Explain the law of variable proportion with the help of a diagram.

The law of variable proportions is sort of three laws in one.

It applies to production where at least one factor of production is fixed and another is variable. Usually we assume that capital is fixed and labor is variable, but it would still apply if we assumed that capital is variable and labor is fixed, or even if we introduced other factors of production such as natural resources or human capital. (It applies to almost all industries. Start trying to include creative industries like art and scientific research where initial cost is huge but marginal cost is virtually zero, and it seems to break down. But it works for most things, from apples to zambonis.)

It says that production occurs in three phases:

In Phase 1, marginal productivity is increasing. Each new unit of the variable factor makes production more efficient.

In Phase 2, marginal productivity is decreasing, but positive. Each new unit of the variable factor makes production less efficient, but does still produce more of the good.

In Phase 3, marginal productivity is negative. Each new unit of the of the variable factor makes production so much less efficient that you actually end up producing less of the good.

I've attached a little color-coded diagram I made. The blue curve is the total production function T(x) (which I set to be 5x^3 - 3x^4 in this particular case). The red curve is the average production function (T(x)/x, so 5x^2 - 3x^3). The green curve is the marginal production function (dT/dx, so 15x^2 - 12x^4).

I've marked with purple vertical lines the divisions between the phases. Notice how as we shift from Phase 1 to Phase 2, marginal productivity hits its maximum and total production is at an inflection point. Notice also how as we shift from Phase 2 to Phase 3, marginal productivity hits zero and total production hits its maximum. Finally, notice how marginal productivity intersects average productivity at the maximum of average productivity.