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### ch09

Course: ECON ECON111, Spring 2009
School: Punjab Engineering...
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9 PROFIT CHAPTER MAXIMIZATION Problems in this chapter consist mainly of applications of the P = MC rule for profit maximization by a price-taking firm. A few of the problems (9.29.5) ask students to derive marginal revenue concepts, but this concept is not really used in the monopoly context until Chapter 13. The problems are also concerned only with the construction of supply curves and related concepts since...

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9 PROFIT CHAPTER MAXIMIZATION Problems in this chapter consist mainly of applications of the P = MC rule for profit maximization by a price-taking firm. A few of the problems (9.29.5) ask students to derive marginal revenue concepts, but this concept is not really used in the monopoly context until Chapter 13. The problems are also concerned only with the construction of supply curves and related concepts since the details of price determination have not yet been developed in the text. Comments on Problems 9.1 9.2 A very simple application of the P = MC rule. Results in a linear supply curve. Easy problem that shows that a tax on profits will not affect the profit-maximization output choice unless it affects the relationship between marginal revenue and marginal cost. Practice with calculating the marginal revenue curve for a variety of demand curves. Uses the MR-MC condition to illustrate third degree price discrimination. Instructors might point out the general result here (which is discussed more fully in Chapter 13) that, assuming marginal costs are the same in the two markets, marginal revenues should also be equal and that implies price will be higher in the market in which demand is less elastic. An algebraic example of the supply function concept. This is a good illustration of why supply curves are in reality only two-dimensional representations of multi-variable functions. An introduction to the theory of supply under uncertainty. This example shows that setting expected price equal to marginal cost does indeed maximize expected revenues, but that, for risk-averse firms, this may not maximize expected utility. Part (d) asks students to calculate the value of better information. A simple use of the profit function with fixed proportions technology. This is a conceptual examination of the effect of changes in output price on input demand. A very brief introduction to the CES profit function. This problem describes some additional mathematical relationships that can be derived from the profit function. 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 38 Solutions 9.1 a. MC = C q = 0.2 q + 10 set MC = P = 20, yields q* = 50 b. = Pq C = 1000 800 = 200 c. 9.2 ( q) = R ( q) C ( q) With a lump sum tax T ( q) = R( q) C ( q) T R C = 0 = 0 MR = MC , no change q q q Proportional tax ( q) = (1 t )[ R( q) C ( q)] MR = MC , no change = (1 t )( MR MC ) = 0, q Tax per unit ( q) = R ( q) C ( q) tq = MR MC t = 0 , so MR = MC + t, q is changed: a per unit tax does affect output. q dP q a 2q a = + q (1/ b) = dq b b 9.3 a. q = a + bP, MR = P + q Hence, q = (a + bMR)/2. Because the distance between the vertical axis and the demand curve is q = a + bP, it is obvious that the marginal revenue curve bisects this distance for any line parallel to the horizontal axis. b. If q = a + bP ; b < 0; P = MR = 2q a b qa b 1 q b P MR = 39 Chapter 9/Profit Maximization y 40 c. Constant elasticity demand curve: q = aPb, where b is the price elasticity of demand. MR = P + q P q = q a 1/ b ( q / a )1 / b + b Thus, vertical distance = P MR = b < 0) ( q / a )1 / b P (which is positive because = b b d. If eq,P < 0 (downward-sloping demand curve), then marginal revenue will be less than price. Hence, vertical distance will be