1. Simple economic model of the private, profit-maximizing firm.
2. Introduction to "Marginal Analysis," important tool for arriving at optimal decision. "Marginal" means "a little more or a little less," and this characterizes the majority of our decisions and is the basis for the next 8 chapters, so understanding the logic of marginal analysis is very important.
Examples: How many classes to take each semester? How many children to have? How many cars for a household? How many hours to work? How many stores to open? How many different products to offer? How many programs/classes/majors to offer for UM-F? How much output (Q) to produce? How many acres of corn to plant? How many cows to milk?
Example of using Marginal Analysis to locate a shopping mall, page 30, Figure 2.1: Assumptions:
1. Developer will build a mall in a town somewhere along the ocean, between towns A and H, to serve customers primarily in those 8 towns.
2. Number of customers per week can be predicted using market surveys (subject to uncertainty of forecasting) as indicated on page 30.
3. Objective/goal is to locate the mall to MINIMIZE the TTM (Total Travel Miles), assuming that minimizing TTM will MAX ________?
One approach to solving the problem would be using "enumeration," or an iterative trial-and-error approach, selecting various locations and computing TTM. For example, we could choose Point X, which is 1 mile west of Town C and calculate TTM = 742.5 miles. We could then choose other locations and calculate TTM. However, this approach does not necessarily guarantee an optimal solution, unless every possible location is considered. Marginal Analysis can identify the optimal site with much less computation and much more certainty.
Marginal Analysis (MA): Make small changes and see if a given change improves the ultimate objective. For example, we start by arbitrarily choosing Point X, 1 mile west of Town C. It is not necessary to even calculate TTM at that original location to perform MA. We make a small (marginal) change by moving the location 1 mile to the east, from Point X to Town C, and calculate the CHANGE in TTM. There are 70,000 people living to the east of Point X who are all now 1 mile CLOSER, so TTM is reduced by -70,000 for those people. There are 25,000 people living to the west who would now be 1 mile FURTHER, so TTM is increased by +25,000 for those people. Therefore, the net change in TTM is: -70,000 + 25,000 = -45,000 miles, which is an improvement, and we know that Town C is a better location for a mall than Point X. We could now move further in the same direction to Town D, which lowers TTM by -50,000 miles. Moving to Town E would lower TTM by -12,500 miles. Moving to Town F would INCREASE TTM by +22.5 miles.
Conclusion of MA: Assuming the mall will be located in a town, the optimal location is Town E.
APPROACH/LOGIC OF MA: Make a small move and see if there is an improvement in the objective. If so, keep moving in the same direction until you have exhausted the benefits of moving in the optimal direction.
Example: If Marginal Revenue from additional output is greater than the Marginal Cost (MR > MC), the firm should continue to expand output and stop when MR = MC.
SIMPLE MODEL OF THE FIRM
Simplifying Assumptions:
1. A profit-maximizing firm produces a single good in a single domestic market at a single price.
2. The firm's decisions are to determine Q (output) and P (price) to maximize profit.
3. The firm has reliable knowledge of its demand curve, so that it can predict with certainty the revenue (P x Q) and production costs of its output and price decisions.
Even though most firms produce and sell many different products, sometimes at different prices, the single-good (product), single-price, single-market assumption is fairly realistic because multi-product firms like General Mills use a product-by-product strategy where they assign product managers to specific brands. Profit-maximizing decisions are made along product lines, so our simple model could represent the production of a single good (Wheaties) within a multi-product firm (General Mills). Maximizing profits for each product will maximize total profit for the firm, as long as products are independent. However, it gets more complicated if a firm's products are substitutes or complements, or are jointly produced, or if a firm uses price discrimination, because changes in one product (price, advertising, output) may affect other products.
Examples:
The single-good, single-market, single-price firm is a basic model that is used as our starting point, before we consider more complicated situations.
CASE STUDY: MICROCHIP MANUFACTURER
We consider a firm that produces computer microchips (integrated circuits) for the domestic market. The firm's goal is to maximize profits (П = TR - TC). We look at: 1) Total Revenue (TR = P x Q) and 2) Total Cost (TC) separately:
1. REVENUE:
FORMULA: SALES = TOTAL REVENUE (TR) = (P x Q)
Revenue is based on the Law of Demand:
The microchip's demand curve is shown in Figure 2.2, p. 34. (Note: there is also an industry demand curve for microchips, not shown). We assume that the microchip industry is: a) competitive, and b) in equilibrium (stable). Then, because the firm faces a downward sloping demand curve, it can increase SLS (TR) if it lowers its price significantly, for three reasons:
a.
b.
c.
Demand curve in Figure 2.2 shows the Law of Demand graphically. Q = 1 lot of 100 chips sold per week. P is measured in units of $1000.
Point A: At a price of $130,000/lot, the firm will sell 2 lots/week.
Point B: At a price of $100,000/lot, the firm will sell 3.5 lots/week.
Point C: At a price of $50,000/lot, the firm will sell 6 lots/week.
