Question

Find the marginal and average functions for the following total functions and graph the results. Total-cost function: (a) \(C=3 Q^{2}+7 Q+12\) Total-revenue function: (b) \(R=10 Q-Q^{2}\) Total-product function: (c) \(Q=a L+b L^{2}-c L^{3} \quad(a, b, c>0)\)

Short answer

Derivatives of total functions yield marginal functions: 6Q+7, 10-2Q, and a+2bL-3cL^2. Average functions are: 3Q+7+12/Q, 10-Q, and a+bL-cL^2.

Step-by-step solution

Understanding the Marginal Functions

The marginal function for a total function is the derivative of that function concerning its variable of interest. This means the marginal cost is the derivative of the total-cost function with respect to quantity \(Q\), the marginal revenue is the derivative of the total-revenue function with respect to \(Q\), and the marginal product is the derivative of the total-product function with respect to labor \(L\).

Find Marginal Cost Function

The total-cost function is given as \(C = 3Q^2 + 7Q + 12\). Compute its derivative with respect to \(Q\):\[M_C = \frac{d}{dQ}(3Q^2 + 7Q + 12) = 6Q + 7.\]So, the marginal cost function \(M_C\) is \(6Q + 7\).

Find Average Cost Function

The average cost is the total-cost function divided by \(Q\):\[A_C = \frac{C}{Q} = \frac{3Q^2 + 7Q + 12}{Q} = 3Q + 7 + \frac{12}{Q}.\]Thus, the average cost function \(A_C\) is \(3Q + 7 + \frac{12}{Q}\).

Find Marginal Revenue Function

The total-revenue function is \(R = 10Q - Q^2\). Compute its derivative with respect to \(Q\):\[M_R = \frac{d}{dQ}(10Q - Q^2) = 10 - 2Q.\]So, the marginal revenue function \(M_R\) is \(10 - 2Q\).

Find Average Revenue Function

The average revenue is the total-revenue function divided by \(Q\):\[A_R = \frac{R}{Q} = \frac{10Q - Q^2}{Q} = 10 - Q.\]Thus, the average revenue function \(A_R\) is \(10 - Q\).

Find Marginal Product Function

The total-product function is \(Q = aL + bL^2 - cL^3\). Compute its derivative with respect to \(L\):\[M_P = \frac{d}{dL}(aL + bL^2 - cL^3) = a + 2bL - 3cL^2.\]So, the marginal product function \(M_P\) is \(a + 2bL - 3cL^2\).

Find Average Product Function

The average product is the total-product function divided by \(L\):\[A_P = \frac{Q}{L} = \frac{aL + bL^2 - cL^3}{L} = a + bL - cL^2.\]Thus, the average product function \(A_P\) is \(a + bL - cL^2\).

Graphing the Results

Using a graphing tool or software, plot the marginal and average functions for cost, revenue, and product against their respective variables. This will provide a visual representation of how these functions behave as the quantity or labor changes.

Key concepts

Marginal Cost

Marginal cost is a key concept in economics that helps businesses understand how much it costs to produce one more unit of a good. It is derived from the total-cost function and is essentially the derivative of this function with respect to the quantity produced. In our exercise, the total-cost function is given by:\[C = 3Q^2 + 7Q + 12\]To find the marginal cost, we differentiate this function with respect to the quantity, \(Q\). The result is:\[M_C = \frac{d}{dQ}(3Q^2 + 7Q + 12) = 6Q + 7\]This means for every additional unit produced, the cost increases by \(6Q + 7\). Understanding marginal cost helps businesses decide pricing and production levels efficiently.

Marginal Revenue

Marginal revenue is the additional revenue that a firm earns from selling one more unit of a good. This concept is crucial for optimizing profit, as it enables a company to evaluate whether producing and selling an additional unit is beneficial. In the exercise, the total-revenue function is expressed as:\[R = 10Q - Q^2\]To calculate the marginal revenue, we take the derivative of the total-revenue function with respect to \(Q\):\[M_R = \frac{d}{dQ}(10Q - Q^2) = 10 - 2Q\]This tells us that the marginal revenue diminishes as quantity increases. Businesses use this information to find the profit-maximizing quantity, which is where marginal cost equals marginal revenue.

Marginal Product

The concept of marginal product examines how much additional output is produced when one more unit of input is added, keeping other inputs constant. It's vital in understanding returns to scale in production. The total-product function is given by:\[Q = aL + bL^2 - cL^3\]We compute its derivative with respect to labor, \(L\), to find the marginal product:\[M_P = \frac{d}{dL}(aL + bL^2 - cL^3) = a + 2bL - 3cL^2\]This formula indicates that the extra output gained from additional labor depends on labor levels due to the quadratic and cubic terms. It helps firms optimize their labor inputs for maximum efficiency.

Average Cost

Average cost represents the cost per unit of output produced. It's useful for determining overall cost efficiency. To find the average cost, divide the total-cost function by the quantity, \(Q\):\[A_C = \frac{C}{Q} = \frac{3Q^2 + 7Q + 12}{Q} = 3Q + 7 + \frac{12}{Q}\]This expression shows how average cost changes with different levels of production. Companies often aim to minimize average costs to improve competitive pricing and profitability. As quantity increases, the impact of fixed costs like 12 diminishes, which is why average cost changes noticeably with scale.

Average Revenue

Average revenue is the revenue earned per unit sold, akin to the price received. It's calculated by dividing the total-revenue function by the quantity:\[A_R = \frac{R}{Q} = \frac{10Q - Q^2}{Q} = 10 - Q\]This result indicates that average revenue decreases linearly as the output increases, reflecting how price adjustments might be necessary as more units are sold. Firms use average revenue alongside marginal revenue to assess their pricing strategies and determine demand elasticity.

Average Product

The concept of average product measures the output generated per unit of input, aiding in evaluating productivity. It is derived by dividing the total-product function by the amount of labor, \(L\):\[A_P = \frac{Q}{L} = \frac{aL + bL^2 - cL^3}{L} = a + bL - cL^2\]This formula helps assess how efficiently labor is being used. If the average product remains high as more labor is added, it suggests effective use of inputs. Conversely, diminishing returns in the average product would signal inefficiencies or the need for input adjustments.