Preliminary Lecture Notes for a Micro Course Based on Microeconomics 8ed by Pindyck & Rubinfeld
Preliminary Lecture notes for a Micro Course Based on Microeconomics 8ed by Pindyck & Rubinfeld Prepared by Houston H - Preliminary Lecture Notes for a Micro Course Based on Microeconomics 8ed by Pindyck & Rubinfeld introduction. Stokes. Goal of the Notes: Allow the student to have an outline of the key ideas and solutions to a number of problems that will be discussed in class. Since the notes are distributed in WORD® format, students can edit the notes. Introduction Quote from Robert Mundell Man and Economics 1968 “Economics is the science of choice. It began with Aristotle but got mixed up with ethics in the Middle Ages. Adam Smith separated it from ethics, and Walrus mathematized it.
Alfred Marshall tried to narrow it, and Keynes made is fashionable. Robbins widened it, and Samuelson dynamized it, but modern science made it statistical and tried to confine it again. But the science won’t stay put. It keeps cropping up all over the place. There is an economics of money and trade, of production and consumption, of distribution and development. There is also an economics of welfare, manners, language, industry, music, and art. There is an economics of war and an economics of power. There is even an economics of love. Economics seems to apply to every nook and cranny of human experience.
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It is an aspect of all conscious action. Whenever decisions are made, the law of economy is called into play. Whenever alternatives exist, life takes on an economic aspect. It has always been so. But how can it be? It can be because economics is more than just the most developed of the sciences of control. It is a way of looking at things, an ordering principle, a complete part of everything. It is a system of thought, a life game, an element of pure knowledge. Chapter 1 Preliminaries • Economics is concerned with scarcity. If something is not scarce, there is no economic problem.
• Microeconomics => Study of behavior of individual economic units. How they react and how they interact to form larger units. • Economics uses theory to explain actions and predict future actions. For example if: the Bulls are in the playoffs, then there will be a larger number of people wanting tickets than if they have a poor season. If there is high unemployment, then at the university more students enroll in business courses because their opportunity costs (pay loss due to being in class) are less. • Theory does not work well all the time.
The market test of theory is how well it does in comparison to another theory. A theory must have the possibility to being proved wrong. The statement “all unmarried men are bachelors” is a tautology and cannot be proved wrong. The statement “If American Airlines lowers the ticket price to Bermuda, an increased number of people will fly this weekend. ” Can be proved wrong or right. • Marginal utility theory can be used to derive the demand curve. However it will be shown later that to get a downward sloping demand curve the only assumption needed is that consumers buy randomly along the budget line.
=> a minimalist approach to deriving an important economic concept – downward sloping demand. • Normative Economics => What should be done. “Microsoft should be allowed to bundle the IE with Windows XP because it benefits consumers. ” • Positive Economics => What will happen, not what should be done. “An increase in the property tax in the area of UIC will tend to lower the price of apartments, everything else equal. ” • Many decisions involve multiple assumptions. Book example of 1985 decision of Ford to produce the Taurus involved: 1. Consumer tastes vs demand “would they like the car?
” 2. How sensitive would demand be to price changes? (elasticity), 3. What would be the production costs? (Depends on assumptions of #’s of cars produced, union demands, inflation, how fast workers learn). 4. How would competitors react (market structure). 5. How much new capital would be needed? (interest rates, engineering). 6. How would the decision be changed if oil prices moved favorably, unfavorably? 7. How should Ford organize the production? 8. How might anticipated Government regulation changes influence the decision (Gas mileage requirements).
• What is a Market. A market is the collection of buyers and sellers that, through their actual or potential interactions, determine the price of a product or set of products. A market includes more than an industry (a collection of firms that sell the same or closely related products). Key business decision: Determining the market for the product! • If prices differ in two markets, arbitrage may be possible. If the price of a T-bill maturing in period t in NY is > than the price of the same T-bill in Chicago => people will buy in Chicago and sell on the NY market.
If the price of a Big-Mac is less in Norfolk VA than Chicago, arbitrage is NOT feasible, even with fast planes. • Markets can be competitive (wheat) or noncompetitive (electric power) due to entry costs. • In a competitive market with many producers no one producer can change the industry price. Product differentiation => power to alter price within limits. Successful advertising => ( in perceived product differentiation. • Extent of the market => the boundaries, both geographical and physical of a market. • Real vs Nominal Price. Economic decisions should be made on the basis of real prices.
Money illusion => consumer looks at the nominal price not the real price. Assume two groups: debtors and creditors. Creditors and debtors set the interest rate depending on their expectations of price movements. Unexpected price increases (decreases) favor debtors (creditors). • Measuring Price The CPI measures price changes by buying a market basket of goods at different times. Three known problems are: 1. Tastes may change over time but same basket is bought. 2. Relative prices may change but the same basket is bought 3. Price changes involve subtle income changes that give rise to income effects that will alter the mix of goods bought.
• The CPI for period t, Pt is calculated with the Laspeyres formula: [pic](1) • If in place of the above equation the quantities could have been adjusted every year as in the Paasche formula [pic] (2) which is not possible to calculate but would avoid the tastes bias and the relative price bias. Table 1. 2 lists the CPI and education and egg prices in both nominal and real terms. How you get data in 2010 prices? See file ch1_1. xls for the answer? Why would one want to make this calculation? Study Questions: 1. Over the past year the price inflation has been 10%.
The price of a used Ford SUV has fallen from $6,000 to $5,000. How much as the real price fallen: ((1. 1 * 6000) – 5000)/(1. 1*6000) = 24. 24 Since last year the price of gold has risen from $120 to $420. What annual rate of inflation would hold the price of gold fixed in rea1 terms? (420-120)/120 = 250% If this was over 5 years and we assume yearly compounding, what would the rate of inflation have to be? 420 = 120(1. + r)5 (1. + r)5 = (420/120)=3. 5, = > ln(1. +r) * 5 = ln( 3. 5), => ln(1. + r) = ln(3. 5)/5 or (1. + r) = exp(ln(3. 5)/5), (1. + r) = 1. 2847 or 28.
