About a decade ago, there was a neat little computer game written by the brilliant but erratic Chris Crawford called The Global Dilemma: Guns or Butter. A solo game, your mission (once you chose to accept it) was to centrally plan production quotas for armaments and food production, move your resulting decimally represented troops around the randomly generated eye-candy map, and conquer your Central Asian-looking computer opponents and thus the world! Mwa-ha-ha! What made the game fun for me was the production functions. Instead of a technology tree to pace the rate of military advance, like Sid Meier used in Civilization, you had a production spreadsheet. In theory you could produce any item on the list, provided you had enough people to make that item (and any other part that item relied on). If you had enough people, you could make enough tanks to conquer every single neighboring province on your next turn! Unfortunately, you started the game with a very low population, usually only enough to make swords and farm tools. But as your agricultural surplus grew, your population grew, and you could put more people into making more complex and productive items, like muskets and iron plows, or rifles and combines, all the way up to tractors and tanks (and engines, and oil, and all the necessary intermediates). Production followed economies of scale, so it was possible to misallocate people in an industry too primitive or too advanced for its number of workers. Yeah, this is basically the Gosplan game. And I got real good at it. My winning strategy was to keep my military just barely strong enough to keep my predatory neighbors at bay, and then, once my population base was large enough, crush their backwards swordsmen-and-musketeer armies with my tanks! At that point, I could expand as quickly as logistics permitted. Kabul, here I come! Only very rarely would I have to engineer a famine to keep up military production. Unsurprisingly, Soviet theoreticians came up with a very similar model, but here's the thing: it was exactly backwards. By investing in the production goods sector (i.e. heavy industry; 'guns') at the expense of the consumer goods sector (food, clothing; 'butter'), they believed that eventually spillovers from the former would benefit the latter. Here's Allen on the background:
G. A. Fel'dman was an economist in Gosplan. In 1928, he published a two-part article in the Gosplan journal Planovoe Khoziaistvo ['Planned Economy' - CY] that developed a mathematical model of capital accumulation. Fel'dman's model focused on internal sources of investment -- exporting wheat to import machinery received scant consideration. Instead, Fel'dman analyzed the situation in which growth required a country to produce its own structures and equipment. The questions were: How could capital be accumulated? Was there a trade-off between rapid accumulation and standard of living? The surprising answer was that you could have your cake and eat it too: by expanding the investment goods industries, high investment and rising consumption could be achieved together. This insight became the basis of socialist economic development. Fel'dman's model elaborated Marx's division of the economy into two sectors, consumer goods and producer goods. The former included food and clothing that sustained workers, while the latter included construction and machinery that could either be invested to expand the capital stock or be consumed as housing, hospitals, bicycles, or military equipment. The split of producer goods output between consumption and investment was the main issue explored by the model.Fel'dman (or Feldman, depending on your transliteration preferences) came up with an algebraically simple yet still interesting mathematical model. Let me adapt Allen's sub- and superscripted presentation to a plainer ASCII approach. First off, Fel'dman uses very simple production functions. A sector produces an output directly proportional to its capital stock:
producer output = a * producer capital stock consumer output = b * consumer capital stockwhere the variables a and b are simple rates of return. No economies of scale or anything fancy like that. The resulting producer output gets allocated as investment to either the producer or the consumer sectors, according to the ratio e:
producer investment = e * producer output consumer investment = (1 - e) * producer outputSo if the variable e equals 0.70, 70 percent of producer output would be re-invested in the producer sector, while 30% would go to the consumer sector. Finally, we introduce time by way of depreciation. If we let the variable d represent the depreciation rate per time period, then:
the current producer capital stock = (1 - d) * the producer capital stock of last period + the current producer investment the current consumer capital stock = (1 - d) * the consumer capital stock of last period + the current consumer investmentwhere if the depreciation rate is 20%, d = 0.20, and the previous capital stock gets multiplied by 0.80, which reduces it to eight-tenths of its original value. I'm afraid I'll have to switch to algebraic shorthand entirely at this point. In algebraic shorthand, the above two equations would be: pcs_t = (1 - d) * pcs_t-1 + pi_t and ccs_t = (1 - d) * ccs_t-1 + ci_t . Fair enough? Anyway, the upshot is that not only will your producer capital stock ('guns') grow exponentially, each year's production capital increasing by a constant ratio from the previous one, like 1, 2, 4, 8, 16, and so on: pcs_t = [(1 - d) / (1 - e * a)] * pcs_t-1 but that eventually, your consumer capital stock will too ('butter'): ccs_t = (1 - d) * ccs_t-1 + [(1 - d) * (1 - e) * a / (1 - e * a)] * pcs_t-1 with that horrific-looking second term I've emphasized in boldface providing the spillover effect that makes this possible. But note the first term in the consumer capital stock equation. That's simply depreciation, and that's exponential decay, like 1, 1/2, 1/4, 1/8, 1/16 and so on. So for consumer goods (like food, clothing, et cetera) you have an increasing term and a decreasing term being added together. While in the long run, the exponentially growing term will dominate, in the short term, it's entirely possible that the declining term will predominate, leading to an extended dip in consumer output. Like food, clothing, et cetera. I suppose I could put the equations into analytic functional form instead of these recurrence relations, but it was more of a conceptual model anyway. Even for the Soviets.