2 1. Tonight is a calculus review. 2. And a review of basic microeconomics. 3. We will do a couple of problems in class.
3 First hour: Calculus Thinking on the margin. Introducing basic differential calculus and its relationship to economics Finding the margin Finding the maximum Two quick problems Second Hour: Micro Simple Demand Curve Perfect Competition P = MR = MC Monopoly Marginal Revenue Curve Profit maximization (MR = MC) Efficiency Deadweight loss & monopoly One quick problem
4 We are going to do a BRIEF calculus review. This is NOT a math class; it s not a particularly math-oriented class. However, in presenting the concepts in this course, it is often less cumbersome and more instructive to be able to use calculus.
5 Economic relationships are often expressed in terms of functions that relate one economic value to another. Examples include Cost Functions, Demand Functions and Production Functions.
6 Economists also talk about the concept of thinking on the margin. (ECON 1/ ECON 100) Rational people often make decisions by comparing marginal benefits to marginal costs. The margin refers to incremental change. Marginal revenue, for example, refers to the incremental change in total revenue; marginal cost refers to the change in total costs.
7 Economic decisions are made on the margin because everything we do of necessity creates change. Some change makes you better off (a benefit), while other change makes you worse off (a cost).
8 Things we might do: Eat cookies Go to work Start a business Study for I/O Watch TV Added (marginal) benefits: Added (marginal) costs
9 Things we might do Industrial Organization Added (marginal) benefits Added (marginal) costs Eat cookies Tastes great!! Fattening Go to work Increase income Wake up early Start a home business Set my own hours Give up my day job Watch TV Mindless fun Study time Study for I/O Real fun????
10 We do things because of the changes these actions induce especially if the added benefits > than the added costs. That s what it means to think on the margin.
11 If decisions are made on the margin, how can economists identify marginal values? The most prominent method is calculus. Calculus was invented by two men Newton and Leibniz in the 17 th century, independently of one another.
12 To understand how calculus works, remember there are two expressions similar to the concept of the margin : rate of change slope of the tangent line of a function
13 P EXAMPLE: Downward sloping Curve The slope of the tangent line tells us the rate of change at that particular point on the line rise run Q
14 Slope of the tangent line, rate of change and margin are (almost) the same. One type of calculus differential calculus allows us to determine the slope of the tangent line for certain functions. This method is called differentiation, and the result that is obtained is called the derivative.
15 Notations for the Derivative: Given y = f(x), then f (x) y dy dx all represent the derivative of f at x
16 Interpretations of the Derivative: Slope of the tangent line Instantaneous rate of change Question: given a function, how do you calculate the derivative?
17 Selected Differentiation Rules General Formulas Constant Rule Factor Rule Factor Rule Sum Rule Product Rule Product Rule Quotient Rule Chain Rule Power Rule Power Rule 11
18 Differentiate: y = 3x 2 + 2x Solution: Step #1 Sum Rule (3x 2 ) + (2x) Step #2 Power Rule 2*3*x *2x 1-1 = 6x + 2 Differentiate f(x) = (3x + 1) 4 Solution Step #1 Gen l Power Rule u = 3x + 1; n = 4 nu n-1 = = 4(3x + 1) 3 du/dx = (3x + 1) = 3 Therefore: nu n-1 *du/dx = 4(3x + 1) 3 * 3 = 12(3x + 1) 3
19 For multivariable functions, we take derivatives for each variable individually, holding the other variable(s) constant. These are partial derivatives, represented by rather than d. If U is a function of (y, x) the partial derivatives are represented by U/ y and U/ x. All the differentiation rules presented for single variable equations apply. Remember, when taking the partial derivative of one variable, the other variable is treated like a constant.