47% annual inflation. Check: (1. 2847)5 = 3. 5, 120*3. 5 = 420 2. Suppose that the Japanese yen raises against the U. S. dollar; that is, it now takes more dollars to buy any given amount of Japanese yen. Explain why this simultaneously increases the real price of Japanese cars for U. S. consumers and lowers the real price of U. S. automobiles for Japanese consumers. Chapter 2 Basics of Supply and Demand The basic model postulates that Supply responds positively to price and Demand responds negatively to price. Qs = Qs(P) and Qd = Qd(P). At equilibrium P (P0) = Q (Q0) and the market clears.
Qs(P) > Qd(P) implies excess supply while Qs(P) < Qd(P) implies excess demand. This is shown below S P P1 P0 P2 D Q Points where price is above P0 represent excess supply, while points below P0 represent excess demand. The laws of motion of Walrus state “If there is excess demand, price will rise. ” The laws of motion Marshall state “If there is excess demand, quantity will rise. ” P0 is the “market clearing price. ” Here supply = demand. Assume that somehow the price goes to P1. Now demand < supply. According to Marshall adjustment will proceed by a reduction in supply, while according to Walrus price will fall.
Similar arguments can be made for P2. In a market for Old Master Paintings clearly Walrus adjustment is the only legal adjustment mechanism. In 1969 the first man walked on the moon. The New York times printed the edition showing this historic moment for three days until everyone got a copy. This was Marshall adjustment. In the above diagram it does not matter which adjustment model you use, price will go to P0. Assume that supply now is negatively sloped or that economics of scale exist. The producer can increase his production and lower his price. Two Models are possible. P DPS D S QQ
The left graph is stable according to Marshall and unstable according to Walrus. The right graph is stable according to Walrus and unstable according to Marshall. Why? There is a shift in supply if one of the variables held constant (not on the axis) changes and that in turn changes the amount supplied. Assume the supply of wheat is a function of P and R where R = rainfall. Qs = a + (1 P + (2 R (3) where (1 > 0 and (2 > 0. An increase in R would, everything else equal move the supply curve right and lower the price for wheat assuming the demand curve is not flat. Assume the demand for wheat is a function of price and income.
Qd = a’ +(3P +(4Y (4) Where (3 < 0 and (4 > 0. An increase in Y moves the demand curve right and results in an increase in price assuming that the supply curve is not flat and that Y( => quantity demanded (. Demand and supply curves hold tastes, prices of all other good, incomes and technology etc constant. Over time supply and demand usually shift right. An exception would be the demand for buggy whips! Why? Key Concept: Given a demand curve, it can be said that there is a change in demand or shift in the demand curve if the price of another good changes, income changes or tastes change.
If the price of the good itself changes, then there is a change in the quantity demanded or a movement along the demand curve. In the above example [pic]will cause a change in demand while [pic] will cause a change in the quantity demanded. For supply, [pic] causes a change in the quantity supplied while [pic] causes a change in supply. It is always important to understand the difference with what moves a curve and what causes a movement along the curve. . Solve simple system: Qs= 1800 + 240P Qd= 3550 – 266P In equilibrium Qd = Qs. or 1800 + 240P = 3550 – 266P 506P = 1750 => P = $3.
46 and Q = 1800 + (1750/506)*240 = 2630 Excel file ch2_1. xls solves this system. Using this technology. Solve the system assuming: – An advertising campaign raises 3550 to 4023. – A lack of rainfall lowers 1800 to 1500 – A change in taste changes 266 to 280 This is the basic template for micro analysis of supply and demand. A major research objective is to determine what will change a, a’, (1,…(4. It will be made more complex as we move forward. Ed = elasticity of demand = (%(Q)/(%(P) =((Q/Q)/((P/P) =((Q/(P)*(P/Q) (5) Usually (d is used in place of Ed. The price elasticity is measured at a point.
On a linear demand curve the slope ((Q/(P) is constant. Q = 8-2P => ((Q/(P) = -2 As Q ( 0 => Ed ( -( As P ( 0=>Ed ( 0 If |Ed| > 1 => P( => Total Revenue (TR) up. TR = P * Q (6) A firm should never operate where |Ed| < 1. Advertising is designed to make |Ed| decrease for every price. After a successful advertising campaign the firm will have more product differentiation and can either raise price or increase sales at the current price. Careful analysis will tell us what to do. S b a P2 P1cd O Q1e f Firm started with demand curve a e which implied price P1 and Q1.
A major advertising campaign moved the demand curve from a e to b f. For purposes of this example a e is | | to b f. Price rises to P2. Proof that |(d | at d is more inelastic than at c. At c (d = ((Q/Q)/((P/P) = (Q1e/OQ1)/(OP1/OP1) = (Q1e/OQ1) = ce / ac (| | lines produce proportionate segments) At d(d = ((Q/Q)/((P/P) = df / bd Since df = ce and bd > ac = > at d |(d| is less than at c Note that at a (d = -( while at e (d = 0. A parallel shift right of a demand curve makes is more inelastic at each price. Income elasticity measures the percent change in the quantity of a good for a percent change in income.
EI ( (I = ((Q/Q)/((I/I) = ((Q/(I)(I/Q)(7) A good is normal if (I > 0 and inferior if (I < 0 . Baked beans are the classic inferior good and the BMW car a classic normal good. Assume an individual faces goods a to n. Let Pa be the price of good a, Pb be the price of good b. Let (A be the change in the amount of A bought and (B be the change in the amount of B bought. Assume the individual gets an increase in income (I. Pa (A + Pb(B + ….. + Pn(N = (I (8) If we assume (I = 0, the above equation defines the budget line which will be used in indifference curve analysis.
Divide (8) through by (I and then multiplying all terms on the left by I A / I A gives a key expression. In steps [pic] [pic] Finally Ka (Ia + Kb (Ib + … + Kn (In = 1 (9) Or the sum of the weighted income elasticities = 1 The cross price elasticity of demand [pic] measures the effect of the price of good B on the quantity of good A. (10) If A is a substitute for B, then [pic] is > 0 or an increase in the price of B => the quantity of A (. If A is a complement of B then [pic] is < 0. Coke is a substitute for Pepsi while gas is a complement for an auto.
