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Profit maximization is a fundamental goal for firms in the realm of mathematical economics. Essentially, it involves finding the best combination of prices and output levels that yield the highest possible profit, which is the difference between total revenue and total cost. In this exercise, the firm sells two products, and the profit function given describes how revenue and costs interact. The formula for profit (") is:

• Total Revenue = Sales Price  Quantity of Product 1 + Sales Price  Quantity of Product 2
• Total Cost = Cost of Producing Quantity 1 plus twice the Cost of Producing Quantity 2, with a fixed cost of 10

To maximize this profit function, we first need to express it in terms of the prices of the products (1 and 2) by substituting the demand functions. This substitution turns the problem into a calculus exercise where first-order and second-order conditions help us find the optimal set points.

Demand functions reflect how much of each product consumers are willing to buy at various price levels. In this exercise, we have two demand functions for two products:

• For product 1: \(Q_1 = 40 - 2P_1 - P_2\)
• For product 2: \(Q_2 = 35 - P_1 - P_2\)

These equations imply a relationship between the price of each product and the quantity demanded. When the price of one product goes up, the demand for it tends to go down, which is a typical downward-sloping demand relationship. The demand functions are critical in setting up the profit maximization problem because they determine how quantities sold will change as prices change, impacting both total revenue and total profit. By substituting these demand functions into the revenue equations, we can express the profit function solely in terms of prices rather than outputs. This makes it easier to apply calculus methods needed to find the maxima or minima.

A cost function represents the total cost of production associated with different levels of output. For the firm in this exercise, the cost function is:\[C = Q_1^2 + 2Q_2^2 + 10\]This equation reveals that costs increase with the quantity produced of each product, but the cost of producing the second product increases at double the rate compared to the first product. The constant term "+10" represents fixed costs that don't change regardless of production levels. Understanding this cost function is crucial for profit maximization because it directly affects the profit equation. The aim is to figure out the levels of output (\(Q_1\) and \(Q_2\)) that balance the revenue obtained from selling these products against the costs incurred in producing them. By determining these levels, we achieve the highest possible profit.

First-order conditions are the set of criteria used in calculus to find the critical points where a function may reach a high or low value. In the context of profit maximization, they involve taking the derivative of the profit function with respect to each price, \(P_1\) and \(P_2\), and setting them to zero:

• \(\frac{\partial \pi}{\partial P_1} = 0\)
• \(\frac{\partial \pi}{\partial P_2} = 0\)

At these points, any small change in the variables \(P_1\) and \(P_2\) has no immediate effect on profit, revealing they could either be maximum or minimum profit points. Solving these equations gives the values of \(P_1\) and \(P_2\) where profit is extremized. Once found, these points must be tested using second-order conditions to determine if they represent a maximum, ensuring that the chosen price and output levels do indeed maximize profits rather than just balance between profits and losses.

Second-order conditions are used to verify whether the critical points found using the first-order conditions are maxima or minima. In this context, second-order conditions involve calculating the second derivatives of the profit function and using the Hessian determinant:

• Second derivatives \(\frac{\partial^2 \pi}{\partial P_1^2}\) and \(\frac{\partial^2 \pi}{\partial P_2^2}\), along with \(\frac{\partial^2 \pi}{\partial P_1 \partial P_2}\)
• Hessian determinant, \(H = \left(\frac{\partial^2 \pi}{\partial P_1^2}\right)\left(\frac{\partial^2 \pi}{\partial P_2^2}\right) - \left(\frac{\partial^2 \pi}{\partial P_1 \partial P_2}\right)^2 > 0\)

If \(H\) is positive, it indicates the presence of a local maximum. These conditions ensure the solutions derived provide a reliable economic interpretation of being the most efficient prices and corresponding sales volumes maximizing profits, offering a form of certainty about the outcomes derived from the critical points.