20 EXAMPLE: U is a function of (y, x) U = y 3 + 3yx 2 + 4x The partial derivatives are: U/ y = 3y 2 + 3x 2 U/ x = 6yx + 4
21 In economics, we use derivatives to find marginal values. The Marginal Cost Function approximates the change in the actual cost of producing an additional unit. Given a cost function C(x), MC = C (x) The Marginal Revenue Function measures the rate of change of the revenue function. It approximates the revenue from the sale of an additional unit. Given the revenue function, R(x), MR = R (x)
22 Total Costs = C(x) = x -.25x 2 Find the marginal cost function. Step 1 Sum Rule (575) + (25x) + (-.25x 2 ) Step 2 Constant & Power Rules (0) + (25) + 2*-.25x 2-1 MC Function = x
23 Why do we say approximate for MC and MR? The formal definition of the derivative is: f (x) = Lim f (x + h) f (x) h 0 h Marginal revenue/cost is the incremental change from adding ONE unit. If C(x) is a cost function, then Marginal Cost = C (x + 1) C (x). Instantaneous Incremental Lim f (x + h) f (x) C (x + 1) C (x) h 0 h 1
24 P e Difference between f (x) & MC MC Curve x x+1 Q
26 Now comes the most important piece of this puzzle. MAXIMIZATION Much of economics involves maximizing some function: Maximize utility Maximize profits
27 At the maximum point of a function, the slope of the tangent line = 0. Therefore, if you find the slope of the tangent line = derivative, and set it to 0, you can find the maximum! P Slope of the tangent line = 0 Max Point Xmax Q
28 Example: Find the maximum revenue Demand equation p = q Revenue (R) = p*q = q*( q) = 10q -.001q2 R (derivative) = q Set = to 0 0 = q Therefore MAX R is reached at q = 5000
29 There are tests to determine if an answer is the absolute maximum. What is important here is the concept of maximizing a function by (1) taking the derivative and (2) equalizing it to 0. This is a very critical technique used by almost all economists.
30 Basic Microeconomics Contrast between two polar opposite cases: Perfect competition Monopoly What is efficiency? (Pareto Optimality) No reallocation of the available resources makes one economic agent better off without making some other economic agent worse off.
31 We will begin with examining the profit maximizing behavior of firms. Assume a standard linear inverse demand curve, P = A BQ P = price Q = quantity A = intercept B = slope
32 $/unit A P 1 Maximum willingness to pay Constant slope Equation: P = A - BQ linear demand Q 1 Demand At price P 1 a consumer will buy quantity Q 1 A/BQuantity
33 We start with Perfect Competition. Firms and consumers are price-takers. A firm can sell as much as it likes at the prevailing market price. Firms believe that their actions will not affect the market price. Therefore, marginal revenue equals price. What about profits?
34 All firms maximize profits. p = profit; R = revenues; C = costs Profit = p(q) = R(q) - C(q) Profit maximization: dp/dq = 0 This implies dr(q)/dq - dc(q)/dq = 0 dr(q)/dq = marginal revenue (MR) dc(q)/dq = marginal cost (MC) So profit maximization implies MR = MC.
35 Therefore, for perfectly competitive, profitmaximizing firms, p = MR = MC. The next question: how does a perfectly competitive market work?
36 (a) The Firm (b) The Industry $/unit $/unit MC AC D 1 S 1 P 1 P 1 S 2 P C P C D 2 q c q 1 Quantity Q C Q 1 Q C Quantity
37 1. With market demand D 1 and market supply S 1 equilibrium price is P C and quantity is Q C 1a. With market price P $/unit C the firm maximizes profit by setting MR (= P C ) = MC and producing quantity q c 2. Now assume Demand increases to D 2 $/unit MC 3a. Firms maximize profits by increasing AC output to q 1 3. Excess profits induce firms to enter the market 5. Firms enter. The supply curve moves right 6. Price falls; entry continues while profits exist D 1 4.With market demand D and market supply S 1 equilibrium price is P 1 and quantity is Q 1 7. Long-run equilibrium is restored at price P C & supply curve S 2 S 1 P 1 P 1 S 2 P C P C D 2 q c q 1 Quantity Q C Q 1 Q C Quantity
38 Definition of normal profit not the same as zero profit implies that a firm is making the market return on the assets employed in the business those returns = opportunity cost of capital
39 Now we come to the other pole: monopoly. A monopoly is the only firm in the market market demand is the firm s demand output decisions affect market clearing price
40 At price P 1 consumers buy quantity Q 1 $/unit Loss of revenue from the reduction in price of units currently being sold (L) Marginal revenue from a change in price is the net addition to revenue generated by the price change = G - L At price P 2 consumers buy quantity Q 2 P 1 P 2 L Q 1 G Q 2 Gain in revenue from the sale of additional units (G) Demand Quantity
41 Derivation of the monopolist s marginal revenue Demand: P = A - BQ Total Revenue: TR = PQ = AQ - BQ 2 Marginal Revenue: MR = dtr/dq $/unit A Therefore: MR = A - 2BQ With a linear demand the marginal revenue curve is also linear with the same price intercept but twice the slope of the demand curve. MR Demand Quantity 41
42 The monopolist maximizes profit by equating marginal revenue with marginal cost This is a two-stage process $/unit Output by the monopolist is less than the perfectly MC competitive Stage 1: Choose output where MR = MC This gives output Q M Stage 2: Identify the market clearing price This gives price P M P M AC M Profit Q M output Q C AC MR Q C Demand Quantity MR is less than price; Price is greater than MC: loss of efficiency Price is greater than average cost Positive economic profit Long-run equilibrium: no entry
43 Having presented two distinct market models perfectly competitive and monopolistic we need to add the method by which economists evaluate these outcomes. Why is one highly regarded and the other highly regulated? The answer is efficiency.
