Growth with perfect capital movements in CES: US Debt Dynamics and model estimation

We derive the central differential equation of the neoclassical growth model for the case of a CES (constant elasticity of substitution) production function with perfect capital movement in terms of the debt/GDP ratio and estimate it in several ways for the United States and in a later step the whole model. Debt data are derived from the accumulation of differences between investment and we show that then valuation effects play a minor role. The result is that the US debt/GDP ratio follows the pattern of a stable differential equation, which will lead to a long-run debtor position. The debt/GDP ratio will approach a value between 50% and 60% (depending on the specification used) unless a structural break increases the world interest rates or, similarly, US spreads reduce the US demand for foreign debt. A value of 50% will be achieved around 2040. We also find short-run deviations from this long-run path, which are characterized by non-sustainable explosive debt growth. These phases are characterized by high interest rates and followed by devaluations of the dollar. Our simple method allows detecting such phases early on. The estimation of the whole model yields an elasticity of substitution for capital and labour of .155 with autocorrelation correction (and 1/3 without), a growth rate of labour-augmenting technical change of 1.65% (1.5%) and a corresponding initial level of labour productivity as of 1959 of about 350 (320).