# Learning Curve Theory

Economics is often said to be the study of human psychology and can be evidenced ever more with the study of Learning Curve Theory. Conceptualized from the works of a 19th century German psychologist Hermann Ebbinghaus and later quantified in 1936 at Wright-Patterson Air Force Base in the United States, this theory is to me is one isthmus between two great subjects – Psychology and Economics.

###### The Theory as seen through the research at Wright-Patterson Air Force Base:

Every time total aircraft production doubled, the required labour time decreased by 10 to 15 percent.

###### The Generalised Theory:

For each doubling of the total quantity of items produced, costs decrease by the same proportion.

###### Sample Graph:

A learning curve graph when a job done in 15 minutes is repeatedly done n number of times and the average time per unit decreases by 15% every time the number n hits a power of 2.

###### Modelling the LC Theory:

Let $f(x)$ be the function that equates to the average time for performing a certain job x times. Then:

$f(1) \; =\; A_1\; -----Eq. 1\;$

Where, $latexA_1$ is the time taken to do the job for the first time.

When the job is done the second time, the formula becomes:

$f(2) \; =\; A_2\; -----Eq. 2\;$

As per the LC Theory, for each doubling of the total quantity of items produced, costs decrease by the same proportion. Therefore:

$A_2\; \div\; A_1\;=\; R\;-----Eq. 3\;$

Where R is that proportion of decrease whenever the quantity is doubled (earlier in our graphical example R was 100% – 15% or 85%).<\p>

Re-writing $Eq. 2\;$ with $Eq. 3\;$ we get: <\p>

$f(2) \; =\; R\; \times\; A_1\; -----Eq. 4\;$

If the quantity of 2 is doubled to 4, we get: <\p>

$f(4) \; =\; R^3\; \times\; A_1\; -----Eq. 5\;$

Therefore, for $2^k\;$ times job done, we get:

$f(2^k) \; =\; R^k\; \times\; A_1\; -----Eq. 6\;$

Therefore, for an $X\;$ arbitrary number of times job done, $R\;$ would be raised to that power $k\;$ that will equal it to $X\;$. This gives us:

$2^k\; =\; X\; -----Eq. 8\;$

$k\; =\; log_2 X\;$

$k\; =\; log X\; \div\; log 2\; -----Eq. 9\;$

Using $Eq. 8\;$ and $Eq. 9\;$ in $Eq. 6\;$ we get:

$f(X) \; =\; R^\frac{log x}{log 2}\; \times\; A_1\; -----Eq. 10\;$

$Eq. 10\;$ represents the mathematical model of LC Theory. This equation can be used to project the future cost savings because of the effects of learning curve theory