I. Introduction, 574. — II. Algebraic formulation, 577. — III. Dual variables — aggregate utility and gradualist maximands, 580. — IV. Dual variables — terminal capital and income distribution maximands, 584. — V. An illustrative example, 589. — VI. Some extensions, 595.

I. Introduction, 214. — II. Time series comparisons, 222. — III. Regional comparisons, 228. — IV. Cracking and recycling data, 230. — V. Conclusions, 235.

[This paper begins by formulating a finite horizon linear programming model for economic development. The formulation allows for heterogeneous capital goods and for nonnegativity constraints upon investment in each sector. It is then proved that a certain set of conditions are sufficient to ensure that an optimal solution to this T period, finite horizon plan will also coincide with an optimal solution during the first T periods of an infinite horizon plan. Among the restrictive conditions imposed to prove this sufficiency theorem are the following: gradualist consumption paths, no primary factors that cannot themselves be produced within the economy, a Leontief technology, and a characterization of the optimal finite horizon solution as one in which the terminal investment and output levels are positive. An illustrative numerical example is provided.]

one is concerned with the interplay between economies of scale and an anticipated persistent growth in demand for capacity. The generalizations discussed here are of two types: (a) the use of probabilities in place of a constant rate of growth in demand; and (b) a study of the economies and the penalties involved in accumulating backlogs of unsatisfied demand. The possibility of accumulating such backlogs raises considerable doubt with respect to Chenery's excess capacity hypothesis. Surprisingly enough, generalization (b) leads to greater difficulties in analysis than (a). The use of probabilities to describe the growth process does little-if anything-to complicate matters. A probabilistic version of Chenery's model turns out to be closely related to the classical problem of gambler's ruin, and a powerful tool can be borrowed from that area-the Laplace transform for the duration of the game. Thanks to this transform, the zero-backlog probabilistic model becomes no more difficult to study than the corresponding deterministic one. A direct implication is that a probabilistic growth course makes it necessary to incur higher expected costs, and also makes it desirable to install plant capacity of a somewhat larger size than would be optimal if demand were growing at a steady rate equal to the expected value of the probabilistic increments. Uncertainty, in this sense, has a stimulating effect upon the magnitude of individual investments.

JN a I948 issue of this REVIEW, Professor Ragnar Frisch detailed an ingenious plan which would, it was hoped, rescue international from the existing chaos of bilateralism and restore it to a truly multilateral basis.2 Few economists will disagree with his ultimate aim of restoring multilateralism, but it appears to the present writer that Frisch's particular policy proposals are thoroughly infected with the cardinal marginal utility heresy, and are consequently of questionable merit. By using the more orthodox notion of an ordinal welfare index, however, it seems quite feasible to attack the multilateral problem, and, at least in principle, solve it. Because of space limitations, it is hardly possible to do justice to Professor Frisch, and to reproduce his entire discussion. What follows is the barest skeleton of his remarks. He points out that the prevalent system of conducting international on an essentially bilateral basis primitive barter is a quite inefficient method of organizing the international division of labor. In place of this patchwork, some comprehensive multilateral arrangement must be set up. Apparently, Frisch envisages a condition of repressed inflation 3 within each trading country, i.e., a condition in which money prices are meaningless as transformation ratios.4 Consequently, each participating nation must group all import and export goods into perhaps ten categories (0,1,2, . . . 9), such that, within each category at the given domestic prices, the same marginal social valuation is put on a (or whatever the local currency unit may be) of one good as on a of any other good. Furthermore, a crown's worth of goods in category 3 is preferred to a crown's worth of goods in category 2, but is in turn inferior to a crown's worth of goods in category 4. Next, an international authority is to ascertain the categories into which its members place various quantities of import and export goods. It is then claimed that there will exist a unique allocation of trade, given two conditions: (i) the requirement that, in terms of any currency unit, the aggregate value of imports of any one nation from all other nations within the system must, within rather close limits, equal the aggregate value of exports of that nation to all other nations within the system; and (2) the condition that the global from trade be maximized. The crucial objection to the Frisch plan arises from the fact that the notion of global gains is not unambiguous.

In recent years, a number of proposals have been advanced for the limitation of carbon emissions. Some have argued that such limits would be costless, but our analysis suggests that there is no free lunch. (See our forthcoming paper.) We have attempted to estimate the costs but not the global benefits of slowing down climate change through carbon limitations. All computations were performed in parallel for five geopolitical regions. Except for oil trade, these regions were treated independently-as though there were no opportunity for international trade in carbon rights. For stimulating ideas on the politics and economics of negotiating an agreement on greenhouse gas emission permits, see M. Grubb (1989). Whatever rule is adopted for the allocation of carbon emission rights, there are likely to be significant interregional differences in the value of these rights. International trade will be needed if economic efficiency is to be achieved. In the absence of such trade, there are likely to be significant distortions in the comparative advantage of individual locations for the production of tradeable basic materials such as primary metals. These distortions could lead to counterproductive regulations and new forms of nontariff barriers to trade. This paper is intended to quantify the potential for international trade in carbon emission rights.

This paper considers optimization problems in which some or all variables must take on integral values. An ability to solve such problems would be valuable in itself and would also allow handling certain kinds of heretofore intractable 'economies of scale'. An automatic algorithm for solving such problems is not given. A general approach susceptible of individual variations, depending upon the problem and the judgment of the user is presented. Two moderate-size examples are presented to illustrate the method. (Author)