In the long run demand is usually more elastic. When the price of gas went up in the 70’s people did not just dump their cars that got poor gas mileage BUT when they replaced them they made sure they got a car with a higher MPG rating. In the long run the supply of a product is usually more elastic than in the short run since in the long run fixed factors can be changed. Given QD = a – bP (11) QS = c + dP (12) And P* and Q* are equilibrium P and Q, then at equilibrium the elasticity of demand (D and elasticity of supply (S are (D = -b(P*/Q*) (13) (S = d(P*/Q*) (14)
=> if we know (D , (S, P* and Q* we can get the supply curve and the demand curve. In the market we observe P* and Q*. Book shows P* and Q* are $. 75 and 7. 5 million (D , (S are -. 8 and 1. 6 respectively. From equation (13) and (14) we get -b = (-. 8) * (7. 5/. 75) = -8. and d = (1. 6)*(75/. 75) = 16. Excel file ch2_2. xls solves the general case of determining the supply and demand functions given the elasticity and P* and Q*. In the long run supply and demand are usually more elastic. When OPEC raised price => discoveries went up. Users shifted to more efficient cars over time.
An effective price control will cause a shortage. If the supply curve is NOT vertical, the total consumed will go down. Demand for E Problem 1. PQdQs 60 2214 80 2016 100 1818 1201620 (d = ((Q/(P)*(P/Q) (Q/(P ( -2/20 => (d = (-. 1)*(80/20) = -. 40 at 100(d = (-. 1)*(100/18) = -. 56 Problem 2. T or F “Since tuition has doubled in real terms in the last 15 years this implies the demand curve for education is vertical. ” FALSE may be seeing shifts of both supply and demand. Problem 3. File Ch2_3. xls contains simulated Data from 25 Families obtained from Stigler Theory of Price Edition 4 page 37.
Variables are expend = expenditure, pincome = permanent income, Tincome = transitory income, oincome = observed income. The data was generated from the model expend = f(1+. 02*pincome). Tincome was randomly added such that pincome + tincome = oincome Run expend=f(constant pincome) Expend=g(constant oincome) Chapter 3 Consumer Behavior Key Assumptions of Theory of Consumer Behavior: Preferences are complete => Consumers can rank market baskets. Preferences are transitive => If consumer prefers basket A to B and B to C then the consumer prefers A to C. More of any good is always better.
Indifference Curve – Locus of points showing different combinations of goods to which the consumer is indifferent. X III a IIIIV bY Consumer starts with a of X and b of Y. Points in I => more satisfaction. Points is III => less satisfaction. Indifference curves must be in II and IV. XComplements XSubstitutes YY Y Usual Case Utility along b > than along a 2 b 1 a X Indifference curves cannot cross unless there is a change in tastes. Ordinal ranking => A is better than B but cannot tell by how much. Cardinal ranking => Can tell how much better A is than B.
Slope of indifference curve measure marginal rate of substitution between goods. In terms of above example = [pic]. At 1 [pic][pic] At 2 [pic][pic] => along an indifference curve there is diminishing marginal rate of substitution. Given you have relatively more X than Y => will give less Y for every additional X. Budget line = locus of points showing different combinations of goods that can be bought given income. Where budget lines is tangent to highest indifference curve => desired point (a). Y A a 3 2 1 BX A = I / Py, B = I / Px. At a on highest indifference curve. MRS = (Y/(X =Px / Py 12
if income increases budget line moves out. If price of X falls budget line rotates Y a I ( => Budget line moves from 1 to 2 above Px[pic]=> Budget line moves from ab to ac on left. bcX Total spending fixed along budget line. Object is to reach highest Indifference curve (not drawn). Unless there is a corner solution, MRS = Px / Py. Of in words slope of the indifference curve = slope of budget line. Corner solution => consume only one good Utility = satisfaction one gets for consuming a good. Marginal Utility declines as consumption of a good increases unless lumpyness problem (4 tires on a car).
MRSxy = MUx / MUy In equilibrium MUx / MUY= PX / PY MUx / PX= MUY / PY (MUx / PX) >(MUY / PY)=> Have too much Y relative to X Marginal utility per dollar last dollar spent must be the same for all goods. Gas rationing => loss of welfare. Rationing done to maintain price. In NYC rent controls led to vast sections of the city being abandoned. Rationing once in place is hard to remove. See Figure 3. 22 Other Goods O A CGas Consumer wants OC of gas. The government forces person to take OA and thus be on a lower indifference curve. The Laspeyrse price index is defined as [pic]
which overstates the amount of a price increase because the person is given enough money to buy their old bundle even though this is NOT what they will buy since relative prices have changed. a Y 1 2 3 II III I X Consumer was on indifference curve 2 and budget line I. The price of X increases and the consumer ends up on indifference curve 3 and budget line II. A subsidy is given to allow consumer to buy old bundle of goods BUT consumer now buys relative more of good Y and less of good X since Px/Py has risen. # 2 page 99 Draw indifference curves for: Al likes beer, hates hamburgers U3 beerU2 U1 U3 > U2 > U1
Hamburger Betty indifferent between 3 beers or 2 hamburgers. Nothing changes no matter what the level of consumption. beer 9 6 3 2 4 6 Hamburger Chris eats one hamburger and one beer in fixed proportions. beer Hamburger 1. Antonio buys 8 new college textbooks at a cost of $50. 00 each. Used books were $30. 00 each. Next year prices for new books go up 20% and used books go up 10%. Antonio’s Dad sent him $80. 00 more. Is he better off? Cost last year for new books 8 * $50. 00 = $400. 00 Cost last year for used books 8 * $30. 00 = $240. 00 Relative price of new and used books last year 50/30 = 1.