POINTS: a) At a lower price, more units are sold, which is the Law of Demand (demand curves slope downward). b) The firm can use the demand curve to estimate TR at different levels of output (Q).
The equation for the demand curve is: P = 170 - 20Q, where the intercept is ________ and the slope of the line is _________. This is called the inverse demand equation.
We can also rearrange the inverse demand equation and solve for Q: Q = 8.5 - .05P. Here we can see that if the firm raises its price, Q will fall, reflecting the inverse relation between P and Q.
To use the demand curve to predict TR, we make two simplifying and unrealistic assumptions:
1. When Q and P change, we make the ceteris paribus assumption to determine TR, and assume all other factors remain constant, such as:
a.
b.
2. We assume the demand curve reflects a stable, linear and deterministic relationship between P and Q, so we can calculate TR with mathematical certainty.
The table and graph on page 37 (Figure 2.3) illustrates the TR at different prices and level of output (Q). When P = 0, Q = 0, TR = 0. When P = $10, Q = 8, TR = $80. As they now raise price by increments of $20, Q falls, TR starts to increase, reaches MAX at P = 90, Q = 4, TR = 360. Further price increases now lead to decreasing TR.
With a downward sloping demand curve, we know that: a) raising price first increases TR, and b) at some point raising price eventually lowers TR, resulting in an upside down U-shaped curve.
We can derive the revenue function, or the equation for TR = f (Q) as follows:
TR = P x Q
TR = (170 - 20Q) x Q
TR = 170Q - 20Q2
CHECK STATION PROBLEM (p. 38): If P = 340 - .8Q, what is TR?
COST
Total Cost (TC) = Fixed Cost (FC) + Variable Cost (VC)
Assume that FC = $100,000 per week and VC = $38,000 per lot of 100, so that the Cost Function is: TC = $100 + 38Q (see Figure 2.4, p. 38). When:
Q = 0 TC = $100,000 (FC only)
Q = 1 TC = $138,000 (FC + VC)
Q = 2 TC = $176,000................etc.
PROFIT
П = f (Q)
PROFIT (П) = TR - TC
П = (170Q - 20Q2) - (100 + 38Q)
П = -$100 + 132Q - 20Q2
See graph on page 39 (Figure 2.5), П = f (Q). At low levels of output, profits are negative, because FC are high, but then become positive at about Q=1, increase to a maximum when Q = 3 to 3.5, and then fall and become negative again when Q is approx. = 6. Logic: Think of the shape of the TR curve and the TC curves.
CHECK STATION 2 (p. 40): If P = 340 - .8Q and TC = $120 + 100Q, solve for Profit (П), as f (Q).
MARGINAL ANALYSIS (MA)
Goal: Find the level of output (Q*) that MAXIMIZES PROFITS (П). We could use enumeration, following the procedure on p. 39, calculating profit for various levels of output. IF the firm can only sell whole (discrete) lots, then we can tell that PROFIT MAX occurs when Q* = __________. However, when the decision involves many options (continuous), enumeration is not practical. Instead, we can use Marginal Analysis (MA).
MA involves looking at the CHANGE in Profit from a small CHANGE in Q. For example, when Q = 3, П = $116,000. When Q = 3.1, П= $117,000, so П has increased by $1000 for a 0.10 change in Q. We can define Marginal Profit (MP):
Marginal Profit = Change in П = Δ П = ( П1 - П0) / ( Q1 - Q0 )
Change in Q Δ Q
where Δ = change.
See Table 2.1 on page 41, for Marginal Profit. Using MA, we apply the following rule: Make small changes in Q, if П increases, keep making changes in that direction until Marginal Profit becomes 0 or negative. For example:
ΔQ = .10 MP (ΔП)
3 to 3.1 $10,000 ($1000 / .10)
3.1 to 3.2 6,000
3.2 to 3.3 2,000
3.3 to 3.4 -2,000
Moving from 3.3 to 3.4 would now DECREASE profits, so the firm should stop producing at Q* = 3.3 to maximize P.
Another approach is to use calculus, using the following optimization rule: The firm will MAX П at the output level where Marginal Profit is 0, which means that the slope of the line is 0 (see Figure 2.6, p. 43). Using calculus we can differentiate the Profit equation:
П = -$100 + 132Q - 20Q2
MP = Δ П / Δ Q = 0 + 132 - 40Q
MP = 132 - 40Q
Remember the simple derivative RULES (p. 67):
1. The derivative of a constant is 0. (-100 becomes 0).
2. The derivative of a constant times the variable (Q), is simply the constant (132 Q becomes 132)
3. The derivative (dy/dx) of: axn is: n axn-1. If y = 4x3 then dy/dx = 12 x2.
Knowing the formula for Marginal Profit (132 - 40Q), and knowing that profits are maximized when MP = 0, we used Table 2.1 to see that Profit MAX occurs when Q* = 3.3. We also know that the slope of the MP line is 0 when profits are at a MAX, so we can set MP = 0, and solve for Q* (see Figure 2.6 on p. 43):
132 - 40Q = 0
132 = 40 Q and solving for Q,
Q* = 3.3
Knowing profit-maximizing Q*, we can also solve for P, and then TR, TC and П:
P = 170 - 20 Q
P = 170 - 20 (3.3)
P = $104 thousand
TR = P x Q = $104 x 3.3 = $343.20 (thousand)
TC = $100 + 38Q = 100 + 38 (3.3) = $225.40 (thousand)
П = TR - TC = $343.20 - 225.40 = $117.8 (thousand)
or using the П equation:
П = -$100 + 132Q - 20Q2 = -100 + 132(3.3) - 20 (3.3)2 = $117.8
See Figure 2.7 (p. 45) of П and MP on p. 45, confirming graphically that MP = 0 when П is at a MAX.
