The Solow Model II: the Inada Condition

Attention Conservation Notice: Another step in the exegesis of the Solow model with technology, Hicks-neutral version. Growth submits to diminishing returns, but production is proportional.

The Story So Far

• The model is of the rate of production of a single good (service) produced by using the services, called capital and labour, of the stocks of capital goods and workers available at the time. (This single good can be called "everything", I guess.)
• The rate of production depends only on the available services of capital and labour: nothing else is scarce. The stocks of capital and workers are assumed (initially) to be fully used, to provide the maximum flow of services that can be got from combining them in a production process.
• Over time, people think of better ways to combine capital and labour (and perhaps make use of non-scarce things more effectively, too), so the same flows of each produce more. This is added to the model in the form of a term $B(t)$, the production method at time $t$. 
• For convenience $B$ is associated with $L$, and has its name changes to $A$ as a result.
• In maths: the production function:   $Y(t) = f(K(t), A(t)L(t))$  , where:

• Explicit assumption: changes in $L$ and $A$ are unaffected by $Y$: they are exogenous.
• A few implicit assumptions:  $Y$, $K$, $A$, and $L$, are all greater than or equal to $0$ for all times $t$, and at time $t=0$ they are all strictly greater than zero - in the beginning, there are  some things, some people, and some ways of making things .

...and for capital accumulation,

• $K$ is decreased by depreciation of the capital stock at a constant rate $\delta$
• $K$ is increased by using a fraction $s$ of the output flow $Y$ to create new capital:

$$\frac{dK}{dt} = sY - \delta K = sf(K, AL) - \delta K .$$ 

There is an equilibrium point ($dK/dt=0$, i.e., $K$ does not change) when $sY = \delta K$.

So: a two-equation model of a single-good economy using two factors of production.

More on the Production Function

What forms can $f(K, AL)$ take?  Quadratic? exponential?

In economic terms we require  a few things.  When either $K$ or $L$ is $0$, $Y$ has to be $0$ too. When $K$ and $L$ are both positive, $Y = f(K, AL)$ has to be positive too, and  bigger amounts of either or both of them should increase $Y$. So far, so obvious.

Diminishing Returns

For economic plausibility, we need diminishing returns to each factor of production. Holding $AL$ constant, when there is an  increase from a small amount of $K$ to a slightly larger amount, this should  have a large positive effect on how much is produced. When there's a lot of $K$, increasing it by the same increment should have a much smaller effect on production.

DeLong New California Economy example: If 100 baristas have one espresso machine for them all, and then you give them another one (or replace the existing one with one that works faster), we expect a large increase in the number of lattes produced per hour.  Similarly with yoga instructors and yoga mats.

At the other extreme, if 100 baristas have 1000 espresso machines and 100,000 cup sets and all the other facilities in excess, adding another espresso machine will have almost no effect on production.

Putting these into mathematical terms:$$\lim_{K \to 0} \frac{\partial f} {\partial K} = \infty  ,  \lim_{K \to \infty} \frac{\partial f}{\partial K} = 0 , $$ and we require $\partial f /\partial K$ to be differentiable throughout, and the second partial derivative of each factor to be negative throughout.

Similarly for $L$, and $AL$: When not much of these services are around, increasing them a bit produces a large increase in output, and when there's a lot (with no change in the capital they can use), the same increase has almost no effect.

So:$f$ is defined, continuous, and $>=0$ for $K >=0$ and $AL >=0$, $f(0,0) = 0$ , $f$ is monotonically increasing, and the partial derivatives of $f$ are concave down (have derivatives that are $< 0$).  Collectively these are called the Inada Conditions.

 These conditions add up to requiring that in terms of each factor of production, $f$ is a "concave down" function passing through $0$.

.... but Constant Returns

Also for economic plausibility, we require proportionality overall:  double the amounts of both capital and labour (by putting the economy into an EconoClone^™ duplicating machine), and you should get twice as much output. 

In maths:  $f(cK, cAL) = cf(K, AL)$ for $c >=0$.

What can satisfy all these conditions? Stay tuned for the next thrilling instalment.