Question

John's Lawn Moving 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 } \operatorname{cost}=.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 in order to maximize profit? b. Calculate John's maximum daily profit. c. Graph these results and label John's supply curve.

Short answer

#Short Answer# John's Lawn Mowing Service should cut 50 acres of lawn daily to maximize his profit. The maximum daily profit at this optimal point is $200. By plotting the profit function and the supply curve on a graph, we can visually identify the optimal acres to cut as the point where the supply curve intersects with the maximum daily profit.

Step-by-step solution

Determine the profit function

To determine the profit function, we'll need to combine the given total cost function with the revenue function. The revenue function can be found by multiplying the prevailing market price, \(20, by the number of acres, \)q$. The profit function is the revenue function minus the total cost function: Profit\((q) = 20q - (0.1q^2 + 10q + 50)\) Now, simplify the profit function: Profit\((q) = 20q - 0.1q^2 - 10q - 50\)

Find the optimal number of acres to maximize profit

To find the optimal number of acres, we'll need to find the critical points of the profit function by taking the first derivative with respect to \(q\) and setting it equal to zero: \(\frac{d(\text{Profit}(q))}{dq} = \frac{d(20q - 0.1q^2 - 10q - 50)}{dq} = 20 - 0.2q - 10\) Now, set the first derivative equal to zero and solve for \(q\): \(0 = 20 - 0.2q - 10 \Rightarrow 0.2q = 10 \Rightarrow q = 50\) So, John should choose to cut 50 acres in order to maximize profit.

Calculate the maximum daily profit

Plug the optimal number of acres found in step 2 back into the profit function to find the maximum daily profit: Maximum Daily Profit \(= \text{Profit}(50) = 20(50) - 0.1(50)^2 - 10(50) - 50\) Calculate the maximum daily profit: Maximum Daily Profit \(= 1000 - 0.1(2500) - 500 - 50 = 1000 - 250- 500 - 50 = 200\) John's maximum daily profit is $200.

Graph the results and label John's supply curve

On a graph, plot the following points: 1. The acres (\(q\)) on the x-axis and the corresponding profit on the y-axis. 2. The maximum profit point should be at \((50,200)\) from step 3. Next, draw a curve connecting the points, which represents the profit function. This curve should have a peak at the point \((50,200)\), representing the maximum daily profit. Label this curve as "Profit". To identify the supply curve, note that it represents the marginal cost of producing each additional acre. The supply curve can be determined by taking the first derivative of the total cost function and plotting the resulting function: \(\frac{d(\text{Total Cost}(q))}{dq} = \frac{d(0.1q^2 + 10q + 50)}{dq} = 0.2q + 10\) Plot the marginal cost function and label this curve as "Supply". The point where the supply curve intersects the point of maximum daily profit represents the optimal number of acres John should cut each day (i.e., 50 acres).

Key concepts

Marginal Revenue

Marginal revenue (MR) is a concept in microeconomics that represents the additional income generated by selling one more unit of a good or service. It's pivotal for understanding how businesses decide on the optimum quantity to produce.

For a price taker, like John's Lawn Mowing Service, the marginal revenue is simply the market price, which is constant. In John's case, MR is \(20, the price he gets for mowing an additional acre. Thus, regardless of how many acres John mows, he always gains an extra \)20 by mowing one more. This assumption rests on him being a small player in the market; his output decisions don't affect the market price.

Total Cost Function

A total cost function represents the overall cost that a company incurs to produce a certain quantity of goods. This cost includes both fixed costs (expenses that do not change with the level of output) and variable costs (expenses that increase with more output).

In John's scenario, the total cost of mowing lawns is given by the equation
\[ \text{Total Cost} = 0.1q^2 + 10q + 50 \]
Here, \(q\) stands for the quantity of acres mowed. The term \(0.1q^2\) could represent costs that grow at a growing rate with each additional acre mowed, while the linear term \(10q\) might be direct costs related to each acre mowed such as labor and fuel. The constant term \(50\) signifies fixed costs like equipment maintenance or a lease payment, which are incurred irrespective of the number of acres mowed.

Profit Function

The profit function is a calculation that helps businesses gauge their profitability at varying levels of output. It is the difference between total revenue and total costs.

For John, the profit function, derived from his revenue (\($20q\)) and his costs (\(0.1q^2 + 10q + 50\)), is
\[ \text{Profit}(q) = 20q - (0.1q^2 + 10q + 50) \]
Simplifying this, we get
\[ \text{Profit}(q) = 20q - 0.1q^2 - 10q - 50 \]
This function will tell us how much profit John makes for different quantities of acres mowed, which is key to determining how many acres he should mow to maximize his profit.

Marginal Cost

Marginal cost (MC) is the cost of producing one additional unit of a good. It is crucial in determining the supply curve, as a rational producer will typically continue to increase production until marginal revenue equals marginal cost.

In the context of John's Lawn Mowing Service, the marginal cost of mowing additional acres is obtained by differentiating the total cost function with respect to \(q\), yielding
\[ MC(q) = \frac{d(\text{Total Cost}(q))}{dq} = \frac{d(0.1q^2 + 10q + 50)}{dq} = 0.2q + 10 \]
Graphically, the supply curve is the upward-sloping curve resulting from plotting the marginal cost against the quantity of acres mowed. Intuitively, this makes sense as the costs typically increase with increased production.

Supply Curve

The supply curve is a fundamental concept in microeconomics outlining the relationship between the price of a good and the quantity supplied. For firms, the supply curve is upward sloping, demonstrating that at higher prices, firms are willing and able to supply more goods, mainly due to higher marginal revenue.

For John, the supply curve can be plotted by drawing the marginal cost function on a graph. The point where John's supply curve intersects with the market price (or his marginal revenue) is crucial—it determines the optimal quantity to supply, which, in John's case, is 50 acres. This point of intersection is where John will make the most profit without incurring additional costs that exceed his marginal revenue.