67 Relative price of new and used books this year (50*1. 20)/(30*1. 1) = 60 / 33 = 1. 82 ? New books are relative more costly. Cost this year for the two choices would be 8*60= $480 8*33= $264 Dad gave him money to buy 8 new books this year. Will he still buy this bundle now that prices of new books are relatively higher? ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 2. Jane has a utility function U(F,c) = FC. Set up a table for U=12 and U = 24 Suppose Jane has $12. 00 to spend and Pf = $1. 00 and Pc = $3. 00 Show budget line What should she buy? (F=6 C =2 ). Ch3_1. xls
|Jane U(Fmc)=FC | | | | | | | | | | | | | |U(F,C) = FC |P food |1 |P Cloth |3 | | | | |U=12 | | | |U=24 | | F| | | | | | | | |C |Cost | |F |C |Cost | | 1 |12 |37 | |1 |24 |73 | |1. 5 |8 |25. 5 | |2 |12 |38 | |2 |6 |20 | |3
|8 |27 | |3 |4 |15 | |4 |6 |22 | |4 |3 |13 | |6 |4 |18 | |6 |2 |12 | |8 |3 |17 | |8 |1. 5 |12. 5 | |12 |2 |18 | |12 |1 |15 | |24 |1 |27 | Budget line is Y = Pf * F + Pc * C MRS = -1/3 Chapter 4 Individual Market Demand Demand Curve holds constant: Prices of other goods Income Tastes [pic] Assume Income available is I: A = I /P1y , B = I / P1x, C = I / P2x,D = I / P3x As Px falls the budget line moves from AB to AC to AD.
The possible set of goods increases. The consumer increases x purchases and Y purchases. The above diagram is same as figure 4. 1. Points to remember: As price of X falls there is a substutution effect that is due to the relative price change and an income effect which is due to the increase in real income due to the price decrease. If the price of X falls, the consumption of Y can increase, or decrease. Income Consumption Curve Traces out points of consumption of X and Y that occurs as income increases. If X and Y are both superior goods, as I increases the consumption of X and Y increases. => Demand curves for both goods shift right. Y ICC X
Here X is inferior at high income = > as I increases, consumption of X eventually falls. Y 3 2 1 X As Income rises X consumption first increases, then falls. As income rises the consumption of Y increases. Y is a normal good. X is a normal good for low incomes, an inferior good for high prices. In an N good world all goods can be normal goods. As a upper limit only N-1 can be inferior. => In a 2 good world cannot have two inferior goods. Income and substitution effects can be drawn 4 ways for a price fall and 4 ways for a price rise assuming normal goods. For the inferior not giffen and inferior giffen there are 8 other ways each. Total number of cases = 24.
Inferior not Giffen = > good is inferior but not so inferior that the income effect out weighs substitution effect. Four ways to draw cases: Hicks method (American assumptions). => Assume base case. Prices change causing the budget line to rotate (See figure 4. 6). Helping budget line drawn tangent to old indifference curve but parallel to new budget line. Slutsky method (American assumptions) => Assume base case. Prices change causing the budget line to rotate . Helping budget line drawn through old basket but parallel to new budget line. Hicks method (European Assumptions). => Assume base case. Prices change causing the budget line to rotate .
Helping budget line drawn tangent to new indifference curve but parallel to old budget line. Slutsky method (European Assumptions) => Assume base case. Prices change causing the budget line to rotate . Helping budget line drawn through new basket but parallel to old budget line. For a price fall Hicks (Slutsky) method with American assumptions looks like a price increase for Hicks (Slutsky) with European assumptions. We assume American assumptions for this course!!! Define AB = substitution effect using Hicks AB* = substitution using Stutsky BC = income effect using Hicks B*C = income efffect using Stutsky AC = Total Effect Assume P X falls. Normal good = > ABB*C
Inferior not Giffen=> ACB*B Inferior Giffen->CB*BA Y a 2 1 IIIII I A B B* C c d X We assume income is fixed. Original budget line is ac. Consumer consumes A of X. Price falls => budget line shifts out to ad. Consumer now obtains C of X. Hicks method draws helping budget line 1 tangent to old indifference curve I but parallel to new budget line. AC broken into AB (Hicks substitution effect) and BC Hicks income effect. Slutsky method draws helping budget line 2 through old point. Consumer is overcompensated and substitution effect is now AB* and income effect B*C. Slutsky American method is how Lasperse price index ([pic]) is constructed.
Slutsky European method is how Paasche price index ([pic]) is constructed. Figure 4. 6 shows Hicks method for a price fall of a normal good. In terms of ABC, F1 = A, E = B and F2 = C. Figure 4. 6 can be used to show the Hicks method using European assumptions for a price increase. Figure 4. 7 shows Hicks method for an inferior good not Giffen good. Figure 4. 8 shows Hicks method for an inferior good that is Giffen. Figure 4. 9 shows the effect of a Gas tax with total rebate. Gas tax => relative price of gas rises => relative consumption of gas falls. BUT the gas tax involves an negative income effect. State elects to rebate the full tax to each person on an average basis.
=> heavy users of gas will be at less utility, low users of gas will benefit since they will be over compensated. Graph shows effect on heavy user. For all users relative price of gas has increased. Market demand = sum of all private demands PriceABCMarket 16101632 2481325 3261018 404 711 5024 6 At price > 3 consumer A is out of the market. Assuming demand curve is elastic (inelastic) => price fall inplies that revenue increases (decreases). Point elasticity = (P/Q)*(1/slope) For straight line demand curves elasticity = point elasticity. Not all demand curves are straight lines. Arc elasticity=[pic][pic][pic] Arc Income elasticity = [pic][pic]
Demand curves have kinks as consumers jump into the market. See figure 4. 11 P Domestic Demand World DMarket Demand Q Consumer surplus = total benefit of consuming = difference between what a consumer is willing to pay for a product and what consumer actually pays. A B E C D Consumer surplus = area ABE = AB * BE / 2 = (P * Q) / ((d * 2) (d= BC / AB, P*Q= BC * CD =>(P * Q) / ((d * 2) = (BC * CD)/(2*BC/AB) =(BC*CD*AB)/(2*BC) =(AB*BE/2) Value of clean air: Would consumers maintain their cars better if the exhaust was on front of car? Why? Bandwagon effect=> You like the good better if others get it.