given by P MR. dP dP dq , vertical distance is q , and since = b is the slope of dq dP dq 1 the tangent linear demand curve, the distance becomes q as in Part (b). b Since MR = P + q e. 9.4 Total cost = C = .25q2 = .25(qA + qL)2 qA = 100 2PA PA = 50 qA/2 qL = 100 4PL PL = 25 qL/4 2 2 R A = P A q A = 50q A q A / 2 RL = PLqL = 25q L q L / 4 MRA = 50 qA MCA = .5(qA + qL) MRL = 25 qL/2 MCL = .5(qA + qL) 40 Set MRA = MCA and 50 qA = .5qA + .5qL MRL = MCL 25 qL = .5qA + .5qL these 2 Solving simultaneously gives qA = 30 qL = 10 9.5 PA = 35 PL = 22.5 = 1050 + 225 400 = 875 C = wl = wq 2 / 4 . a. Since q = 2 l , q 2 = 4l Profit maximization requires P = MC = 2wq/4. Solving for q yields q = 2P/w. b. Doubling P and w does not change profit-maximizing output level. = Pq TC = 2P2/w P2/w = P2/w, which is homogeneous of degree one in P and w. c. It is algebraically obvious that increases in w reduce quantity supplied at each given P. 9.6 a. Expected profits = E() = .5[30q C(q)] + .5[20q C(q)] = 25q C(q). Notice 25 = E(P) determines expected profits. For profit maximum set E(P) = MC = q + 5 so q = 20 E() = E(P)q C(q) = 500 400 = 100. b. In the two states of the world profits are P = 30 = 600 400 = 200 P = 20 = 400 400 = 0 and expected utility is given by E (U ) = .5 200 + .5 0 = 7.1 c. Output levels between 13 and 19 all yield greater utility than does q = 20. Reductions in profits from producing less when P is high are compensated for (in utility terms) by increases in profits when P is low. Calculating true maximum expected utility is difficultit is approximately q = 17. d. If can predict P, set P = MC in each state of the world. When P = 30 q = 25 = 212.5, P = 20 q = 15 = 12.5 E() = 112.5 E (U ) = .5 212.5 + .5 12.5 = 9.06 a substantial improvement. 9.7 a. In order for the second order condition for profit maximization to be satisfied, marginal cost must be decreasing which, in this case, requires diminishing returns to scale. 41 Chapter 9/Profit Maximization y 42 b. q = 10k 0.5 = 10l 0.5 so k = l = q 2 100 C = vk + wl = q 2 ( v + w) 100 Profit maximization requires P = MC = q( v + w) 50 or q = 50 P (v + w) . ( v, w, P ) = Pq C = 50 P 2 ( v + w) [50 P ( v + w)]2 ( v + w) 100 = 25P 2 ( v + w) . c. If v = 1000, w = 500, P = 600 then q = 20, = 6000 . d. If v = 1000, w = 500, P = 900 then q = 30, = 13500 . e. 9.8 a. With marginal cost increasing, an increase in P will be met by an increase in q. To produce this extra output, more of each input will be hired (unless an input is inferior). b. The Cobb-Douglas case is best illustrated in two of the examples in Chapter 9. In Example 9.4, the short-run profit function exhibits a positive effect of P on labor demand. A similar result holds in Example 9.5 where holding a third input constant leads to increasing marginal cost. c. l P = [ w] P = 2 Pw = q w . The sign of the final derivative may be negative if l is an inferior input. 9.9 b. Diminishing returns is required if MC is to be increasingthe required second order condition for profit maximization. c. determines how easily firms can adapt to differing input prices and thereby shows the profitability obtainable from a given set of exogenous prices. d. q = P = K (1 ) 1 P 1 ( v1 + w1 ) (1 )( 1) . This supply function shows that does not affect the supply elasticity directly, but it does affect the shift term that involves input prices. Larger values for imply smaller shifts in the supply relationship for given changes in input prices. e. See the results provided in Sydsaeter, Strom, and Berck. 42 9.10 a. l v = 2 vw = 2 wv = k w . This shows that cross price effects in input demand are equal. The result is similar to the equality of compensated cross-price effects in demand theory. b. The direction of effect depends on whether capital and labor are substitutable or complementary inputs. c. q w = 2 wP = 2 Pw = l P . This shows that increases in wages have the same effect on reducing output that a fall in the product price has on reducing labor demand. This is, the effects of wages and prices are in some ways symmetrical. d. Because it seems likely that l P > 0 (see Problem 9.8), we can conclude that q w < 0 that is, a tax on labor input should reduce output. 43
