The AK model with Physical and Human capital

The AK model with Physical and Human capital

Setup:

The preference of the representative household at time t =0 are given by

In particular, the aggregate production function takes the form

Y(t) = F(K(t), H(t)) ………….(2)

Where H(t) denotes the human capital, which is accumulated in same way as the physical capital.

Suppose that there is no population growth (for simplicity). So n=0 , where n is the population growth rate.

The budget constraint of the representative household is given as

Where,

h(t) denotes the effective units of labour of the representative household.

W(t) is wage rate per unit of human capital.

ih(t) is investment in human capital.

Human capital evolves according to the equation

δh is the depreciation rate of human capital.

Evolution of capital stock is given from the observation that k(t) = a(t) and depreciation rate of physical capital is denoted byδk.

In this model, the representative household maximises its utility by choosing the paths of consumption, human capital investments, and asset holdings.

Competitive factor markets imply that

R(t) = f’(k(t)) and w(t) = f(k(t)) – k(t)f’(k(t)) ………..(5)

The effective capital-labour ratio is given as

k(t) = K(t) / H(t), (which is ratio of capital stock and human capital.)

A competitive equilibrium of this economy consists of paths of per capital consumption, capital-labour ratio, wage rates and rental rate of capital.

S.t representative households maximises utility (eqn 1) subject to budget constraint (eqn 3) and human capital (eqn 4).

The current-value Hamiltonian for the representative household is given as :

So, equating the rate of return on human and physical capital, we get

As the LHS is decreasing in k(t), while the RHS is increasing,

The effective capital labour ratio must satisfy

K(t)= k* for all t.

Limitations and Conclusion

Here, we have shown that the AK model can explain long-run growth using the same basic assumptions as the neoclassical model but adding knowledge externalities among firms that accumulate physical capital. However, the AK model does not provide a convincing explanation for convergence.
In our view the underlying source of the difficulties faced by the AK model is that it does not make an explicit distinction between capital accumulation and technological progress. In effect it just lumps together the physical and human capital whose accumulation is studied by neoclassical theory with the intellectual capital that is accumulated when technological progress is made.

Reference: Daron Acemoglu, chapter 11,page- 393-94.

Vivek Sharma

Hi. I’m Vivek Sharma. I do write Blog on different issues and topics related to Economy, Exams and Political issues. Inspired to make things looks better.