Bandwagon effect makes market more elastic. Retail marketing based on creating bandwagon effect. MJ paid to wear a special shoe. Snob Effect=> You like the good less the more others have it. Snob effect makes market demand less elastic. Table 4. 5 data is in Excel file Table4_5. Estimate [pic] and [pic] Do you get what the book gets? (Hint I do not!!! ) Math Treatment Assuming goods X and Y and income I, Customer wants to (1)Max U(X,Y) Given budget constraint (2)I = PxX + PyY Form Lagrangian or the function to be maximized (3)[pic] If the budget constraint is satisfied => second term = 0. Differentiate Lagrangian with respect to X, Y and [pic]and set to zero.
Obtain (4)[pic] (5)[pic] (6)[pic] where [pic] etc Equations (4) and (5) indicate that the consumption of X and Y will proceed until the marginal utility is a multiple [pic] of P or (7)[pic] or (8)[pic] Equation (8) expresses an equilibrium condition. This equation is satisfied where the indifference curve is tangent to the budget line. Along an indifference curve the utility is fixed or (9)[pic] As we move along the indifference curve (10)[pic] can express the MRSXY as (11)[pic] or the ratio of the marginal utilities. Practical case using Cobb-Douglas utility function. This section based on Appendix to Chapter 4. (12)[pic] which can be estimated as
(13)[pic] forming the Lagrangian and solving we find (14)[pic] (15)[pic] (16)[pic] (17)[pic] Solving (15) and (16) for PXX and PYY and substituting in the budget equation gives (18)[pic] which implies that [pic]. Form the first two equation we get (19)[pic] and 20) [pic]. The Cobb-Douglas utility function implies that the consumption of each good depends only on the price of that good and income, not the price of the other good or that cross elasticities of demand are zero. The Lagrange multiplier [pic]represents the extra utility generated by one more dollar. Taking the total derivative (21)[pic] since any increase in income is spent (22)[pic]
since from (4) and (5) [pic] and [pic] then (23)[pic] substituting for [pic] gives (24)[pic] An example. Assume [pic], PX=$1. 00 and PY=$2. 00 and I = $100. From (19) & (20) [pic] and [pic]. Thus X = 50 and Y = 25. Since [pic], then [pic]=1/100. If income were to increase to $101, X=50. 5 and Y=25. 25. The original utility was [pic] while the new utility was [pic]. [pic] was . 01 which is the exact difference in utility. (12) implies that if the consumption of X and Y were to increase by [pic], then total utility goes up by [pic]. In a more general case 25) [pic] where in general [pic]. Here if X and Y increase by [pic], then utility goes up by [pic].
Direct differentiation of (25) with respect to X and Y gives 26) [pic] 27) [pic] Equations (27) and (28) show that if Y is fixed and X is increased the marginal utility of each additional X will decline. An additive utility function will not have this property. Duality in Consumption Theory states that maximizing utility for a given budget is equivalent to minimizing the cost of obtaining a given level of utility. This can be best seen with indifference curves and budget lines. Discussion Problem # 1 Sally Henin has an arc price elasticity of demand for gasoline of -. 8. Her income elasticity for gas is . 5.
Sally has a current income of $40,000 per year and spends $800 per year for gas. The current price of gas is $1. 00. a. The government is concerned about energy usage and is contemplating an excise tax which will cause gas prices to increase to $1. 40. What will happen to Sally’s consumption? b. The Government is considering a $200. 00 tax rebate. How will this impact Sally? c. Assume both the tax and the rebate are implemented. Is Sally worse off or not? a. Solve arc elasticity formula for new Q b. Using new Q and arc income elasticity formula sub in new income and recalculate Q c. See how much she spends before and after [pic][pic] [pic][pic] [pic][pic] [pic] [pic] [pic] [pic]
[pic] On final indifference curve she spends 613. 53 * 1. 40 = $858. 94 . With her income and the rebate she would have 40200-858. 94 = $39,341 to spend on goods. If she bought 614 at the old price she would have had 40000-614 = $39386 Sally is NOT happy with the government!! Problem # 2 The San Francisco Chronicle reported that the toll on the Golden gate bridge was raised from $2. 00 to $3. 00. Following the toll increase traffic fell 5%. What is the point price elasticity of demand? Steve Leonoudakis chairmen of the bridge’s finance auditing committee warned that the toll increase could cause toll revenues to decrease by $2. 8 million per year.
Is this statement consistent with Economic Theory? Elasticity = [pic] = -. 05/. 50 = -. 1 or inelastic demand Steve is wrong. Inelastic demand => revenue increases as price increases. Chapter 5 Uncertainty and consumer behavior not covered in detail. Fines: Individual balances “gain from breaking the law” against expected fine Double park=> utility gain $5. 00 Fine=> $50. 00 where probability of getting caught = . 1 Fine=> $500. 00 where probability of getting caught = . 01 These cases imply enforcement costs = 0. 0. Most drivers who do not like risk will not double park. In case of music piracy difficult to catch. => can have very high fines. Chapter 6 Production
Firms turn inputs or factors of production into outputs via a production function Q=F(K,L). A production function implies a given technology. Firm uses each input as efficiently as possible. Isoquant (production indifference curves) show different quantities of inputs (say L and K) such that output is fixed. L Q=30 Q=20 Q=10 K In short run not all inputs can be changed. In the long run all inputs can be changed. Given K, Average product = Q/L, Marginal product = (Q/(L. Q = Total product. When Marginal Product > Average Product => average product is increasing. Figure 6. 1 shows relationship between AP, TP and MP. Why would a firm never operate where MP < 0?
