A Note on the Euler Equation of the Growth Model

The neoclassical Euler equation provides the necessary conditions for households to maximize lifetime utility by allocating income between consumption and investment, and is the core equation for solving the steady-state of the neoclassical growth model. The existing textbooks (Barro and Sala-i-Martin, 2004, ch6.3; Acemoglu, 2009, ch13.2, ch15.6; Aghion and Howitt, 2009, ch3.2.2) ignore the premise of this equation and directly apply it to solve the steady state of other growth models, which not only leads to incorrect results but also limits the ability of growth models to analyze the steady-state technological progress direction. This note first points out and rigorously verifies the errors in existing textbooks; Then, by replacing the capital accumulation function with exogenous growth rate with the generalized capital accumulation function considering adjustment costs of investment in the Acemoglu (2009, ch15.6) model, the note put forward the generalized Euler equation and steady-state equilibrium including capital-augmenting technological progress, which reveals the necessary conditions for the neoclassical Euler equation and Uzawa’s (1961) steady-state theorem; Finally, it is pointed out that the possible reasons for the misuse of the neoclassical Euler equation in existing textbooks maybe confuse the rental price of capital and the interest rate of investment.