The closed-form solutions for finance-extended Lucas–Uzawa model

Abstract

In this paper, I establish the closed-form solutions for all the variables in the finance-extended Lucas–Uzawa model under no parameter restrictions by utilizing the partial Hamiltonian approach and second by direct utilization of the standard methods. I provide an overview of the model and properties of the balanced growth path (BGP) as discussed in Bucci and Marsiglio (Scot J Polit Econ 2018). I have provided sufficient conditions as well. First, the model is solved by utilizing the partial Hamiltonian approach and I obtain two first integrals. I utilize both first integrals to find the closed-form solutions for all variables of finance-extended Lucas–Uzawa model. I also establish the growth rates of all the variables in the model to fully understand the dynamics of the model. The properties of BGP as described by Bucci and Marsiglio (2018) are analyzed and some additional properties are provided as well. Next, I establish a closed-form solution for all the variables of the model directly by utilizing the standard methods and these can be, alternatively, derived by utilizing only one first integral. This closed-form solution also satisfies all properties of the BGP. For the special case when inverse of intertemporal elasticity of substitution is equal to the share of physical capital, both approaches provide same closed-form solution for all the variables of the model which is presented in Bucci and Marsiglio (2018). The relationship between financial development and imbalance effects is analyzed in detail.