MP reaches a peak at point of inflection. of TP curve. Figure 6. 2 shows how output per time period goes up as technology improves. Figure 6. 4 shows that cereal yield and the food price index appear to be negatively related. Why might this be so? Except for a brief period in the 70’s, food prices have declined. Law of diminishing returns => with other inputs fixed, MP of an input declines as the amount of that input used increases. Malthus believed that the earth would not support population growth. To date he has been wrong. Table 6. 3 shows Annual Growth of Labor Productivity in various countries. It is shown below: YearUSJapanFranceGermanyUK 1960-19732. 297. 864. 703. 982. 84
1974-1982 . 222. 291. 732. 281. 53 1983-19911. 542. 641. 502. 071. 57 1992-20001. 941. 081. 401. 642. 22 2001-2009 1. 90 1. 50 . 90 . 80 1. 30 GDP per hour worked in 2009 dollars $56. 90$38. 20$54. 70$53. 10$45. 80 A technological change raises the production function. Here it is occurring over time. [pic] In any particular year the aggregate value of goods and services produced by en economy is equal to the payments made to all factors of production. We can estimate a production function in log form with Excel. Take Data from 6. 4 and fit model. See ch6_1. xls [pic] |Production Function data from Table 6.
4 of Pindyck-Rubinfeld (2005) | | | | | | | | | | | | | | | | | | | | | |Multiple R |0. 961637 | | | | | | | | | | | | | | | | |Observation |Predicted lnq |Residuals | |Yhat |Actual |Error | | | |1 |3. 394232 |-0. 3985 | |29.
79175 |20 |9. 79175 | | | |2 |3. 72906 |-0. 04018 | |41. 63997 |40 |1. 639968 | | | |3 |3. 924923 |0. 08241 | |50. 64917 |55 |-4. 35083 | | | |4 |4. 063889 |0. 110498 | |58. 20024 |65 |-6. 79976 | | | |5 |4. 17168 |0. 145808 | |64. 82428 |75 |-10. 1757 | | | |6 |3. 72906 |-0. 04018 | |41. 63997 |40 |1. 639968 | | | |7 |4. 063889 |0. 030455 | |58. 20024 |60 |-1. 79976 | | | |8 |4. 259752 |0. 057736 | |70. 79241 |75 |-4. 20759 | | | |9 |4. 398718 |0. 043933 | |81. 34655 |85 |-3. 65345 | | | |10 |4. 506509 |-0. 0067 | |90. 60499 |90 |0. 604987 | | | |11 |3. 924923 |0. 08241 | |50. 64917 |55 |-4. 35083 | | | |12 |4. 259752 |0. 057736 | |70. 79241 |75 |-4.
20759 | | | |13 |4. 455614 |0. 044195 | |86. 10902 |90 |-3. 89098 | | | |14 |4. 594581 |0. 010589 | |98. 94665 |100 |-1. 05335 | | | |15 |4. 702372 |-0. 04841 | |110. 2082 |105 |5. 208238 | | | |16 |4. 063889 |0. 110498 | |58. 20024 |65 |-6. 79976 | | | |17 |4. 398718 |0. 043933 | |81. 34655 |85 |-3. 65345 | | | |18 |4. 594581 |0. 010589 | |98. 94665 |100 |-1. 05335 | | | |19 |4. 733547 |-0. 03307 | |113. 6982 |110 |3. 698178 | | | |20 |4. 841338 |-0. 09641 | |126. 6387 |115 |11. 63871 | | | |21 |4. 17168 |0. 145808 | |64. 82428 |75 |-10. 1757 | | | |22 |4. 506509 |-0. 0067 | |90. 60499 |90 |0. 604987 | | | |23 |4. 702372 |-0. 04841 | |110. 2082 |105 |5.
208238 | | | |24 |4. 841338 |-0. 09641 | |126. 6387 |115 |11. 63871 | | | |25 |4. 949129 |-0. 16164 | |141. 0521 |120 |21. 05206 | | | | The above Excel output shows how data can be used. Figure 6. 4 graphs this data. Marginal rate of technical substitution = [pic] for a given Q (K on vertical axis). [pic] since along isoquant [pic] When inputs are substitutes => isoquants are straight lines. When inputs are complements => isoquants are right angles Assuming a Cobb-Douglas production function sum of coefficients determines returns to scale. Assume all inputs increase by [pic]. Get [pic] Example: MPL= 50, MRTS = . 25. What is MPK? [pic]. =>[pic]
Example: Determine whether decreasing, constant or increasing returns. A. [pic] or increasing returns. B. [pic] or constant returns. Example: Assuming Q=100(K. 8L. 2) starting K=4 and L=49 show MP of labor and K are declining [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] Chapter 7 Cost of Production Accounting Cost – concerned with historical expenditures. Economic Cost takes into account opportunity cost that has to be taken into consideration in the decision process (It is more costly for a doctor to mow the lawn on Monday at 10:00 than a janitor). Sunk costs – costs that cannot be recovered. Valuable Alternative use of land => high opportunity cost of present use.
Zoning in many cases reduces opportunity cost. Total Cost (TC) = Variable Cost (VC) + Fixed Cost (FC) Marginal Cost (MC) =[pic] Average Cost (AC) = [pic] Table 7. 1 illustrates FC, AC VC MC, AFC, AVC, ATC In the short run can change labor to increase output. Given the wage [pic], [pic] Assuming labor is the only variable input [pic] where[pic] is the average product of labor. Figure 7. 1 The MC curve goes through the minimum of the ATC and AVC but not the AFC. As [pic]. In the long run all inputs are variable. Isocost line (= production budget line) is locus of points shows different combinations of inputs such that cost is fixed. [pic] solving for K gives [pic] Isocost slope [pic]
of ratio of wage rate to rental cost of capital. Want to maximize production for a given total cost TC. K A=TC/Pk AB=TC/PL Q B L The[pic]. In equilibrium the slope of the isocost [pic] = MRTS [pic] or [pic] Figure 7. 5 show effect of an effluent fee raising the relative price of the input water. Production moves to be less water intensive. Expansion path shows the amounts of labor and capital used as firm grows (figure 7. 6). This is a long run concept. K Long Run Short Run L Economies of scale includes increasing returns to scale as a special case. Increasing returns to scale requires inputs be used in fixed proportions. Economies of scale allows the ratio of inputs to change. Figure 7.
