A Defense of AK Growth Models

AK growth models predict that permanent changes in government policies affecting investment rates should lead to permanent changes in a country’s GDP growth. Charles Jones (1995) sees no evidence for this prediction in data for 15 OECD countries after World War II: rates of investment, especially for equipment, have risen while GDP growth rates have not. This article provides evidence supporting the AK models’ prediction. Data back to the 19th century show a strong positive relationship between investment rates and growth rates and short-lived deviations from trends. A strong positive relationship also exists between average rates of investment and growth in postwar data for a large cross-section of countries. To account for the short-run deviations in rates that Jones highlights, the model he used is extended to allow policies to affect not only investment/output ratios but also capital/output ratios and labor/leisure decisions. The views expressed herein are those of the author and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. Over the past 200 years, many countries have experienced sustained growth in gross domestic product (GDP) per capita. Accounting for this sustained growth has been a central goal of modern economic growth theory. Early models simply assumed some positive rate of technological progress which translated into positive GDP growth. Now models have been developed that generate growth endogenously. One class of such models, commonly called AK models, relies on the assumption that returns to capital do not diminish as the capital stock increases. Without diminishing returns, a country with a high stock of capital is not deterred from continued investment and, therefore, continued growth. The AK class of models has been heavily criticized. Most critics have attacked the main assumption, the absence of diminishing returns, as having little empirical support. However, such criticisms are themselves difficult to support if capital is viewed broadly to include human capital and intangible capital, both of which are difficult to measure. More serious critiques analyze the testable predictions of AK models. Jones (1995), for example, argues that a key prediction of AK models is inconsistent with the data. Unlike the earlier exogenous growth models, AK models predict that permanent changes in government policies affecting investment rates should lead to permanent changes in a country’s GDP growth. Jones tests this prediction by comparing investment as a share of GDP and the growth rate of GDP for 15 countries that belong to the Organisation for Economic Co-operation and Development (OECD). Using data for the post–World War II period, Jones (1995) argues that AK models are inconsistent with the time series evidence because during the postwar period, rates of investment, especially for equipment, have increased significantly, while GDP growth rates have not. Here I defend AK growth models against that critique: I demonstrate that the key prediction of AK theory is consistent with the data. Using historical data going back to the 19th century, I show that the patterns Jones points to— episodes in which investment rates rose while growth rates remained constant or fell—were short-lived. Yet the simple model Jones tests predicts not short-run patterns, but longrun trends. The longer time series show that periods of high investment rates roughly coincide with periods of high growth rates, just as AK models predict. This is true for OECD countries and for three Asian non-OECD countries for which historical data are available. A positive relationship is also clear in the data for a larger number of countries than Jones examines. Cross-sectional data for a range of countries at different stages of development reveal a strong positive relationship between average investment rates and average growth rates, again, just as AK models predict. To account for the short-run deviations that Jones finds in investment and growth trends, I consider a version of an AK-style model that is slightly more general than the one he tests. The version Jones tests assumes that government policies affecting investment and growth do not affect key factors like capital/output ratios or labor/leisure decisions. Since those factors are not changing, the model predicts a stark relationship between the rate of physical investment and growth: they should move in lockstep. If, instead, the model assumes that these factors are affected by changes in government policies, then the model does not necessarily predict that growth rates will change one-for-one with investment rates. I construct simple examples in which changes in policy variables directly affect capital/output ratios and the labor/leisure tradeoff. These AK-style models can predict deviations in trends of investment rates and growth rates consistent with the patterns in postwar data. Basic Theory To start, let’s look at several simple AK growth models to highlight the link between investment rates and growth rates that this class of models predicts. As we shall see, the simplest versions of AK models imply a tight positive relationship between investment as a share of output and the growth rate of output. Consider a simple AK model of growth. The model has a representative household that chooses per capita consumption c and per capita investment x in each period to maximize lifetime utility U; that is, (1) max{ct, xt} t=0β U(ct) for 0 < β < 1, where t is an index for time. The optimization problem (1) is subject to a resource constraint, a capital accumulation constraint, and inequality constraints: (2) ct + xt = Akt (3) kt+1 = (1−δ)kt + xt (4) ct ≥ 0 and xt ≥ 0 given k0, where kt is the stock of capital at time t, A is the level of technology, and δ is the rate of depreciation of the capital stock. Per capita output in this model is simply