Tuesday, February 26, 2019

The Model Thinker #7 : Concavity and Convexity

Convexity are functions with an increasing slope.

Most of us are familiar with the eighth wonder of the world compound interest, without which the financial markets would not exist :

V(t) = V(0)(1+R) ^ t

Another variant of convexity is exponential decay. In these models, the slow begins at a negative value and increases towards zero. A half-life model can predict that your engineering degree would be halve its value every 4-5 years.

Concave models are the opposite being functions with a decreasing slope. One possible function is the log function :

V(t) = N log Rt

Where the slope of a concave function is positive, it models diminishing returns. When concavity is assume, there is also a preference for diversity and risk aversion. One day in Kulai JB is great, but 10 days in Kulai will bore the shit out of you.

The Solow Growth model predicts that output in a given economy :

Output = A(LK)^0.5

  • Is linear with innovation
  • Is linear with the square root of labor and capital. ( Concavity )

If so, the long-run equilibrium output :

  • Increases linearly with labor and savings rate.
  • Exponentially with innovation ( Convexity )
  • Decreases with a rise in depreciation rate ( Convexity) 

Qualitatively speaking, the Solow Growth model may shed some light on Chinese GDP growth figures. Nations can receive a huge growth by adopting the technology of other countries. This explains China's explosive growth for the past 20 years.

However, sustainable growth requires innovation.

Notwithstanding the trade-war, the question is whether the patents filed in China are large paradigm shifting inventions or simply small incremental improvements.

If you agree with this week's copy of the Economist, then you will believe that you cannot innovate without an open mind and in a few years, China will more likely go the way of Japan and the GDP will trend lower over the next 20 years.

If this is true, then China will grow older before it grows richer.

Perhaps, more controversially, factoring the savings rate into long term Solow Growth Model, we may finally understand why the Oxford PPEs who run Singapore are so reluctant to have Singaporeans draw down on their CPF.


  1. As mentioned before, can compensate for CPF withdrawals by increased foreigner floodgates (labor).

  2. Right now we are reducing L !

  3. That's coz right now they are trying to count on A (innovation, productivity) to make it work. But it's the hardest of the 3 main Solow variables to ramp up in S'pore.

    Hopefully the national strategy works. But when SHTF increasing L is the quickest option in a "damn if you do, damn if you don't" situation. :)

  4. Well, once I learnt about this model, I begin to have a perverse insight on how our government works.

    There seems to be no reasonable strategy to cut L, so that has to be an election gimmick. Otherwise, we will have to delay giving out CPF even longer.

    The trick is to expand L and get as much engineering talent from overseas as we can so that L goes up and A goes up at the same time.

    This means that the gravy train enjoyed by Computer Science majors right now will not last very long.