Profit from Two Products A company manufactures and sells...

Profit from Two Products A company manufactures and sells two products, 1 and 11 , that sell for $S p_{1}$ and $S p_{2}$ per unit, respectively. Let $C(x, y)$ be the cost of producing $x$ units of the product I and $y$ units of product II. Show that if the company's profit is maximized when $x=a, y=b,$ then $$\frac{\partial C}{\partial x}(a, b)=p_{1} \quad \text { and } \quad \frac{\partial C}{\partial y}(a, b)=p_{2}.$$

Profit from Two Products A company manufactures and sells two products, 1 and 11 , that sell for $S p_{1}$ and $S p_{2}$ per unit, respectively. Let $C(x, y)$ be the cost of producing $x$ units of the product I and $y$ units of product II. Show that if the company's profit is maximized when $x=a, y=b,$ then
$$\frac{\partial C}{\partial x}(a, b)=p_{1} \quad \text { and } \quad \frac{\partial C}{\partial y}(a, b)=p_{2}.$$

Key Concepts

Profit Maximization

This concept involves choosing the production levels that yield the highest possible profit, where profit is defined as the difference between total revenue and total cost. It is a fundamental principle in economics and business decision-making, guiding firms to operate where the benefits from producing one additional unit equal the costs incurred.

Marginal Cost

Marginal cost refers to the additional cost incurred by producing one extra unit of output. In the context of optimal production decisions, equating the marginal cost to the selling price ensures that producing one more unit neither increases nor decreases total profit, which is a necessary condition for profit maximization.

First Order Optimality Conditions

These conditions are derived from taking the derivatives of an objective function—in this case, the profit function—with respect to decision variables and setting them equal to zero. They provide necessary criteria for identifying potential optimal points in a multivariable optimization problem, indicating where marginal revenue equals marginal cost for each product.

Transcript

0:00 Hello.

00:01 And today we are given a little bit more of a theory heavy question.

00:05 So we are given the following information.

00:08 We are given that there are two products who sell for these prices, as are given in this box here, that have a cost function.

00:21 We're not giving the details of the function, but the function is denoted as such.

00:26 And then we are asked to show that if the profits are maximized at x equals a and y equals b then this these two things these two equations will hold true now how do we show that well first of it might be a good idea to actually go right out a formula for profits so we know that profit is equal to revenue minus cost and revenue we know is simply and i'm not sure exactly wish to use because here they use a p1 and they never use these two again they say sp1 and sp2 but i assume that they're the same thing so we're just going to be using p1 and p2 so then x times p1 or p1 x rather because p1 is a constant p1x plus p2y that is the revenue since we know that p1 is what x sells for and p2 is what y sells for and x and y is just the amounts of sold of product one and product two.

01:46 So then from that we subtract the cost, and the cost is simply the cost function in x and y.

01:57 And then i'm not gonna, you can basically just drop the brackets there.

02:05 So then let's just write p in the point x y.

02:10 So we define a new function p in the point x y, which denotes profit.

02:16 So then to maximize profit, what do we do? well, profit is maximized when the derivative of the profit function in terms of the amount sold of product 1 is equal to 0.

02:35 So let's write that out.

02:38 That's just p1...

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