Optimal Decisions using Marginal Analysis Outline Locating a shopping mall A model of the firm Profit maximization Marginal analysis Sensitivity analysis

Locating a shopping mall in a coastal area Villages are located East to West along the coast (Ocean to the North) Problem for the developer is to locate the mall at a place which minimizes total travel miles (TTM). Number of Customers per Week (Thousands) 15 10 10 10 5 20 10 15 West East x A B C D E F G H 3.0 3.5 2.0 2.5 4.5 2.0 4.5 Distance between Towns (Miles)

Minimizing TTM by enumeration The developer selects one site at a time, computes the TTM, and selects the site with the lowest TTM. The TTM is found by multiplying the distance to the mall by the number of trips for each town (beginning with town A and ending with town H). For example, the TTM for site X (a mile west of town C) is calculated as follow: (5.5)(15) + (2.5)(10) + (1.0)(10) + (3.0)(10) + (5.5)(5) + (10.0)(20) + (12.0)(10) + (16.5)(15) = 742.5

Marginal analysis is more effective Enumeration takes lots of computation. We can find the optimal location for the mall easier using marginal analysis—that is, by assessing whether small changes at the margin will improve the objective (reduce TTM, in other words).

Illustrating the power of marginal analysis Let’s arbitrarily select a location—say, point X. We know that TTM at point X is equal to 742.5—but we don’t need to compute TTM first. Now let’s move in one direction or another (We will move East, but you could move West). Let’s move from location X to town C. The key question: what is the change in TTM as the result of the move? Notice that the move reduces travel by one mile for everyone living in town C or further east. Notice also that the move increases travel by one mile for everyone living at or to the west of point X..

Computing the change in TTM To compute the change in total travel miles (TTM) by moving from point X to C: TTM = (-1)(70) + (1)(25) = - 45 Reduction in TTM for those residing in and to the East of town C Increase in TTM for those residing at or to the west of point X. Conclusion: The move to town C unambiguously decreases TTM—so keep moving East so long as TTM is decreasing.

Rule of Thumb Make a “small” move to a nearby alternative if, and only if, the move will improve one’s objective (minimization of TTM, in this case). Keep moving, always in the direction of an improved objective, and stop when no further move will help. Check to see if moving from town C to town D will improve the objective. Check to see if moving from town E to town F will improve the objective.

Model of the firm Assumptions: A firm produces a single good or service for a single market with the objective of maximizing profit. The task for the firm is to establish price and output at levels which achieve the objective. Firms can predict the revenue and cost consequences of its price and output decisions with certainty.

A microchip manufacturer A microchip is a piece of semiconducting material that contains a large number of integrated circuits. The problem for the microchip manufacturer is to determine the quantity of chips to manufacture, as well as their price. The objective of management is to maximize profits—the difference between revenue and cost. In algebraic terms, we have:  = R – C where  is profit; R is revenue, and C is cost.

Definitions Demand: The quantities of a good or service (or factor of production) buyers are willing and able to buy at various prices, other things being equal. Quantity demanded: The quantities of a good or service (or factor of production) buyers are willing and able to buy at a specific price, other things being equal. Law of demand: Other things being equal, price and quantity demanded of a good or service (or factor of production) are inversely related.

The demand for microchips ( T h o u s a n d f D l ) The firm uses the demand curve to predict the revenue consequences of alternative pricing and output policies 2 1 5 A B 1 C 5 2 4 6 8 1 Q u a n t i y ( L o s ) 2 2 2

Algebraic representation of demand The demand curve for microchips is given by: Q = 8.5 - .05P, [2.1] Where Q is the quantity of lots demanded per week, and P denotes the price per lot (in thousands of dollars). We see, for example, that if the P = 50, then according to [2.1] Q = 6. This corresponds to point C on our demand curve.

Inverse demand and the revenue function (R) By rearranging equation [2.1], we obtain the following inverse demand equation for microchips: P = 170 – 20Q [2.2] Note that revenue from the sale of microchips (R) is given by price (P) times quantity sold (Q) or: R = P  Q Substituting [2.2] into this equation yields the revenue function (R); R = P  Q = (170 – 20Q)Q = 170Q – 20Q2 [2.3]

The revenue function (R) a l R e v n u ( h s d f D r ) 4 3 2 1 2 4 6 8 1 Q u a n t i y ( L o s ) 3 3 3

Find the revenue function: Check Station 1 The inverse demand function is given by: P = 340 - .8Q Find the revenue function: Thus the revenue function is given by: R = P  Q = (340 - .8Q)Q = 340Q - .8Q2

The cost function (C) To produce microchips, the firm must have a plant, equipment, and labor. The firm estimates that for each chip produced, the cost of labor, materials, power, and other inputs is $38. This converts to a variable cost of $38,000 per lot. In addition, there are $100,000 in cost the firm could not avoid even if it shut down—that is, fixed cost = $100,000. Thus, the cost function is given by: C = 100 + 38Q [2.4]

C = 100 + 38Q T o t a l C s ( h u n d f D r ) 4 5 4 3 5 T o t a l c s 4 3 5 T o t a l c s 3 2 5 2 1 5 1 5 2 4 6 8 1 Q u a n t i y ( L o s ) 4 4 4

The profit function () Given a revenue function (R) and a cost function (C), we can derive a profit function ():  = R – C [2.5] = (170Q – 20Q2) – (100 + 38Q) = -100 + 132Q – 20Q2

Profit from microchips a l P r f i ( h u s n d D ) 1 5 1 5 T o t a l p r f i - 5 - 1 - 1 5 1 2 3 4 5 6 7 8 5 5 5

Check Station 2 Suppose the demand function is P = 340 - .8Q And the cost function is: C = 120 + 100Q Write the profit() function:

Marginal Analysis--Again OK, we have a profit equation. Now we want to find the profit maximizing quantity (Q). One method is enumeration—that is, we substitute different values for Q into [2.5] until we find the Q that gives the highest profit (). But this is too cumbersome. Marginal analysis is better.

