \ \mathrm{all}\ \ \ \left(x,y\right)\in J $$

(9)

A programme (x, y, c) is stationary if

$$ {x}_i={x}_0,\ \ \ \ \ {y}_i={y}_0,\ \ \ \ \ {c}_i={c}_0 $$

for all t ∈ N. Define

$$ D\equiv \left\{\left(x,y\right)\in J:y-x\ge 0\right\} $$

and

$$ C\equiv \left\{c\in {R}_{+}^m:c=y-x,\left(x,y\right)\in D\right\}. $$

C and D are non-empty, compact, convex sets. Define u_* = max {u (c):c ∈ C} There is a unique triplet (x^*, y^*, c^*) such that (x^*, y^*) ∈ D,c^* ∈ C and u(c^*) = u_*.

For simplicity of exposition I assume that C^* ≫ 0. The following price support property of (x^*, y^*, c^*) is useful:

• R.3. There is p^* ≫ 0 such that

$$ u\left({c}^{\ast}\right)-{p}^{\ast }{c}^{\ast}\ge u(c)-{p}^{\ast }c\ \ \ \ \ \mathrm{for}\ \ \ \mathrm{all}\ \ \ c\in {R}_{+}^m $$

(10)

$$ {p}^{\ast}\left({y}^{\ast }-{x}^{\ast}\right)\ge {p}^{\ast}\left(y-x\right)\ \ \ \ \ \mathrm{for}\ \ \ \mathrm{all}\ \ \ \left(x,y\right)\in J $$

(11)

Using (R.3), one can show that the stationary programme ${x}_t={x}^{\ast },{y}_t={y}_t^{\ast },{c}_t={c}^{\ast }$ for t ∈ N in F(y^*) is competitive: the price system p = (p_t) is the stationary sequence p_t = p^* (satisfying (8) and (9) for t ∈ N). Furthermore, this stationary programme (known as the golden rule programme) is optimal in F(y^*). Not all competitive programmes from any initial y₀ ≫ 0 are optimal. The link between optimality (7) and competitive conditions ((8) and (9)) is made precise by the following (for a proof, see Brock and Majumdar 1985):

• R.4. Let c^* ≫ 0 and y₀ ≫ 0 A programme $\left(\overline{\mathbf{x}},\overline{\mathbf{y}},\overline{\mathbf{c}}\right)$ in F(y₀) is optimal if and only if it is competitive at prices $\overline{\mathbf{p}}=\left({\overline{p}}_t\right)$ and

$$ {V}_t=\left(\overline{p}-{p}^{\ast}\right)\left({\overline{y}}_t-{y}^{\ast}\right)\le 0\ \ \ \mathrm{for}\ \ \ t\in N. $$

(12)

One can show that any competitive programme $\left(\overline{\mathbf{x}},\overline{\mathbf{y}},\overline{\mathbf{c}}\right)$ satisfying (12) has a ‘turn-pike’ property:

$$ \underset{t\to \infty }{\lim }{\overline{x}}_t={x}^{\ast },\ \ \ \ \ \underset{t\to \infty }{\lim }{\overline{y}}_t={y}^{\ast },\ \ \ \ \ \underset{t\to \infty }{\lim }{\overline{c}}_t={c}^{\ast }. $$

Furthermore, from (10),

$$ u\left({\overline{c}}_t\right)-u(0)\ge {\overline{p}}_t{\overline{c}}_t\ge {\overline{p}}_t\left({c}^{\ast }/2\right) $$

for all sufficiently large t. Since $c=\left({\overline{c}}_t\right)$ is a bounded sequence, ${\overline{p}}_t$ is also bounded. Earlier characterization of (‘overtaking’) optimality of competitive programmes was cast in terms of such a boundedness property of $\overline{\mathbf{p}}=\left({\overline{p}}_t\right)$ or $\left({\overline{p}}_t{\overline{x}}_t\right)$. Once again, whether such sequences are bounded cannot be determined by agents in period t if they are allowed to observe only a finite number of prices and quantities. The verification of (12) by agents in period t requires a knowledge of ${\overline{p}}_t,{\overline{y}}_t$ and the vectors p^*, y^* (which can be computed if J is known). Finally, we note that in this model:

• R.5. There exists a unique optimal programme from y₀ ≫ 0.

See Also

• General Equilibrium

• Intertemporal Equilibrium and Efficiency

• Turnpike Theory

• Von Neumann Ray