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Punjab Engineering College - ECON - ECON111
Chapter 9Production FunctionsProduction Function The firms production function for a particular good (q) shows the maximum amount of the good that can be produced using alternative combinations of capital (k) and labor (l)q = f(k,l)Marginal Physical
Punjab Engineering College - ECON - ECON111
CHAPTER 10THE PARTIAL EQUILIBRIUM COMPETITIVE MODELThe problems in this chapter focus on competitive supply behavior in both the short and long runs. For short-run analysis, students are usually asked to construct the industry supply curve (by summing f
Punjab Engineering College - ECON - ECON111
Chapter 10Cost FunctionsDefinitions of Costs Accounting and economic costs are different Accountants stress out-of-pocket expenses, depreciation, and other bookkeeping entries economists focus more on opportunity cost Labor Costs to accountants, labor
Punjab Engineering College - ECON - ECON111
CHAPTER 11APPLYING THE COMPETITIVE MODELThe problems in this chapter are intended to illustrate the types of calculations made using simple competitive models for applied welfare analysis. Usually the problems start from a supply-demand framework much l
Punjab Engineering College - ECON - ECON111
Chapter 11Profit MaximizationThe Nature of Firms A firm is an association of individuals who have organized themselves for the purpose of turning inputs into outputs Different individuals will provide different types of inputs the nature of the contrac
Punjab Engineering College - ECON - ECON111
CHAPTER 12GENERAL EQUILIBRIUM AND WELFAREThe problems in this chapter focus primarily on the simple two-good general equilibrium model in which supply is represented by the production possibility frontier and demand by a set of indifference curves. Beca
Punjab Engineering College - ECON - ECON111
Chapter 12The Partial Equilibrium Competitive ModelMarket Demand Assume that there are only two goods (x and y) An individuals demand for x isMarket demand for X = x i ( px , py , I i )i =1nMarket DemandXpxIndividual 1s demand curvepxIndividu
Punjab Engineering College - ECON - ECON111
CHAPTER 13MONOPOLYThe problems in this chapter deal primarily with marginal revenue-marginal cost calculations in different contexts. For such problems, students primary difficulty is to remember that the marginal revenue concept requires differentiatio
Punjab Engineering College - ECON - ECON111
Chapter 13General Equilibrium and WelfarePerfectly Competitive Price System We assume all markets are perfectly competitive a large number of homogeneous goods both consumption goods and factors of production each good has an equilibrium price there
Punjab Engineering College - ECON - ECON111
CHAPTER 14TRADITIONAL MODELS OF IMPERFECT COMPETITIONThe problems in this chapter are of two types: analytical and essay. The analytical problems look at a few special cases of imperfectly competitive markets for which tractable results can be derived.
Punjab Engineering College - ECON - ECON111
Chapter 14MonopolyMonopoly A monopoly is a single supplier to a market This firm may choose to produce at any point on the market demand curve A monopoly exists because other firms find it unprofitable or impossible to enter the market Barriers to entr
Punjab Engineering College - ECON - ECON111
CHAPTER 15GAME THEORY MODELS OF PRICINGThe first six problems for this chapter are intended to illustrate the concept of Nash equilibrium in a variety of contexts. Many of them have only modest economic content, but are traditional game theory problems.
