10.7 Marginal Analysis in Business and Economics

This is the non-linear cost function. It is an increasing function because it costs more if you produce more and more items The cost function increases quickly at first and then slowly because producing larger quantities is often more efficient then producing smaller quantities. Fixed cost The sharp increases may occur when new factories have to be built and when resources become scarce Quantity q Cost C

This is the non-linear revenue function. In this section, however, there are problems that we will use linear revenue function It is an increasing function because the company receives more money if they sell more and more items In reality, for a large values of q, the price is dropping Quantity q Revenue R

For what production quantities does the company make a profit? When revenue function is above cost function Production between 150 and 520 items will generate a profit q $$$ R C

Estimate the maximum profit Arrow goes up represent a profit Arrow goes down represent a loss Maximum Profit can occur where R’=C’ q $$$ R C

Marginal Cost Remember that marginal refers to an instantaneous rate of change, that is, a derivative. Definition: If x is the number of units of a product produced in some time interval, then Total cost = C(x) for producing x units Marginal cost = C’(x) Marginal cost is the instantaneous rate of change of cost relative to a given production level

Marginal Revenue and Marginal Profit Definition: If x is the number of units of a product sold in some time interval, then Total revenue = R(x) for selling x units Marginal revenue = R’(x) If x is the number of units of a product produced and sold in some time interval, then Total profit = P(x) = R(x) – C(x) Marginal profit = P’(x) = R’(x) – C’(x)

Marginal Cost and Exact Cost Assume C(x) is the total cost of producing x items. Then the exact cost of producing the (x + 1)st item is C(x + 1) – C(x). The marginal cost is an approximation of the exact cost. C’(x) ≈ C(x + 1) – C(x). Similar statements are true for revenue and profit.

Review

Application The total cost of producing x electric guitars is C(x) = 1, x – 0.25x 2. 1.Find the exact cost of producing the 51 st guitar. The exact cost is C(x + 1) – C(x). C(51) – C(50) = 5, – 5375 = $ Use the marginal cost to approximate the cost of producing the 51 st guitar. The marginal cost is C’(x) = 100 – 0.5x C’(50) = $75.

Example 1: A company manufactures automatic transmissions for automobiles. The total weekly cost (in dollars) of producing x transmissions is given by C(x) = x -.75x 2 (A) Find the marginal cost function C’(x) = 600 – 1.5x (B) Find the marginal cost at a production level of 200 transmissions per week and interpret the results. C’(200) = (200) = = 300 At a production level of 200 transmissions, total costs are increasing at the rate of $300 per transmission. (C) Find the exact cost of producing the 201st transmission C(201) – C(200) = – =

Example 2: Price-demand equation: x = p or p = x where x is the demand at price p (or x is the number of headphones retailers are likely to buy at $p per set). Cost function: C(x) = x where $7000 is the fixed costs and $2 is the estimate of variable costs per headphone set (materials, labor, marketing, transportation, storage, etc.) (A)Find the domain of the function defined by the price-demand equation. p = x ≥ 0; -.001x ≥ -10; so 0 ≤ x ≤ 10,000 (B) Find the marginal cost function C’(x) and interpret. C’(x) = 2 means it costs an addition $2 to produce one more headset (C) Find the revenue function (R = xp) as a function of x, and find its domain. R = xp = x( x) = 10x -.001x 2 Domain: x( x) ≥ 0; so 0 ≤ x ≤ 10,000 (D) Find the marginal revenue at x = 2000, 5000, and Interpret the results Use G.C or find R’(x), R’(2000) = 6; revenue is increasing at $6/headphone R’(5000) = 0; revenue stays the same with an increase in production R’(7000)= -4; revenue is decreasing with an increase in production

(E) Graph the cost function and the revenue function in the same coordinate system, find the intersection points of these two graphs, and interpret the results. R(x) = 10x -.001x 2 C(x) = x G.C window: -10,10000,-10,30000 To find the intersections using GC: you can trace the cursor or press 2 nd trace, 5, enter, enter, then move the cursor to the intersection, then press enter again. You can also Solve algebraically, set R=C continue R C Break-even-points: points where revenues and costs are the same. In this problem, they are: (1000, 9000) and (7000, 21000)

continue (F) Find the profit function, and sketch its graph. R(x) = 10x -.001x 2 C(x) = x P(x) = R(x) – C(x) P(x) = -.001x 2 + 8x – 7000 Use GC to find the production level to maximize the profit: 2 nd, Trace, 4, move cursor to the left, press enter, move cursor to the right, press enter, move to the maximum point then press enter again. You should get 4000 (G) Find the marginal profit at x = 1000, 4000, and Interpret these results. P’(x) = -.002x + 8 P’(1000) = 6 profit is increasing if produce more P’(4000) = 0 profit stays the same if produce one more P’(6000) = -4 profit is decreasing if produce more Maximum point

Marginal Average Cost Definition: If x is the number of units of a product produced in some time interval, then Average cost per unit = Marginal average cost =

If x is the number of units of a product sold in some time interval, then Average revenue per unit = Marginal average revenue = If x is the number of units of a product produced and sold in some time interval, then Average profit per unit = Marginal average profit = Marginal Average Revenue Marginal Average Profit

Example 3: The cost function for the production of headphone sets : C(x) = x (A)Find C(x) and C’ (x) (B)Find C(100) and C’(100) and interpret C(100) = 72; the average cost per headphone is $72 C’(100) = -0.70; the average cost is decreasing at a rate of 70 cents per headphone (C)Use the result in part B to estimate the average cost per headphone at a production level of 101 headphone sets. 72 – 0.7 = 71.30; about $71.30 per headphone