MATH MODULE 5. Total, Average, and Marginal Functions. 1. Discussion M51


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1 MATH MODULE Total, Average, and Marginal Functions 1. Discussion A very important skill for economists is the ability to relate total, average, and marginal curves. Much of standard microeconomics involves comparisons at the margin, for the purpose of maximizing the value of firm profits or of individual utility. In the output market for a firm, for example, Profits = Total Revenue minus Total Cost, or TR TC. If the TR and TC functions are wellbehaved, and there is some positive level of output at which the firm can cover all of its total variable costs, then the level of output Q at which it maximizes profits will be given by the condition MR =MC, Marginal Revenue = Marginal Cost. [This is known as the firstorder condition for profit maximization. The secondorder condition is covered in footnotes 9 and 10 on pages of your text.] 1.1 SOME IMPORTANT EXAMPLES In this subsection we list five important sets of total, average, and marginal curves. In the next subsection we focus on the important special case where the total curve is quadratic and the corresponding average and marginal curves are linear. We will use the following units of measurement in our illustration, and assume that the functions refer to a single period: output of good X (Q X ) is in kilograms, the price of good X (P X ) is in $/kg, labour input (L) is in labourdays, and the daily wage rate or price of labour (P L ) is in $/labourday. The sets of equations for the input market (#4 and #) refer to a situation in which there is one variable input (labourdays), and so it refers to the short run, when not all factors are variable. You should not aim at memorizing these relations, but rather at learning them. The best way to learn them is to use them and play with them, along the lines of the exercises M1
2 M2 MATH MODULE : TOTAL, AVERAGE, AND MARGINAL FUNCTIONS at the end of this Module. Thinking about them in terms of the units in which they are measured will make them much more concrete and easy to use and to remember. OUTPUT MARKET 1. Total Revenue: TR = P X Q X [TR($) = Price ($/kg) x Quantity (kg)] Average Revenue: AR = TR/Q X = (P X x Q X )/Q X = P X [AR = P X ($/kg)] Marginal Revenue: MR = TR/ Q X [MR = ($)/(kg) = $/kg] 2. Total Cost: TC = FC + VC [TC($) = Fixed Cost($) + Variable Cost($)] = ATC Q X [TC($) = Average Total Cost ($/kg) Quantity (kg)] Average Total Cost: ATC = TC/Q X = (ATC Q X )/Q X [ATC = ($)/(kg) = $/kg] = AFC + AVC [Average Fixed Cost($/kg) + Average Variable Cost($/kg)] Average Fixed Cost: AFC = FC/Q X [AFC = ($)/(kg) = $/kg] Average Variable Cost: AVC = VC/Q X [AVC = ($)/(kg) = $/kg] Marginal Cost: MC = TC/ Q X = VC/ Q X [MC = ($)/(kg) = $/kg] PRODUCTION 3. Total Product of Labour: TP L Q X = AP L L [TP L (kg) = (kg/lday) Ldays] Average Product of Labour: AP L = TP L /L Q X /L [APL(kg/labourday)] Marginal Product of Labour: MP L = TP L / L Q x / L [MP L (kg/labourday)] INPUT MARKET 4. Total Revenue Product of Labour: TRP L P X Q X = ARP L L [TRP L ($) = ($/kg)(kg) = ($/Lday) Ldays] Average Revenue Product of Labour: ARP L = P X AP L = TRP L /L P X Q X /L [ARP L ($/Lday) = ($/kg)(kg/lday)] Marginal Revenue Product of Labour: MRP L = TRP L / L = MR MP L [MRP L ($/labourday) = ($/kg)(kg/labourday)]. Total Factor Cost of Labour: TFC L P L Q L = AFC L L [TFC L ($) = ($/kg)(kg) = ($/Lday) Ldays] Average Factor Cost of Labour: AFC L = P L = TFC L /Q L [AFC L ($/Lday) = ($)/(Ldays)] Marginal Factor Cost of Labour: MFC L = TFC L / Q L = MC MP L =( TC/ Q X ) ( Q x / L) [MFC L ($/labourday) = ($/kg)(kg/labourday)] 1.2 AN IMPORTANT SPECIAL CASE: LINEAR AVERAGE AND MARGINAL CURVES In general, there is no reason to assume that the functions with which economists are concerned, such as the average and marginal functions we outlined in Section 1.1 of this Module, are in fact linear. We use linear functions so frequently in our illustrations and examples basically because of their mathematical simplicity. Using them, we can often understand fairly difficult points in economic theory without requiring any more mathematics than high school algebra. We do need to be alert to the fact that in some cases, results that are valid for linear functions do not necessarily hold if the functions have a more general, nonlinear form. But we will still continue to use linear functions extensively, because they have a relatively high economicstomathematics ratio. It is therefore important to understand clearly the relationships among linear average and marginal curves and the quadratic total curves to which they correspond. We will focus here on an example based on a demand curve for a good, but the rules we derive apply to all of the sets of functions outlined in Subsection 1.1 of this Module.
