Piketty's fundamental inequality in an AK growth model with heterogeneous agents

Inequality of income tends to be studied under static or oversimplified dynamic assumptions, while economic growth tends to be studied under homogeneity or oversimplified heterogeneity assumptions. This article finds a parsimonious but not oversimplified solution to the economic growth accounting problem with heterogeneous agents subject to unequal incomes by applying a log-linear aggregation method to a simple Cass-Koopmans-Ramsey AK growth model with heterogeneity, bridging the two perspectives and therefore filling a literature gap. It then evaluates Piketty’s fundamental inequality of capitalistic accumulation (r > g) in the context of an AK growth model with productivity heterogeneity across households. In this model, a stable level of capital or consumption inequality is achieved through the existence of social capital spillovers. It is found that the relations among r, g and inequality of wealth or income in reality depend on the values of the model’s fundamental parameters such as the time discount rate, the intertemporal elasticity of substitution, the output elasticity of social capital, and the household productivity levels. On the other hand, a relation notionally similar to the one proposed by Piketty can be derived when heterogeneous rates of return on capital are assumed, but it will necessarily involve other fundamental parameters of the economy and will only hold during specific episodes of transitory and unequal growth that positively affect the richest households in the economy. Numerical simulations illustrate the validity of the relationships between parameter values and the fundamental inequality that is obtained when aggregation theory is applied to dynamic macroeconomic models.