MARGINAL REVENUE AND MARGINAL COST
MARGINAL REVENUE
TR = f (Q)
MR = ΔTR = TR1 - TR0
ΔQ Q1 - Q0
For example, using TR = $170Q - 20 Q2, when
Q = 2.0 TR = $260
Q = 2.1 TR = $268.80
Therefore, the MR from increasing output from 2.0 to 2.1 is:
ΔTR = $268.80 - 260.00 = $8.80 = $88 thousand / lot
ΔQ 2.1 - 2.0 .10
TR graphically is represented in Figure 2.8, on p. 48. MR for any Q is the slope of the tangent line touching the TR curve. Using calculus, we can solve for MR by taking the derivative of TR:
TR = f (Q)
TR = $170 Q - 20 Q2
MR = Δ TR / Δ Q
MR =
For example, when Q = 3, MR = 170 - 40 (3) = $50 thousand per lot. At this sales quantity (3), increases in Q will increase TR by $50,000 per lot.
MARGINAL COST
TC = f (Q)
MC = ΔTC / Δ Q
TC = $100 + 38Q
Δ TC = $38, which is the VC per unit
Δ Q
MC is constant in this case at $38, see Figure 2.8 (Panel b) on page 48.
PROFIT MAXIMIZATION
П = TR - TC
MP = MR - MC
As Q changes, Marginal profit is equal to the difference between MR and MC. If Q increases by 1 unit, the MP is equal to the MR from that unit, minus the MC of that unit.
Profit MAX occurs when MP = 0, which is also the level of output (Q*) where MR = MC. Logic: Firm increases output as long as MR > MC, since profits will increase. When MR = MC, that is the PROFIT MAX LEVEL OF OUTPUT, see Figure 2.8 on p. 48.
Also, MR = 170 - 40Q and MC = 38, setting MR = MC, and solving for Q*:
170 - 40Q = 38
Q* = 3.3
SENSITIVITY ANALYSIS
See Figure 2.9 (a) on p. 51, showing the MR curve and the MC curve. The firm should expand output (Q) as long as MR > MC, and stop at Q* = 3.3, which is the level of output that maximizes profit.
ASKING WHAT IF?
1. Increased Overhead, what effects will that have on P, Q, П, etc.? Suppose that FC increase from $100,000 to $112,000, so that now TC = 112 + 38Q. Note that MC has NOT changed, it is still $38 thousand. MR has also NOT changed, so Panel a still represents the firm's new situation. The firm still produces Q* = 3.3, because that is where MR = MC. Illustrates how MA always leads to the correct solution. The firm should NOT change P or Q, it will just have to absorb the increased overhead cost. Trying to raise price, will only reduce overall profits!!
2. Increased VC or input prices. Suppose that the price of silicon rises so that the VC goes up from $38,000 to $46,000 per lot (+21% increase in materials), what should the firm do? TC = 100 + 46Q. MC = $46, so the MC curve shifts up, see Panel (b) of Figure 2.9 on page 51. Set MR = MC:
170 - 40Q = 46
Q* = 3.1, and P = 170 - 20 (3.1) = $108 thousand, up from $104,000 per lot. Cost went up by $8000 and the firm was able to pass along 50% of that in the form of a price increase to buyers.
3. Increased Demand. Suppose that demand increases, demand curve shifts out, new equation is P = 190 - 20Q, see Panel (c) of Figure 2.9 on page 51. Now,
TR = (190 - 20Q) x Q
TR =
MR =
Set MR = MC:
Q* =
P* =
The firm benefits by selling more at a higher price.
REVIEW NUTS AND BOLTS on Page 55:
1. The basic building blocks for the firm's price and output problem are: 1) its demand curve and 2) its cost function. From the demand curve, we can calculate Total Revenue: TR = P x Q. We can then solve for PROFIT: П = TR - TC.
2. Knowing П, TR and TC, we can solve for MP, MR and MC by taking the first derivative of П, TR and TC.
3. We can solve for the level of output (Q*) that will maximize profits by either 1) setting MP = 0, solve for Q* or 2) setting MR = MC, solve for Q*. Once we know Q*, we can solve for P* using the demand equation, and solve for П using the profit equation.
4. Setting MR = 0 and solving for Q will determine the level of output that will maximize SALES REVENUE.
