Question

John's Lawn Mowing Service is a small business that acts as a price-taker (i.e., \(M R=P\) ). The prevailing market price of lawn mowing is \(\$ 20\) per acre. John's costs are given by \\[ \text { total cost }=0.1 q^{2}+10 q+50 \\] where \(q=\) the number of acres John chooses to cut a day. a. How many acres should John choose to cut to maximize profit? b. Calculate John's maximum daily profit. c. Graph these results, and label John's supply curve.

Short answer

Answer: John should choose to cut 50 acres to maximize his profit, and his maximum daily profit is $0.

Step-by-step solution

Write down the cost function

The total cost function is given by \(C(q) = 0.1q^2+10q+50\).

Find the revenue function

As John's business is a price-taker, the price per acre is constant at \(P=\$20\). The revenue function is given by \(R(q)=P\cdot q = 20q\).

Find the profit function

The profit function is the difference between the revenue and cost functions: \(\pi(q)=R(q)-C(q) =20q - (0.1q^2+10q+50)\).

Differentiate the profit function

To find the maximum profit, we need to find the critical points by taking the first derivative of the profit function with respect to \(q\): \(\frac{d\pi(q)}{dq} = \frac{d}{dq}(20q - 0.1q^2-10q-50)\).

Calculate the first derivative

Using basic calculus rules, we get: \(\frac{d\pi(q)}{dq} = 20 - 0.2q-10\).

Find the critical points

To find the critical points, we need to equate the first derivative to zero and solve for \(q\): \(20 - 0.2q-10 = 0\).

Solve for \(q\)

Solving the above equation for \(q\), we get: \(0.2q = 10 \Rightarrow q = 50\).

Determine the optimal number of acres

Since \(\frac{d^2\pi(q)}{dq^2} = -0.2 < 0\), the second derivative is negative and indicates that the profit function has a maximum at \(q = 50\). Thus, John should choose to cut \(50\) acres to maximize his profit.

Calculate John's maximum daily profit

Substitute the optimal value of \(q = 50\) back into the profit function: \(\pi(50) = 20(50) - (0.1(50)^2+10(50)+50) = 1000 - 1000 = \$ 0\). So, John's maximum daily profit is \(\$ 0\).

Graph the results

Plot the cost function, revenue function, and profit function on a graph, labeling the axes and supply curve. The cost function is a quadratic function with a minimum at \(q = 0\), while the revenue function is a linear function with a slope of \(20\). The profit function is also a quadratic function with a maximum at \(q = 50\). The supply curve will be a vertical line at \(q = 50\) acres.

Key concepts

Price-Taker

A price-taker is an individual or company that is not able to influence the prices in a market. This situation occurs in markets that are highly competitive where goods are perceived as identical, often referred to as perfect competition. Price-takers accept the prevailing market price and have no power to change it.

In our exercise, John's Lawn Mowing Service is described as a price-taker in the lawn mowing market. He must accept the current market price for lawn mowing, which is $20 per acre. The concept of being a price-taker is crucial to determine his revenue, as it directly affects how he generates income from his services. For price-takers, revenue depends solely on the quantity of service or product sold since the price is fixed.

Profit Maximization

Profit maximization is the process by which a firm determines the price and output level that returns the greatest profit. The key to maximizing profit lies in comparing marginal revenue (the revenue from selling one additional unit of output) to marginal costs (the cost to produce one additional unit of output).

In the context of John's Lawn Mowing Service, to maximize his profits, John would need to find the number of acres he can cut that will give him the biggest difference between his total revenue and total costs. The solution involves setting the derivative of the profit function (which represents marginal profit) to zero and solving for the quantity that yields this maximum profit. In our example, John determines that mowing 50 acres will achieve his profit maximization goal.

Cost Function

The cost function represents the total cost of production as a function of the quantity produced. It incorporates all variable and fixed costs into a single equation. In microeconomic theory, understanding the cost function is essential for a firm to determine how its costs will change with varying levels of output.

In the exercise, John's total cost function is given by the equation \(C(q) = 0.1q^2+10q+50\), where \(q\) represents the number of acres cut. This quadratic function includes both fixed costs (50 in this case, which are costs that do not vary with output), and variable costs (0.1q^2+10q\), which do vary with the level of production. John uses his cost function to calculate the most effective way to allocate his resources to maximize profitability.

Revenue Function

The revenue function is the total income a firm receives from selling its goods or services, typically calculated as the product of price per unit and the number of units sold. This function is central in understanding a firm's income potential at varying levels of output.

In our step-by-step solution, the revenue function is defined as \(R(q)=P\cdot q = 20q\), with \(P\) representing the fixed price per acre. For John, who is a price-taker, this function is particularly simple because the price per unit remains constant regardless of the quantity. Therefore, his revenue grows linearly as the number of acres mowed increases, unlike his costs, which increase at a different rate as described by his cost function.