44 Economists use a concept called Pareto Optimality or Pareto Efficiency. Pareto Optimal: no one can be made better off without someone being made worse off. We can put this in the form of a question: Can we reallocate resources to make some individuals better off without making others worse off? To answer the question, we need a measure of well-being = surplus.
45 There are three aspects to the concept of the surplus in economics. First is consumer surplus: consumer surplus: difference between the maximum amount a consumer is willing to pay for a good and the amount actually paid. Aggregate consumer surplus is the sum over all units consumed by all consumers.
46 Then there is producer surplus: producer surplus: difference between the amount a producer receives from the sale of a unit and the amount that unit costs to produce. The aggregate producer surplus is the sum over all units produced by all producers Together they comprise total surplus = the consumer surplus + the producer surplus.
47 The demand curve measures the consumers willingness to pay for each unit Consumer surplus is the area between the demand curve and the equilibrium price $/unit Competitive Supply The supply curve measures the marginal cost of each unit Producer surplus is the area between the supply curve and the equilibrium price P C Consumer surplus Producer surplus Equilibrium occurs where supply equals demand: price P C quantity Q C Demand Aggregate surplus is the sum of consumer surplus and producer surplus The competitive equilibrium is efficient Q C Quantity 47
48 Assume that the industry is monopolized The monopolist sets MR = MC to give output Q M The market clearing price is P M $/unit Deadweight loss of Monopoly Competitive Supply This is the deadweight loss of monopoly Consumer surplus is given by this area And producer surplus is given by this area P M P C The monopolist produces less surplus than the competitive industry. There are mutually beneficial trades that do not take place: between Q M and Q C Q M Q C MR Demand Quantity Chapter 2: Basic Microeconomic 48
49 Example: Water Produced (2.5) Demand: P = 25 ½ Q TC = Q; dtc/dq = MC = 10 Price at efficient allocation? Price to maximize profits? What is the deadweight loss of monopoly?.
50 Efficient allocation: P = MC; 25 ½ Q = 10. Q = 30, P = 10 Monopoly pricing: P = 25 ½ Q PQ = 25Q = ½ Q 2 MR = d(pq)/dq = 25 2*1/2Q = 25 Q MR = MC = 25 Q = 10; Q = 15, P = 17.5
52 Why can t the monopolist appropriate the deadweight loss? Increasing output requires a reduction in price. this assumes that the same price is charged to everyone. The monopolist does create a surplus. some goes to consumers some appears as profit The monopolist bases its decisions purely in its surplus alone. The monopolist undersupplies relative to the competitive outcome.
53 Recall the quote from Smith: The monopolists, by keeping the market constantly understocked, by never fully supplying the effectual demand, sell their commodities much above the natural price. Adam Smith, The Wealth of Nations (1776)
54 We have covered most of Chapter 2, except Part 2.2 on discounting. Next week we will do discounting, and also cover Chapter 3 on Market Structure and Market Power and Chapter 4 on Technology and Cost. Please keep reading: PRN, Chapters 1 4
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