John's Lawn Mowing Service is a small business that acts as...

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 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.

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
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.

Key Concepts

Price Taker

A price-taker is a firm that accepts the prevailing market price without having any influence over it. In perfectly competitive markets, individual firms are too small to affect market prices, so they base their production decisions on the market price, taking it as given when determining revenue and output levels.

Marginal Cost

Marginal cost represents the additional cost incurred from producing one additional unit of output. It is calculated by taking the derivative of the total cost function with respect to quantity. This concept is critical for decision-making in production, as it helps firms determine the cost implications of increasing production and is used to set production levels where profit can be optimized.

Profit Maximization

Profit maximization is the process by which a firm determines the level of output that yields the highest possible profit, which is the difference between total revenue and total cost. For a price-taking firm, profit maximization typically occurs when the marginal revenue (which equals the market price) equals the marginal cost, ensuring that producing one more unit would not add to profit.

Supply Curve

The supply curve of a firm shows the relationship between the price of a good and the quantity of that good the firm is willing to supply. In a perfectly competitive market, a firm’s short-run supply curve is typically derived from the portion of the marginal cost curve that lies above the average variable cost, indicating the prices at which the firm will increase production.

Graphical Analysis in Economics

Graphical analysis is an essential tool in economics used to visually represent relationships between variables such as cost, revenue, and output. By plotting functions like the total cost, total revenue, and marginal cost curves, one can identify key points of equilibrium, such as where marginal cost equals marginal revenue, and accurately illustrate a firm’s supply curve and profit maximization point.

Transcript

00:01 Hey guys leads to problem 41.

00:03 In this problem we need to find the average profit, the marginal average profit per more if 50 more are produced and using the result from part a and b we need to estimate the average profit per more if 51 mores are produced and the profit function is given as 30x minus 0 .03 x squared minus 750.

00:26 We know that when we divide the profit function by x we get the average profit function p bar x and we are dividing 30x minus 0 .03x square minus 750 by x and when we divide it we get the average profit function as 30 minus 0 .03x minus 750 x because 30x by x here we are canceling out x and then 0 .03 x squared by x we are canceling out the square by x we are canceling out the square only one x will be left then minus 750 by x.

01:05 Okay, then we need to find what is the average profit per more.

01:10 If 50 more are produced in this average profit function equation, we need to substitute x with 50.

01:19 Then if the 50 mores are produced, the average profit per unit will be p bar 50 is equal to 30 minus 0 .03 multiplied by x, which we substituted with 50.

01:31 Minus 750 divided by 50.

01:34 And when we do the calculation, we get 13 .5.

01:39 Therefore, if 50 mores are produced, the average profit per unit is 13 .5.

01:45 And then next we need to find out what is the marginal average profit at a production level of 50 mores.

01:55 The marginal average profit function we can find by taking the derivative of the average profit function...

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