3 MATH MODULE : TOTAL, AVERAGE, AND MARGINAL FUNCTIONS M3 These basic rules are also discussed at a number of points in your text, including page 343, footnote 14; pages 389, Figures 127 and 128; and (as they relate to elasticity), page 111, Figure 423 and page 89, Figure A.41. Consider the demand curve with the form P = 10 Q, data for which are in Table M.1. Total Revenue is given by TR = P Q = (10 Q) Q = 10Q Q 2. Total revenue, as Table M.1 and Figure M.1 show, thus has a quadratic form: its graph is a parabola opening downward, with a peak or maximum value of $2 when Q = kg, and a value of 0 at Q = 0 and Q =10. TABLE M.1 The Demand Curve P = 10 Q Price ($/kg) Quantity (kg) Total Revenue ($) Marginal Revenue ($/kg) The Marginal Revenue (or TR/ Q) curve may be derived in three ways. For those with calculus, it is simply the slope or derivative of the TR curve at any value of Q: dtr/dq = 10 2Q. Those without calculus should glance at Module 9, which contains some basic rules for calculating derivatives, including the one used here. Yet it is also possible to derive it from Table M.1. Note that the values for Marginal Revenue are located at the midpoints of the relevant values of Q. For example, the value of MR as we go from Q = 2 to Q = 3 (or viceversa) is equal to +$/kg, and is situated at Q = 2. kg, since it is the change in TR in moving between 2 and 3 kg. Note also that the MR declines by 2 for each increase of Q by 1 unit: its slope is 2. Its equation is therefore MR = 10 2Q. The third way of calculating it is to use The Rules. The Rules apply to any related linear average and marginal curves and the corresponding total function. THE RULES 1. From Average to Marginal Curve: Same intercept, twice the slope. [If Average Revenue = P = a + bq, then MR = a + 2bQ. In our example, if P = AR = 10 Q, then MR = 10 2Q.] 2. From Marginal to Average Curve: Same intercept, half the slope. [If MR = a + bq, then P = AR = a + 0.bQ. In our example, if MR = 10 2Q, then P = AR = 10 Q.] 3. From Average to Total Curve: Average times Q = Total. [If Average Revenue = P = a + bq, then TR = aq + bq 2. In our example, if P = AR = 10 Q, then TR = 10Q Q 2.] 4. From Total to Average Curve: Total divided by Q = Average. [If Total Revenue = P Q = aq + bq 2, then AR = a + bq. In our example, if TR = 10Q Q 2, then P = AR = 10 Q.]. From Total to Marginal and From Marginal to Total Curve: Either use calculus or use a 2 step procedure: Total Average Marginal or Marginal Average Total.
4 M4 MATH MODULE : TOTAL, AVERAGE, AND MARGINAL FUNCTIONS TR ($) 2 TR = P Q 0 10 Q (kg) P = AR, MR ($/kg) 10 D = AR 0 10 MR While The Rules have been expressed in terms of Demand and Marginal Revenue curves, they apply equally as well for any of the sets of functions in Section 1.1 of this Module. The only set that poses any problems is the Total Cost/ Marginal Cost one. The reason is that Fixed Costs complicate the situation slightly. The needed adjustments to The Rules in this case are covered in your text, on page 343, footnote 13, and you will get some practice in the Exercises.
5 MATH MODULE : TOTAL, AVERAGE, AND MARGINAL FUNCTIONS M 2. Exercises 1. For each of the following cases, provide the Total Revenue, Average Revenue, and Marginal Revenue equations and give the value for each of the 3 equations when Q = 10 tonnes: (a) P = AR = 30 Q (b) TR = 10Q 0.1Q 2 (c) MR = 30 6Q (d) MR = 40 4Q (e) TR = 2Q 0.2Q 2 (f) P = AR = 60 2Q (g) MR = 6 0.6Q (h) MR = 10 (i) TR = 60Q 3Q 2 (j) TR = 60Q 2. Maximum Total Revenue is reached when MR = 0. For cases (a) to (j) in Exercise 1, find Q*, the value for Q that maximizes Total Revenue, and give the value of Total Revenue at that point. 3. The Total Cost function this period for a firm producing Electric Cheese ( the cheese with shockingly good taste! ) is given by the equation TC = Q + Q 2, where TC is in $ and Q is in kilograms. (a) Give the following functions for Electric Cheese: Fixed Cost (FC), Variable Cost (VC), ATC, AFC, AVC, and MC. [If you do not have calculus, and even if you do, you may want to check page 343, n. 13 of the text.] (b) By using calculus or by using the rule that MC = ATC at the minimum point of ATC, calculate the level of Q at which ATC is at its minimum. (c) If Electric Cheese sells at the jolting price of $24/kg, and profits are maximized when P = MR = MC, calculate the profitmaximizing level of Q, Q*, and the level of profits (= TR TC) this period, when the firm produces at this level of output. 4. A company s Total Revenue function this period is given by the equation TR = 60Q 0.Q 2 and its Average Total Cost function is given by the equation ATC = 10/Q Q, where TR is in $, Q is in units, and ATC is in $/unit. (a) Profitmaximization occurs where MC = MR. Give the equations for the MC and MR curves and calculate Q*, the profitmaximizing level of Q. (b) Calculate the company s profits this period at Q*.. A company s Average Factor Cost of Labour (or labour supply) function this period is given by the equation AFC L = 20 + L and its Marginal Product of Labour function is given by the equation MP L = 40 L,
6 M6 MATH MODULE : TOTAL, AVERAGE, AND MARGINAL FUNCTIONS where AFC L is in $/labourday, MP L is in kg/labourday, and L is in labourdays. It can sell as much output as it wants at a constant price of $3/kg. It has no fixed costs. (a) Profitmaximization occurs where MFC L = MRP L. Give the equations for the MFC L and MRP L curves and calculate the profitmaximizing level of L, L*. (b) Calculate the company s profits this period at L*.
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