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Eudey, Econ 302, March 2019 Chapter 7, Solow’s Exogenous Growth Model Reading Things you can skip from this chapter: The section in the textbook on the Malthusian model. The detail goes way beyond what I want you to know about that old model. You should instead read up on the Malthusian Trap in the supplemental reading on the https://ourworldindata.org/economic-growth site. The Solow Exogenous Growth model In Chapters 4 and 5 we used the logic of utility maximization and of profit maximization to find the equilibrium relationships between variables. In Chapters 6-8 we instead use logical relationships and accounting identities to carefully define characteristics we must see in equilibrium. Predictions in Chapters 6-8 are less precise because the exact details of utility and profit functions aren’t specified, but this accounting foundation approach gives some logical insights that sometimes can get blurred in the math of utility and profit maximization. Of course one can layer maximization problems on top of what we will do in Chapters 6-8, but we won’t do that in this course.
Equations in the Solow model Resource constraint: Y = C + I Where I is Investment spending into new capital goods K Savings rate: Y – C = sY Production function is Cobb-Douglas and CRS: Y = zf(K,N) Evolution of capital: K’ = (1-d)K + I Where a prime indicates a future variable and d is the rate of depreciation of capital goods and measures the reduction in usefulness of the capital good either through time, use, or obsolescence. Evolution of N: N’ = (1+n)N Where n is the population growth rate. Substitution gives us a steady-state flow relationship (**): Y-C = I = sY K’ – (1-d)K = I = sY = szf(K,N) K’/N – (1-d)k = szf(k) where k = K/N (as in the previous set of notes) K’/N’ * (N’/N) – (1-d)k = szf(k) ** k’(1+n) – (1-d)k = szf(k)
Solow’s work coincided with growth and expansion of first the General Agreement on Tariffs and Trade, which later in the 1990s because the World Trade Organization, and to the growth of the European Union and Common Market, and to NAFTA and other trade agreements. The data strongly show a correlation between these agreements and trends and economic growth, but of course there have been income distributional effects and although competition is good for economic growth it is also hard work; GDP and welfare are only correlated up to a point. Speed limit on contributions from the capital stock Forecasters often talk about the “speed limit” of the economy—there are limits to how quickly it can grow without technological progress. That way of thinking about the economy comes from the Solow model and is a very “macro” way of thinking; individual firms have no speed limit because they can eat up resources form other firms (hire away workers, get a better deal on a loan or contract than another company can, etc). The limits to the contribution from labor come from population growth and immigration. The limits to contribution from capital, and from policies designed to increase capital investments, can be seen graphically in the Solow model but here is the math: Y = z K . N . Totally differentiating (you don’t need to know how to do this) we get y ˙= z ˙ +.3 K ˙ +.7 N ˙ Which means a tax cut for example that gets K to rise by 1% causes GDP to rise by 0.3% Using that and assuming: If K increases by $1m every period forever more because of a tax incentive Then GDP increases by $0.3m the first year; assume a whopping 90% of that is saved and put into new capital goods—that could be considered an upper bound and a definite “speed limit”
Then we have ∆K1 = $1m ∆K2 = (1-d)$1m + .9.3$1m = [(1-.3) +.27]$1m = .97$1m ∆K3 = [(1-.3)+.27].97$1m = .97.97$1m And so on so although K increases each period with the increased spending on capital goods, it increases by less and less each additional period because as the capital stock grows an increasingly large share of capital spending goes to replacing depreciated capital goods (when K is bigger there is more to replace and it gets harder to maintain that higher capital stock). These additions follow a Taylor series expansion, with the total eventual change in the capital stock infinite years from now being equal to ∆ K = $ 1 m ∙ 1 1 −. = $ 33 m And the change in output being about 1/3 as big as that. Thus if the savings rate is 90% we could get output increasing about 10 times as much as the impact of a tax cut on capital spending. Of course, no one expects savings rates to be anything like 90% and the smaller the savings rate the smaller the impact of the capital tax cut on the economy. And of course the “0.3” contribution of the capital stock applies to the historical capital stock—you might expect diminishing returns to kick in as the K stock increases, so that contribution to output might fall. And so on. The point is that there is a predictable mathematical upper limit and so any reasonable discussion about the effect of capital gains tax cuts on the economy is constrained by math and data. Solow thought we should focus attention on productivity, but of course that doesn’t mean that capital investments and savings policies don’t matter. The “golden rule” level of savings is the one that the Social Planner would find and that the market should also be driven to so long as there are no imperfections in capital markets. That “golden rule” level of savings maximizes the steady-state level of per-capita consumption.
