In production, increasing returns to factors operate when the ideal factor is achieved, however, when the ideal factor ratio is crossed, diminishing returns to factors start to operate. However, it should always be kept in mind that this law is applicable only in the short - run. 

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Price Elasticity of Supply is a percentage change in quantity supplied in response to percentage change in price of the commodity.

\(E_s={Percentage\ Change\ in\ Quantity\ Supplied\over Percentage\ change\ in \ Price}or{\Delta Q\over\Delta P}\times{P\over Q}\)
here,
P=Initil Price,
\(\Delta P\)=Change in price
Q=Initial quantity,
\(\Delta Q\)=Change in quantity supplied

As resources have alternative uses, the quantity supplied of a commodity depends not only on its price, but also on the prices of other commodities.Increase in the prices of substitute good makes them more profitable in comparison to the given commodity.As a result, the firm shifts its limited resources from production of other good.e.g. increase in the price of wheat will induce the farmer to use land for cultivation of wheat in place of rice.
Decrease in price of substitute good will shift the supply curve to the right and vice-Versa.
Incase of complementary goods, if price of one good increases, then supply of its complementary good also increases, conveying a direct relationship.So, rise in the price of car, will cause the supply of petrol to also rise and the supply curve shifts to the rightward ad vice-versa.

In case of increase in input prices, cost of production tends to rise.accordingly, producers will supply less of the commodity at its existing prices as there is a decrease in producer,s profit.
On the other hand, in case of fall in the prices of inputs, the cost of production tends to fal, leading to an increase in producer's profit.This induces him to increase his supply.

Producer's equilibrium refers to a situation, where a producer is producing that level of output, at which its profits are maximum. In other words, it is a situation of profit maximisation or cost minimisation (under MR and MC approach). Following schedule explain the producer's equilibrium:

Conditions of producers equilibrium Following are the two conditions of producer's equilibrium:
(i) MR=MC (Marginal Revenue = Marginal Cost)
(ii) MC must be rising at the point of equilibrium or MC curve must cut MR curve from below. In the given schedule MR = MC, both at 2 units and 10 units of output, but the second condition of rising MC is fulfilled only at 10th unit of output. So, the producer is in equilibrium when he is producing io units.

The law of variable proportions states that if we go on using more and more units of a variable factor (labour), keeping other inputs fixed, the total product increases at an increasing rate in the beginning, then increases at a diminishing rate and after a level of output, it ultimately falls. In accordance with law, the marginal product increases in the beginning, then it starts falling but remains positive and ultimately it continues to fall and becomes negative also. The following schedule and diagram illustrate the law:

Elasticity of supply may be defined as the degree of responsiveness of the quantity supplied of a commodity to change in its price.

Under geometric method, the elasticity of supply is measured at a point on the supply curve. To explain this method, let us draw three straight line supply curves.

To measure elasticity of supply at a point, we extend the supply curve so that its meets the x-axis at point B in its negative range, positive range and exactly at the point of origin. Elasticity of supply at a point (say A) is equal to the horizontal segment BC divided by the quantity supplied at point A. Thus,

\({ e }_{ s\quad }=\quad { BC }/{ OC }\)

In Fig.  (i), \({ e }_{ s }=\frac { BC }{ OC } >1\)                                              \(\left[ \because \quad BC\quad >\quad OC \right] \)

In Fig,  (ii), \({ e }_{ s }=\frac { BC }{ OC } <1\)                                                 \(\left[ \because \quad BC\quad <\quad OC \right] \)

In Fig,  (iii), \({ e }_{ s }=\frac { OC }{ OC } =1\)                                                  \(\left[ \because \quad BC\quad =\quad OC \right] \)

Thus, it can be concluded that:

a straight line supply curve which intersects the x-axis in its negative range implies e_s > 1

a straight line supply curve which intersects the x-axis in its positive range implies e_s < 1.

a straight line supply curve passing the origin implies e_s = 1irrespective of how steep or flat it is.

Under percentage method,  elasticity of supply is measured by dividing the percentage change in quantity supplied of a commodity by percentage change in its price.

                                          \({ e }_{ s }=\frac { Percentage\quad change\quad in\quad quantity\quad supplied }{ Percentage\quad change\quad in\quad price } \)

                                              \(=\frac { \triangle { q }/{ p\quad }\times \quad 100 }{ \triangle { p }/{ q\quad \times \quad 100\quad } } =\frac { \triangle { q }_{ s } }{ \triangle { p } } .\frac { p }{ { q }_{ s } } \)

 \(\therefore\)                                       \({ e }_{ s }=\frac { \triangle { q }_{ s } }{ \triangle { p } } .\frac { p }{ { q }_{ s } } \)

Example. 

                                           \({ e }_{ s }=\frac { \triangle { q }_{ s } }{ \triangle { p } } \times \frac { p }{ { q }_{ s } } =\frac { 100 }{ 10 } \times \frac { 10 }{ 100 } =1\)

Thus, \({ e }_{ s }\) is unity of 1.