What are the applications of partial derivatives in economics field?

Answer 1

By no means do I know every single application of partial derivatives in economics, but the most common one used as an example in Calc III classes is the study of what price to sell goods for.

If you had a revenue function that would give the total revenue you expect to receive from selling two goods at set prices x and y, you can find out what will happen to the total revenue if you changed the price of one good. Suppose you have a revenue function:

R(x,y)=5x2-4y2+6x3 where x and y are the prices of two goods

The partial derivatives would look like so:

Rx(x,y)=10x+18x2

Ry(x,y)=-8y

These partial derivatives can be interpreted to determine the impact a price change of one product will have on total revenue. Suppose you wanted to change the price of Good X. If you kept the price of Good Y constant, the partial derivative Rx tells you that you can expect total revenue to increase at a rate of 10x+18x2 for every unit of price that you increase the price of Good X. So if the initial price of Good X was $16.00, and you wanted to increase the price to $20.00, you could use the partial derivative Rx to realize that with your initial price, every unit sold of Good X would bring in $4768.00 of revenue, while at your new price, every unit sold of Good X would bring in $7400.00 of revenue. Similarly, if you wanted to study the changes of a price change in Good Y, you can do the same thing with the partial derivative Ry. If Good Y was initially $8.00, but you wish to change its price to $10.00, you can see that at the initial price, each unit of Good Y sold would lose you $64.00 in revenue, while at the new price, each unit sold would lose you $80.00 in revenue.

From this analysis, you can see that you want to sell as much of Good X as possible while limiting your sales of Good Y in order to maximize revenue.

This works because the revenue function R(x,y) is arbitrarily defined. It could even model real situations. Suppose your company makes two competing goods, and the more of Good Y, the cheaper product, that is sold causes total revenue to fall.

Extending this to further Calc III principles, you can even employ maximization and minimization to locate the optimal price point to sell each good at. That would require more explanation and is outside the scope of this question, but feel free to message me if you would like me to explain that, too.