Type or paste a DOI name into the text box. Please forward this error screen to 209. Attention conservation notice: Over 7800 words about optimal planning for a socialist economy modeling monetary economies solutions manual pdf its intersection with computational complexity theory.
This is about as relevant to the world around us as debating whether a devotee of the Olympian gods should approve of transgenic organisms. Red Plenty mostly as a launching point for a tangent. It’s basically a work of speculative fiction, where one of the primary pleasures is having a strange world unfold in the reader’s mind. The early chapter, where linear programming breaks in upon the Kantorovich character, is one of the most true-to-life depictions I’ve encountered of the experiences of mathematical inspiration and mathematical work. It should be clear by this point that I loved Red Plenty as a book, but I am so much in its target demographic1 that it’s not even funny. My enthusing about it further would not therefore help others, so I will, to make better use of our limited time, talk instead about the central idea, the dream of the optimal planned economy. But could it even have been tried?
Let’s think about what would have to have gone in to planning in the manner of Kantorovich. We need a quantity to maximize. In Kantorovich’s world, the objective function is linear, just a weighted sum of the output levels. Equivalently, the planners could fix the desired output, and try to minimize the resources required. In some contexts these might be physically comparable units.
We need complete and accurate knowledge of all the physical constraints on the economy, the resources available to it. We need complete and accurate knowledge of the productive capacities of the economy, the ways in which it can convert inputs to outputs. For instance, there is nothing in the formalism which keeps it from including constraints on how much the production process is allowed to pollute the environment. The shadow prices enforcing those constraints would indicate how much production could be increased if marginally more pollution were allowed. We still need this unbiased knowledge about everything, however, and aggregation is still a recipe for distortions.
More serious is the problem that people will straight-up lie to the planners about resources and technical capacities, for reasons which Spufford dramatizes nicely. There is no good mathematical way of dealing with this. Nonlinear optimization is possible, and I will come back to it, but it rarely makes things easier. Computing time must be not just too cheap to meter, but genuinely immense. It is this point which I want to elaborate on, because it is a mathematical rather than a practical difficulty. It was no accident that mathematical optimization went hand-in-hand with automated computing.
There’s little point to reasoning abstractly about optima if you can’t actually find them, and finding an optimum is a computational task. Computer science, which is not really so much a science as a branch of mathematical engineering, studies questions like this. A huge and profoundly important division of computer science, the theory of computational complexity, concerns itself with understanding what resources algorithms require to work. Those resources may take many forms: memory to store intermediate results, samples for statistical problems, communication between cooperative problem-solvers. The way computational complexity theory works is that it establishes some reasonable measure of the size of an instance of a problem, and then asks how much time is absolutely required to produce a solution. 1 of Ben-Tal and Nemirovski’s Lectures on Modern Convex Optimization .
Truly intractable optimization problems — of which there are many — are ones where the number of steps needed grow exponentially2. A good modern commercial linear programming package can handle a problem with 12 or 13 million variables in a few minutes on a desktop machine. Let’s be generous and push this down to 1 second. Or let’s hope that Moore’s Law rule-of-thumb has six or eight iterations left,and wait a decade. To handle a problem with 12 or 13 billion variables then would take about 30 billion seconds, or roughly a thousand years.
There are close to 50,000 industrial establishments, plus, of course, thousands of construction enterprises, transport undertakings, collective and state forms, wholesaling organs and retail outlets. Nove’s sources, whether it included the provision of services, which are a necessary part of any economy. Let’s say it’s just twelve million. LP solver, if someone had given it one, couldn’t Gosplan have done its work in a matter of minutes? Maybe an hour, to look at some alternative plans?
The difficulty is that there aren’t merely 12 million variables to optimize over, but rather many more. A beautiful paper at the end of last year had skewered Academician Glushkov’s hypercentralized rival scheme for an all-seeing, all-knowing computer which would rule the physical economy directly, with no need for money. The author had simply calculated how long it would take the best machine presently available to execute the needful program, if the Soviet economy were taken tobe a system of equations with fifty million variables and five million constraints. This alternative vision, the one which Spufford depicts those around Kantorovich as pushing, was to find the shadow prices needed to optimize, fix the monetary prices to track the shadow prices, and then let individuals or firms buy and sell as they wish, so long as they are within their budgets and adhere to those prices. The planners needn’t govern men, nor even administer things, but only set prices. So far as our current knowledge goes, no.