Notes on an Endogenous Growth Model with two Capital Stocks II: The Stochastic Case

The function V implies the following controls along the welfare-maximizing consumption
and human capital allocation paths:

Ct = (1 - αβ) yt   and   ut = (¹-^)+1-^β) := u .                (22)

Note that 0 ≤ u ≤ 1 is satisfied even in the strict sense. Furthermore, the allocation of
human capital is constant regardless of the respective endowments of human and physical
capital. The central planner simply devotes a constant share of goods production to
consumption. Surely, findings (22) must also hold in period T such that it is easy to
see that the Euler equations (17) and (19) and the transversality conditions (20) are
satisfied. This remark closes the discussion of the centralized case. In the next section
we turn to the decentralized economy.

4 The decentralized solution of the model

In the decentralized case, we assume a representative agent with rational behavior. The
agent knows that her stock of human capital equals the average level of human capital
in the economy. Furthermore, she knows that the external effects of human capital
in goods production, captured by the term h^γ_a,t , may increase her and all the other
agents’ wealth. But here, in the decentralized case, the market mechanism prevents
a coordination of agents’ actions. This can be understood as a Nash game producing
the prisoner’s dilemma. For this reason, we introduce a government that taxes and
subsidizes the respective factor compensations. In the first subsection we write down the
representative agent’s optimization problem. Then, the second subsection characterizes
the agent’s optimal behavior. Finally, the third subsection determines the government’s
optimal taxation policy.

4.1 The representative agent’s optimization problem

Although the external effect of the economy’s average human capital stock in period t
may be not exploited, the whole path of ha,t is predictable and is therefore treated as
given by the agents. The representative agent’s DOP is given by:

The variable ua stands for the average human capital allocation in the decentralized
economy, the value of which cannot be influenced by the representative agent.

We start the analysis of the decentralized economy with the definition of the value
function as the solution to the representative agent’s problem:

V (kt, ht; At, ha,t) ≡ sup Et
{cs ,us }_s^∞_=t

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