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aters and wine. You can tell because it takes France less labor to produce a unit of the good. 4. a. In Germany, it takes fewer workers to make either a television or a video camera. Germany has an absolute advantage in the production of both goods. b. Producing an additional television in Germany requires three workers. Shifting those three German workers will reduce video camera production by 3/4 of a camera. Producing an additional television set in Poland requires six workers, and shifting those workers from the other good reduces output of video cameras by 6/ 12 of a camera, or 1/2. Thus, the opportunity cost of producing televisions is lower in Poland, so Poland has the comparative advantage in the production of televisions. Note: Do not let the fractions like 3/4 of a camera or 1/2 of a video camera bother you. If either country was to expand television production by a significant amount—that is, lots more than one unit—then we will be talking about whole cameras and not fractional ones. You can also spot this conclusion by noticing that Poland’s absolute disadvantage is relatively lower in televisions, because Poland needs twice as many workers to produce a television but three times as many to produce a video camera, so the product with the relatively lower absolute disadvantage is Poland’s comparative advantage. c. Producing a video camera in Germany requires four workers, and shifting those four workers away from television production has an opportunity cost of 4/3 television sets. Producing a video camera in Poland requires 12 workers, and shifting those 12 workers away from television production has an opportunity cost of two television sets. Thus, the opportunity cost of producing video cameras is lower in Germany, and video cameras will be Germany’s comparative advantage. In this example, absolute advantage differs from comparative advantage. Germany has the absolute advantage in the production of both goods, but Poland has a comparative advantage in the production of televisions. e. Germany should specialize, at least to some extent, in the production of video cameras, export video cameras, and import televisions. Conversely, Poland should specialize, at least to some extent, in the production of televisions, export televisions, and import video cameras. d. 5. There are a number of possible advantages of intra-industry trade. Both nations can take advantage of extreme specialization and learning in certain kinds of cars with certain traits, like gas-efficient cars, luxury cars, sportutility vehicles, higher- and lower-quality
cars, and so on. Moreover, nations can take advantage of economies of scale, so that large companies will compete against each other across international borders, providing the benefits of competition and variety to customers. This same argument applies to trade between U.S. states, where people often buy products made by people of other states, even though a similar product is made within the boundaries of their own state. All states—and all countries—can benefit from this kind of competition and trade. 6. 554 Answer Key a. Start by plotting the points on a sketch diagram and then drawing a line through them. The following figure illustrates the average costs of production of semiconductors. The curve illustrates economies of scale by showing that as the scale increases—that is, as production at this particular factory goes up—the average cost of production declines. The economies of scale exist up to an output of 40,000 semiconductors; at higher outputs, the average cost of production does not seem to decline any further. b. At any quantity demanded above 40,000, this economy can take full advantage of economies of scale; that is, it can produce at the lowest cost per unit. Indeed, if the quantity demanded was quite high, like 500,000, then there could be a number of different factories all taking full advantage of economies of scale and competing with each other. If the quantity demanded falls below 40,000, then the economy by itself, without foreign trade, cannot take full advantage of economies of scale. c. The simplest answer to this question is that the small country could have a large enough factory to take full advantage of economies of scale, but then export most of the output. For semiconductors, countries like Taiwan and Korea have recently fit this description. Moreover, this country could also import semiconductors from other countries which also have large factories, thus getting the benefits of competition and variety. A slightly more complex answer is that the country can get these benefits of economies of scale without producing semiconductors, but simply by buying semiconductors made at low cost around the world. An economy, especially a smaller country, may well end up specializing and producing a few items on a large scale, but then trading those items for other items produced on a large scale, and thus gaining the benefits of economies of scale by trade, as well as by direct production. 7. A nation might restrict trade on imported products to protect an industry that is important for national security. For example, nation X and nation Y may be geopolitical