6 Long-Run Cost with Constant Returns to scale Figure 7. 10 Long-Run Cost with first Economies then Diseconomies of Scale. The LAC traces tangency points of SAC curves. At minimum point of LAC the LMC and SMC intersect. Economics of scale => MC < AC. Diseconomies of scale => MC > AC. Define [pic]= elasticity of total cost [pic] Economics of scale => [pic] of MC < AC. Economies of scope are present when joint output of a single firm producing two products is greater that the output of two firms each producing one product. Define [pic] as cost of producing [pic] of the ith good. Economies of scope => SC > 0 [pic] Diseconomies of scope SC < 0
Learning Curve => Over time labor costs go down as more units are produced. Define L as labor per unit, A> 0, B>0, 0 A to C A B CAC1 AC2 Q In the early life of a product the learning curve is steep. After while the “learning” slows down. In chip business had a 20% learning curve. => 10% increase in cumulative production => costs fall 2%. In aircraft industry learning curve rate was 40%. Estimating a cost curve Linear Model used if MC is constant at [pic]. [pic] [pic] Quadratic Model used if have straight line MC. [pic] [pic] Cubic cost function used if have U shaped MC curve of form [pic] [pic] Can use Excel and other software to estimate the MC curve. Problem 12 page 272.
Computer firm cost is [pic] where CQ is cumulative quantity, and Q is quantity per year. a. There is a learning curve effect since -. 1 b. There is decreasing returns to scale? [pic] [pic] Since [pic] => decreasing returns to scale c. CQt=40,000, Qt = 10,000, Qt+1= 12,000 [pic] Problem 13 page 272. Assume [pic] [pic]. AC not defined for Q=0. For a U shaped cost curve we want c0. At minimum point of AC, MC=AC. Set a=0 to normalize y axis and equate MC=AC. => [pic] or [pic]. If c=-1 and d=1 then Q=1/2 Key Question: How do we use this theory to solve a practical problem? Firm has a production function [pic] which could be estimated using Excel when we note that [pic].
Want to determine a general case that shows the optimum K and L given the cost of capital r and the wage w. The following is a simpler treatment than 273-278. In equilibrium we note that (1)[pic] Given the production function (2)[pic] (3)[pic] (4)[pic] (5)[pic] Substituting (5) into the production function (2) gives (6)[pic] (7)[pic] The equilibrium amount of capital is (8)[pic] If the cost of capital, r, increases or the marginal product of capital, [pic], falls K will fall. Direct substitution of (8) into (5) gives (9)[pic][pic] If the wage rate, w, increases or the marginal product of labor, [pic], falls then less labor will be used. The firms cost function is 10) [pic]
At equilibrium 11) [pic] Equation (11) shows how C is related to wages, w, cost of capital, r, and returns to scale [pic]. MATLAB can be used to calculate the appropriate derivatives Equation (11) shows how costs are related to the production function of the firm as well and input costs. See ch7_1. xls for solution. Chapter 8 Profit Maximization and Competitive Supply Firms want to maximize revenue. Competitive supply => many firms, all price takers, no product differentiation, free entry Profit is the difference between revenue and cost for a given level of output q. 1) [pic] To maximize profit differentiate and set to zero. => MR(q) = MC(q)
2) [pic] For a competitive firm P is given => [pic] and P=MR. It can be proved that 3) [pic] 4) [pic] [pic] Figure 8. 1 shows profit maximization. Figure 8. 2 shows firm in competitive market is a price taker. Figure 8. 3 shows a competitive firm (MR=AR) making extra normal profits and operating with decreasing returns. In long run new firms will enter and firm will earn normal profits or those profits such that no firms enter or no firms leave industry. Figure 8. 5 show that in the short run the firms MC curve is the firm supply curve. If an input increases in price => AC curve shifts up => MC curve shifts up and firm produces less output.
Book indicates that short run MC of petroleum has flat sections until other refineries get on the line. See figure 8. 8. ComEd’s costs are constant until they start jet engine generators. Industry supply curve is the sum of firm MC curves in the short run. Producer Surplus = area above firm supply curve and below price. See figure 8. 11. In long run firm can alter all inputs. If industry is making extra normal profits in the short run => new firms will enter such that in the long run all firms are making normal profits. The long run competitive equilibrium point is P=SMC=LMC. Firm is at the minimum point of its AC curve. Given the firm’s technology, it is most efficient. It earns economic profit (normal profit). See figure 8. 13.
In long run firm tries to equate LRMC to price BUT new firms can enter the industry and drive down price. Economic profit = R – wL – rK where rK is the opportunity cost of capital. A firm making zero economic profits may still stay in business. Economic rent = return to factors of production in limited supply. MJ earns economic rent. Songs and books are copyrighted to allow economic rent to be earned for a limited number of years. The long run supply of a constant cost industry is a straight line. This is a partial equilibrium argument. As Q increases input prices will increase at some time. General Equilibrium has all assumptions variable. Some industries are so small that they can be assumed to have no effect on input prices. See figure 8. 15
Increasing Cost Case => If extra normal profits are earned new firms will enter the industry and input prices will increase. See figure 8. 17. Taxes. An output tax raises the MC curve. A Lump Sum Tax (such as a liquor license) does not change the MC since it is paid no matter what. The effect of a n output tax in the long run is to reduce industry supply. Perfectly Competitive Markets Assume: – Price Taking – Product Homogeneity – Perfect Mobility of Assets – Perfect Information Even if only one firm in the market it may pay to act as if the firm was in a competitive market since extra normal profits => others firms may enter to bid away profit. Problem # 1 The Conigan box company is in a competitive market. It sells boxes in batches of 100 for $100. 00. [pic].
