Economics and Such LRT 02/19/2018

Transcription

1 Economics and Such LRT 02/19/ / 14

2 Marginal as used in economics Marginal is a word used in economics as a synonym for instantaneous rate of change. Because marginal means some sort of derivative there are two parts that need to be nailed down. What is the quantity we are discussing? What is the variable that is changing? 2 / 14

3 Cost d C Marginal cost = if cost is a function of number of d x items. d C Marginal cost = if cost is a function of price. d p Revenue d R Marginal revenue = if revenue is a function of d x number of items. d R Marginal revenue = if revenue is a function of d p price. Profit d P Marginal profit = if profit is a function of number d x of items. d P Marginal profit = if profit is a function of price. d p 3 / 14

4 Occasionally you will see the term marginal gross national product. For example, here. The output of G.N.P. is usually given in dollars or some other currency. The input is time. So if GNP (t) is the gross national product function, the marginal gross national product is d GNP (t) d t 4 / 14

5 Revenue Revenue is the amount of money being earned by selling things at a price. Often we have a situation where the number of things sold are sold at a fixed price. Then R = px Sometimes we have a supply function p = f(x), and then R(x) = xf(x). Sometimes we have a supply function x = s(p), and then R(p) = ps(p). Depending on which version we have we get two similar formulas R (x) =f(x) + xf (x) R (p) =s(p) + ps (x) 5 / 14

6 Cost Cost is what it costs to produce a given number of items. It typically comes in two parts. You have a fixed cost which is independent of how much you produce. Fixed cost is a constant and is usually positive since you need a business location, often employees, taxes, etc. You also have a variable part since it costs money to produce each unit of something. Hence typically, C(x) is an increasing function of the number of units produced. 6 / 14

7 Profit Profit is revenue minus cost. There are usually two versions: P (x) = R(x) C(x) P (p) = R(p) C(p) In English the marginal formulas are both the same. Marginal profit equals marginal revenue minus marginal cost. 7 / 14

8 Average Cost If the cost is given by C(x) where x is the number of items produced, the average cost is C(x) = C(x) x The marginal average cost is d C(x) d x C (x). or usually written By the Quotient Rule we have C (x) = xc (x) C(x) x 2 8 / 14

9 Since profit is revenue times cost, the average cost is interesting when we observe that P (x) = px C(x) = xp xc(x) = x ( p C(x) ) Since x, the quantity being produced, is typically positive, we see that to make a profit you must get your average cost below your unit price. 9 / 14

10 Relative Rate of Change Given any function f of a variable x, the instantaneous rate of change, d f(x), is one measure of how f is changing. d x Sometimes a better measure of how a quantity is changing is the instantaneous relative rate of change defined by f (x) f(x) An equivalent way to present the same information is as the instantaneous percentage rate of change defined by f (x) f(x) 100% The book gives several examples. 10 / 14

11 Elasticity of Demand Suppose x = f(p) is a demand function. Then the elasticity of demand is defined by E(p) = pf ( ) (p) percentage rate of change in f(p) f(p) = percentage rate of change in p In English, the elasticity of demand is minus the relative rate of change in supply multiplied by the unit price. Demand is said to be elastic if E(p) > 1. Demand is said to be unitary if E(p) = 1. Demand is said to be inelastic if E(p) < 1. One reason for these strange names is on the next page. 11 / 14

12 Since R(p) = pf(p), R (p) =f(p) + pf (p) ( =f(p) 1 + p f ) (p) f(p) =f(p) ( 1 E(p) ) Since f(p) is positive, elasticity of demand determines if an increase in price results in an increase in revenue. 12 / 14

13 R (p) = f(p) ( 1 E(p) ) If a quantity has elastic demand, an increase in price results in a decrease in revenue. If a quantity has unitary demand, an increase in price results in no change in revenue. If a quantity has inelastic demand, an increase in price results in an increase in revenue. 13 / 14

14 The demand is inelastic (E(p) < 1) if small increases in price result in more revenue. Said another way, inelastic demand means that the revenue is not adversely affected by small price increases: the demand may go down but it stays high enough that the revenue still increases. 14 / 14