Punjab Engineering College - ECON - ECON111
Chapter 15Imperfect CompetitionShort-Run Decisions: Pricing &amp; Output When there are only a few firms in a market, predicting output and price can be difficult how aggressively do firms compete? how much information do firms have about rivals? how ofte
Punjab Engineering College - ECON - ECON111
CHAPTER 16LABOR MARKETSBecause the subject of labor demand was treated extensively in Chapter 9, the problems in this chapter focus primarily on labor supply and on equilibrium in the labor market. Most of the labor supply problems (16.116.6) start with
Punjab Engineering College - ECON - ECON111
CHAPTER 17CAPITAL MARKETSThe problems in this chapter are of two general types: (1) those that focus on intertemporal utility maximization and (2) those that ask students to make present discounted value calculations. Before undertaking the PDV problems
Punjab Engineering College - ECON - ECON111
CHAPTER 18UNCERTAINTY AND RISK AVERSIONMost of the problems in this chapter focus on illustrating the concept of risk aversion. That is, they assume that individuals have concave utility of wealth functions and therefore dislike variance in their wealth
Punjab Engineering College - ECON - ECON111
CHAPTER 19THE ECONOMICS OF INFORMATIONThe problems in this chapter stress the economic value of information and illustrate some of the consequences of imperfect information. Only a few of the problems involve complex calculations or utilize calculus max
Punjab Engineering College - ECON - ECON111
CHAPTER 20EXTERNALITIES AND PUBLIC GOODSThe problems in this chapter illustrate how externalities in consumption or production can affect the optimal allocation of resources and, in some cases, describe the remedial action that may be appropriate. Many
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Week 1 Lecture 1 An overview Introduction: Parametric Estimation vs. Nonparametric Estimation I: Parametric density estimation : Let Y1 , Y2 , . . . , Yn i.i.d. with density f (x), 2 R (or R2 , or R10 ). For instance, &quot; # 2 (x ) 1 exp ; 2 R; &gt; 0. f ; (x)
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Week 13 Lecture 24 Adaptive Wavelet Estimation Donoho and Johnstone (1995, JASA). Sketch of the proof. Consider the sequence model where yi = i + zi , i = 1; :; d and zi are independent normal N (0; 1) variables. Set r( ) = d 1 Pk^ k2 . The stein s 2 unbi
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Week 2 Lecture 3 General m . n Model: Let 2 Y1 ; Y2 , o. . , Yn i.i.d. R f; f (m) (x) dx M b Goal: Find f such that Z b f (x) f sup Ef 2Fon [0; 1] with density f 2 F, F =2CM n2m=(2m+1)(Note that K may not be nonnegative). The bias part is Z b Efn (x
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Week 3 Lecture 4 . n Model : Let Y1 ; Y2 , o. . , Yn i.i.d. on [ 1; 1] with density f 2 F , F = R 2 f; f (m) (x) dx M . b We have shown there is a kernel estimator fn such that supf 2FZb E fn (x)2f (x)Cn2m=(2m+1).Because it is hard to analyze the
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Week 4 Lecture 7 Optimal rate of convergence in the sup-norm Model : Let Y1 ; Y2 , . . . , Yn i.i.d. on [0; 1] with density f 2 F (M ), Hlder ball of order . Minimax rate : It can be shown that b inf sup E f^ f F (M ) 2f1Cn log n2 =(2 +1).For simp
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Week 7 Lecture 12 Fourier estimation and Linear Minimaxity An orthonormal basis for L2 ([0; 1]) is1 2k(x) (x)= =1 p2 cos (2 kt) ;2k(x) =p2 sin (2 kx) ; k1.f The periodic Sobolev class W2 (M ) is dened as F= f:jZ1f (m)2M ; f (j ) (0) = f (
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Week 9 Lecture 16 Quadratic functional estimation Model: Observe the sequence model: yi =i:i:d: i+n1=2ziwhere zi N (0; 1). The model comes from the white noise model (or many other models): dy (t) = f (t) dt + n 1=2 dB (t) , t 2 [0; 1] . Let f i (t)
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Week 12 Lecture 22 A group of students present Donoho and Johnstone (PTRF, 1994)?1Lecture 23 Review from the presentation Suppose we observe yi = wherei+ zi ; i = 1; :; n,=1is constrained to lie in a ball of radius C dened by lp norm, n o = ; k kp C
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