What is profit maximizing output? Want MC=MR=AR. => [pic] What is profit? [pic] The firm should stay in business in the short run since AVC < P: [pic] [pic] [pic] Example: Restaurant stays open in “off hours” if it can cover labor costs. Problem # 2 Competitive market has a market demand curve of [pic] and supply curve of [pic] where Q = where a typical firm has [pic]. What is market price and rate of sales? We set S = D or [pic] Each firm sells where MR = MC . Since perfect competition MR=AR=P=$37. 50 [pic] Note: If you use ch2_1. xls template to solve system remember that the template is set up as: [pic] Chapter 9 Analysis of Competitive Markets
Want to look at welfare and efficiency effects of a competitive market. Figure 9. 1 shows producer (area between supply curve and market price) and consumer (area between demand curve and market price) surplus. Price Controls: Figure 9. 2 and 9. 3 show changes in producer and consumer surplus when the government put on a price control. The more inelastic the demand curve the more the deadweight loss. If supply is fixed there is no dead weight loss. Looking at figure. Producer surplus loss = A+C. Consumer net gain = A-C-B. Unless demand is totally inelastic a price control => a shortage. Price controls can have a serious long term effect on market as was case with NY housing.
Figures 9. 4 & 9. 5 contrast the effects to price being held below and above the market price. Kidney Market. (See figure 9. 6) Supply function of [pic] implies that if the price is held to $0. 0 by the 1984 National Organ Transplantation Act, and you cannot sell an organ, only 16000 will be supplied. The demand function is [pic][pic]. Equating [pic] On figure 9. 6 suppliers lost A+C. If supply is reduced due to not being able to sell kidneys, the market clearing price is $40,000 and many consumers are rationed on a “willingness to pay” A+D = total value of kidneys at new market clearing price. Middlemen and hospitals get gain NOT donators.
Minimum Prices (Government restricts market from lowering prices. Fair trade laws. Tariffs. ) (See figure 9. 7). P1 A B P2 C Q3 Q0 Q1 Government raises price from P2 to P1. Supplier produces Q1 not Q3 > “glut” From free market consumer surplus loss = B. Producer surplus loss = C (Assuming producer produces Q3) Minimum Wage Figure 9. 8 shows the analysis for minimum wage which causes unemployment of L2-L1. Price Supports. In contrast to minimum price laws, price supports require that government buy farm products. Figure 9. 11 shows that as a result of the purchase program production when from Q0 to Q2 and price from P0 to Ps.
Consumer surplus reduced by = B+A. Producer gain = A+B+D. Government cost = Ps(Q2-Q1). Change in welfare (A+B+D)-(A+B)-[Ps(Q2-Q1)]. Production Quotas. Government reduces supply (taxi licenses, bar permits). NYC has 113,150 taxi licenses. Roughly same as 1937. Medallion costs $880,000. Cost in 1947 was $2,500. In 1980 was $55,000. If supply went up current medallion owners would see a loss of value of their medallions. Drives cab ride prices very high. See Figure 9. 13. In NYC cabs are run 24/7, not shut off at night. Acreage limitation programs give farmers money incentive to leave land idle. Causes farmers to farm their land more intensively. Figure 9.
11 suggests cost to government must at least equal B+C+D of the gain from planting since farmer sees new higher price as given. [pic] Thus [pic] Total change in welfare [pic] Society would be better off if Government just gave producers A+B+D since what government would lose farmers would gain for a zero net effect. Book Example page 336 figure 9. 12 shows how to use above theory: In 1981[pic] [pic] Government must increase demand to raise the price. Let [pic] be government demand [pic] Given desired P, [pic][pic] is determined. If P=$3. 70, => government buys 122 million 2688-566) bushels at cost of $451. 4 million. Inspect figure 9. 12. Customers lose A((3. 7-3. 46)*2566) + B(. 5*(3. 70-3.
46)(2630-2566) or 616+8=624 million. Note that A is substantially bigger that B in this problem. Total cost = 624+451. 4=1075. 4 Exact answers for this problem and related ones are in ch9_1. xls. As an exercise run the problem where the desired price is $3. 80. Adjust demand and supply and see what happens. If P = $4. 00, => government buys 506*4-1750 = 274 at a cost of $1096 million. Calculate A and B here. Quotas and Tariffs. Figures 9. 14 and 9. 15 shows basic setup. (9. 14 shows a tariff that eliminates all imports). Consider figure 9. 15. With free trade the price is Pw domestic supply is Qs, domestic demand is Qd. Imports are (Qd. -Qs). After a tariff (or quota).
Domestic supply jumps to Q’s, domestic demand falls to Q’d and imports fall to Q’d-Q’s. Trapezoid A = gain to producers. Due to higher price consumer loss is A+B+C+D. If a tariff is imposed, Gov gains revenue D. Net domestic loss is B+C+D. Excel file ch9_2 shows setup to analyze Sugar Case. A tariff of 83% gives book case. Can adjust tariff to eliminate imports and thus tariff revenue (area D). Product Taxes. Figure 9. 16 and 9. 17 shows effect on producers and consumers of a per unit tax on producers. Area A + B = what buyers lose. Area D + C = what sellers lose. Are A + D = what government collects. [pic] = elasticity of supply. [pic] = elasticity of demand. A tax falls mostly on the buyer (seller) if [pic] is small (large).
Pass-through (to consumers) fraction is [pic]. Fraction of tax producers bear is [pic] NOTE: These formulas make use of fact that [pic]. Most luxury goods have inelastic demand => luxury good tax hits consumers hard. Example 9. 7 Tax on Gas. See figure 9. 20 and Excel 9_3. xls We are given that [pic]. From Chapter 2 and excel ch2_2. xls we know that Given QD = a – bP (11) QS = c + dP (12) And P* and Q* are equilibrium P and Q, then at equilibrium the elasticity of demand (D and elasticity of supply (S are (D = -b(P*/Q*) (13) (S = d(P*/Q*) (14) This implies [pic]. With no governmental tax [pic] After Government places a $1. 00 tax the new price is $2.
44 and the new quantity is 89. The producer price falls to $1. 44. The steeper the supply curve the less quantity falls and the more the producer price falls. The more inelastic the demand curve the more the consumer price increases. In this case a greater proportion of the tax is passed on. The following calculations replicate the figure 9. 20 [pic] Problem # 12 page 354 has the demand and supply of hula beans. See p354_12. xls [pic]. The world trade price is $. 60. Congress puts on a tariff of $. 40. Use of ch9_2. xls implies that Equilibrium P and Q are $1. 00 and 50 if there was no foreign supply. If the tariff is imposed at . 666666 then the domestic