THE TWO-SECTOR AK MODEL

THE TWO-SECTOR AK MODEL

In previous sections of AK model the production technologies were linear, which makes it difficult to apply this model to real world situation. This model generates constant constant factor shares in national income without introducing human capital accumulation.

To simplify the analysis this also assumes that there is no population growth, that is, n = 0, and that the total amount of labour in the economy, L, is supplied inelastically.  

Setup

In this two sector model, the first sector produces consumption goods with technology:

Where subscript ‘C’ denotes that these are capital and labour used in consumption sector.

The capital accumulation equation is given by 

Where I(t) denotes investment. Investment goods are produced in the second sector, which has different technology form,

I(t) = AKI (t) …………(3)

The distinct feature of the technology for the investment good sector is that it is linear in the capital stock and does not feature labour.

Market clearing

It implies that for all t, we have

 

At equilibrium capital is allocated between two sectors. Since the two sectors are producing two different goods – consumption and Investment goods – there is a relative price between the two sectors that adjusts endogenously.

Let k(t) denote the share of capital used in the investment sector, so that

Also, Let the price of the investment good is denoted by pI (t) and that of the consumption good by pC (t).

Then from eqn 1 and 3 we get,

 

Now, define a balanced growth path (BGP) as an equilibrium path in which k(t) is constant and equal to some k*. let us choose consumption good as numeraire for all t. then differentiating equation 5 implies that in the BGP,

 

Where gk is the BGP growth rate of capital.

Let rC(t) denote the interest rate that measures how many units of consumption good an individual will receive tomorrow by giving up one unit of consumption today.

Therefore the formula of the rate of return denominated in consumption goods in terms of the rate of return denominated in investment goods is

Differentiating  equation (1) and using the fact that labour is always constant gives,

   

Also, at BGP wage grows at the same rate as of consumption.

Reference: Daron Acemoglu, chapter 11,page- 395-98.