Activity: Solving for the Profit-Maximizing Quantity (Q*) and Price (P*) Using Known Functions
Once the first-order condition for profit maximization (dΠ/dQ = 0) is established, the subsequent step is to determine the explicit values for the optimal quantity and price. [2] If the specific algebraic forms of the firm's cost function, C(Q), and inverse demand function, P = f(Q), are known, one can algebraically solve the first-order condition equation to find the profit-maximizing quantity, Q*. [2] After Q* has been identified, the corresponding profit-maximizing price, P*, is calculated by substituting Q* back into the inverse demand function: P* = f(Q*). [2]
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Ch.7 The firm and its customers - The Economy 2.0 Microeconomics @ CORE Econ
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Activity: Solving for the Profit-Maximizing Quantity (Q*) and Price (P*) Using Known Functions
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Figure 7.4b: Cheerios Profit Function Graph (Profit-Quantity Diagram)
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Profit Maximization at the Intersection of Marginal Revenue and Marginal Cost Curves
Algebraic Profit Maximization via Π'(Q)=0 vs. MR=MC
Profit Maximization for a Custom T-Shirt Business
A firm's profit (Π) is a function of the quantity (Q) it produces. The firm calculates the derivative of its profit function with respect to quantity, dΠ/dQ, at its current output level of 500 units and finds that the value is positive. Assuming the profit function is concave (meaning it has a single peak), what does this result imply about the firm's current production level?
Economic Rationale for the First-Order Condition
A firm's profit is maximized at the output level where the rate of change of its total revenue with respect to quantity is equal to the rate of change of its total cost with respect to quantity.
A firm's profit is depicted as a concave function of the quantity (Q) it produces, meaning the profit curve first rises to a peak and then falls. Three points are identified on this profit curve. Match each point's description with the correct mathematical statement about the first derivative of the profit function (dΠ/dQ) at that point.
Setting Up the Profit Maximization Problem
A company's profit (Π) as a function of the quantity (Q) it produces is given by the equation Π(Q) = -2Q² + 120Q - 500. To find the quantity that maximizes profit, the firm must first find the first derivative of the profit function with respect to quantity and set it equal to zero. The resulting equation, known as the first-order condition, is ____ = 0.
Comparing Profit Maximization Methods
A firm has an equation that expresses its profit (Π) solely as a function of the quantity (Q) it produces. To find the specific quantity that maximizes this profit, the firm must follow a set procedure. Arrange the following mathematical steps into the correct logical sequence.
A company's profit (Π) is described by a standard concave function of the quantity (Q) it produces, meaning the profit curve has a single peak. An analyst is tasked with finding the profit-maximizing output level. They correctly calculate the first derivative of the profit function with respect to quantity (dΠ/dQ). They then evaluate this derivative at two different output levels:
- At Q = 1,000 units, they find dΠ/dQ = +$15.
- At Q = 2,000 units, they find dΠ/dQ = -$10.
Based only on these two calculations, which of the following is the most logical conclusion about the profit-maximizing quantity, Q*?
Learn After
Profit Maximization with Cost Function C(Q)=320+2Q+0.2Q^2 and Inverse Demand P=44−0.5Q
Profit Maximization with Cost Function C(Q) = 50 + 4Q + Q^2 and Inverse Demand P = 100 - 2Q
Profit Maximization for a Bicycle Manufacturer
A company's production costs are described by the total cost function C(Q) = 10Q + 50, and the market demand for its product is represented by the inverse demand function P = 70 - Q. To maximize its profit, what quantity (Q*) should the company produce, and what price (P*) should it charge?
Calculating Optimal Price and Quantity
A firm knows its total cost function and the market's inverse demand function. To find the specific price and quantity that will maximize its profit, it must follow a specific sequence of calculations. Arrange the following steps in the correct logical order.
A firm's production costs are described by the function C(Q) = 20 + 10Q + Q², and the market demand for its product is represented by the inverse demand function P = 70 - 2Q. An analyst correctly determines that the profit-maximizing quantity is Q* = 10. However, in their final step, they calculate the profit-maximizing price to be P* = $220. Which statement best describes the error made in calculating the price?
Evaluating a Suboptimal Production Decision
A firm's production process is described by the total cost function C(Q) = 100 + 4Q². The firm faces a linear inverse demand curve of the form P = A - 2Q, where 'A' is a positive constant representing the maximum possible price. If the firm sets its price at the profit-maximizing level of P* = $80, what must be the value of the constant 'A'?
Impact of a Demand Shift on a Firm's Pricing Strategy
A firm operates with a total cost function C(Q) = 5Q² + 10Q + 50 and faces an inverse demand function P = 100 - 5Q. A consultant advises the firm that to maximize profit, they should produce a quantity of 10 units.
For each firm described by a unique pair of cost and inverse demand functions, match it with its correct profit-maximizing output quantity (Q*).