The marginal profit (M) function Marginal profit (M) is the change in profit resulting from a small change in a managerial decision variable, such as output (Q). The algebraic expression for marginal profit is Change in profit Marginal profit = Change in output where the term “” stands for “change in,” Q0 is the initial level of output (0 is the corresponding level of profit) and Q1 is the new level of output.

Marginal profit To compute M when Q increases from 2.5 to 2.6 lots:

Marginal profit (M) is equal to the slope of a line tangent to the profit function d D l ) Slope at point A = M = 8,000 Slope at point B = M = 0 1 9 B 1 8 A 1 7 Notice that profit () is maximized when the slope of the  function is equal to zero 1 6 3 . 3 . 1 3 . 2 3 . 3 . 4 3 . 5 Q u a n t i y ( L o s ) 6 6 6

Maximum profit is attained at the output level at which marginal profit is zero. Again our profit function is given by = -100 + 132Q – 20Q2 [2.6] Marginal profit (the slope of the profit function) can be found by taking the first derivative of the profit function with respect to output: [2.7]

Set M = 0 and solve for Q M = 132 – 40Q = 0 Solving for Q yields: We know that profit is maximized when the slope of the profit function is equal to zero. So set the first derivative of the function equal to zero to find the optimal output M = 132 – 40Q = 0 Solving for Q yields: Q = 132/40 = 3.3 lots

Check Station 4 Suppose the demand function is P = 340 - .8Q And the cost function is: C = 120 + 100Q Write the marginal profit (M) function. Set M = 0 to find the optimal output:

Marginal revenue Marginal revenue is the additional revenue that comes from a unit change in output and sales. The marginal revenue (MR) of an increase in sales from Q0 to Q1 is given by: Change in revenue Marginal revenue = Change in output

Marginal cost Marginal cost is the additional cost of producing an extra unit of output. The marginal revenue (MC) of an increase in output from Q0 to Q1 is given by: Change in cost Marginal cost = Change in output

Profit maximization revisited We know that  = R – C. It follows that: M  = MR – MC [2.9] We also know that profit is maximized when M  = 0. Another way is say this is that profit is maximized when MR – MC = 0. This leads to profit maximizing rule of thumb: The firm’s profit-maximizing level of output occurs when the additional revenue from selling an extra unit just equals the extra cost of producing it, that is, when MR = MC

Equating MR and MC (a) Total Revenue, Cost, and Profit (Thousands of Dollars) 400 MR is given by the slope of a line tangent to the revenue function MC is given by the slope of a line tangent to the cost function. Profit is maximized when the slopes of the revenue and cost functions are equal 300 T otal cost 200 Re v en ue 100 Profit – 100 2 4 6 8

Microchip example--again We know that MR = 170 – 40Q. We know also that MC = 38. To solve for the profit maximizing output set MR = MC and solve for Q: 170 – 40Q = 38 Thus: 40Q = 132, therefore Q = 3.3 lots

Check station 5 Suppose the inverse demand and cost functions are given by: P = 340 - .8Q C = 120 + 100Q Solve for the profit maximizing level of output using the MR = MC approach. Step 1 is to obtain the revenue function (R): R = P · Q = (340 - .8Q)Q = 340Q - .8Q2 Now find MR by taking the first derivative of R with respect to Q: MR = dR/dQ = 340 - 1.6Q Now find MC by taking the first derivative of C with respect to Q: MC = dC/dQ = 100 Now set MR = MC and solve for Q: 340 – 1.6Q = 100 1.6Q = 240. Thus Q = 240/1.6 = 150

Sensitivity analysis In light of changes in the economic facts of a given problem, how should the decision maker alter his or her course of action? Marginal analysis is a big help. For any change in economic conditions, we can trace the impact (if any) on the firm’s marginal revenue or marginal cost. Once we have identified this impact, we can appeal to the MR = MC rule to determine the new, optimal decision.

Changing economic facts Using sensitivity analysis, we can determine the change in the optimal output resulting from the following: A change in overhead (fixed) costs; A change in materials (variable) cost; and A change in demand

Initial optimum output of microchips Marginal Revenue and Cost (Thousands of Dollars) MR = 170 - 40Q 150 MR = MC when Q = 3.3 lots 100 50 MC = 38 3.3 Quantity (Lots)

Overhead increases by $12,000 Marginal Revenue and Cost (Thousands of Dollars) MR = 170 - 40Q 150 MR = MC when Q = 3.3 lots 100 50 MC = 38 3.3 Quantity (Lots) Notice that the optimal output does not change, since MC is unaffected by a change in overhead cost. Profit, however, decreases by $12,000—whatever the level of output

Silicon prices rise from $38,000 to $46,000 per lot (b) Marginal Revenue and Cost (Thousands of Dollars) MR = 170 - 40Q 150 Note this will shift the MC function up the vertical axis 100 MC = 46 50 MC = 38 3.1 3.3 Setting MR = MC we obtain: 170 – 40Q = 46 Q = 124/40 = 3.1 lots Quantity (Lots)

Increased demand for microchips Marginal Revenue and Cost (Thousands of Dollars) MR = 190 - 40Q Note the increase in demand is manifest in a shift to the right of the MR function 150 100 50 MC = 38 3.3 3.8 Quantity (Lots) Setting MR = MC, we obtain: 190 – 40Q = 38 Q = 152/40 = 3.8 lots