rivals, each with ambitions of increased political and economic strength. Even if nation Y has comparative advantage in the production of missile defense systems, it is unlikely that nation Y would seek to export those goods to nation X. It is also the case that, for some nations, the production of a particular good is a key component of national identity. In Japan, the production of rice is culturally very important. It may be difficult for Japan to import rice from a nation like Vietnam, even if Vietnam has a comparative advantage in rice production. Chapter 20 1. This is the opposite case of the Work It Out feature. A reduced tariff is like a decrease in the cost of production, which is shown by a downward (or rightward) shift in the supply curve. 2. A subsidy is like a reduction in cost. This shifts the supply curve down (or to the right), driving the price of sugar down. If the subsidy is large enough, the price of sugar can fall below the cost of production faced by foreign This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Answer Key 555 producers, which means they will lose money on any sugar they produce and sell. 3. Trade barriers raise the price of goods in protected industries. If those products are inputs in other industries, it raises their production costs and then prices, so sales fall in those other industries. Lower sales lead to lower employment. Additionally, if the protected industries are consumer goods, their customers pay higher prices, which reduce demand for other consumer products and thus employment in those industries. 4. Trade based on comparative advantage raises the average wage rate economy-wide, though it can reduce the incomes of import-substituting industries. By moving away from a country’s comparative advantage, trade barriers do the opposite: they give workers in protected industries an advantage, while reducing the average wage economywide. 5. By raising incomes, trade tends to raise working conditions also, even though those conditions may not (yet) be equivalent to those in high-income countries. 6. They typically pay more than the next-best alternative. If a Nike firm did not pay workers at least as much as they would earn, for example, in a subsistence rural lifestyle, they many never come to work for Nike. 7. Since trade barriers raise prices, real incomes fall. The average worker would also earn less. 8. Workers working in other sectors and the protected sector see a decrease in their real wage. 9
. If imports can be sold at extremely low prices, domestic firms would have to match those prices to be competitive. By definition, matching prices would imply selling under cost and, therefore, losing money. Firms cannot sustain losses forever. When they leave the industry, importers can “take over,” raising prices to monopoly levels to cover their short-term losses and earn long-term profits. 10. Because low-income countries need to provide necessities—food, clothing, and shelter—to their people. In other words, they consider environmental quality a luxury. 11. Low-income countries can compete for jobs by reducing their environmental standards to attract business to their countries. This could lead to a competitive reduction in regulations, which would lead to greater environmental damage. While pollution management is a cost for businesses, it is tiny relative to other costs, like labor and adequate infrastructure. It is also costly for firms to locate far away from their customers, which many low-income countries are. 12. The decision should not be arbitrary or unnecessarily discriminatory. It should treat foreign companies the same way as domestic companies. It should be based on science. 13. Restricting imports today does not solve the problem. If anything, it makes it worse since it implies using up domestic sources of the products faster than if they are imported. Also, the national security argument can be used to support protection of nearly any product, not just things critical to our national security. 14. The effect of increasing standards may increase costs to the small exporting country. The supply curve of toys will shift to the left. Exports will decrease and toy prices will rise. Tariffs also raise prices. So the effect on the price of toys is the same. A tariff is a “second best” policy and also affects other sectors. However, a common standard across countries is a “first best” policy that attacks the problem at its root. 15. A free trade association offers free trade between its members, but each country can determine its own trade policy outside the association. A common market requires a common external trade policy in addition to free trade within the group. An economic union is a common market with coordinated fiscal and monetary policy. 16. International agreements can serve as a political counterweight to domestic special interests, thereby preventing stronger protectionist measures. 17. Reductions in tariffs, quotas, and other trade barriers, improved transportation, and communication media have made people more aware of what is available in the rest of the world. 556 Answer Key
18. Competition from firms with better or cheaper products can reduce a business’s profits, and may drive it out of business. Workers would similarly lose income or even their jobs. 19. Consumers get better or less expensive products. Businesses with the better or cheaper products increase their profits. Employees of those businesses earn more income. On balance, the gains outweigh the losses to a nation. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 References 557 REFERENCES Welcome to Economics! Bureau of Labor Statistics, U.S. Department of Labor. 2015. "The Employment Situation—February 2015." Accessed March 27, 2015. http://www.bls.gov/news.release/pdf/empsit.pdf. Williamson, Lisa. “US Labor Market in 2012.” Bureau of Labor Statistics. Accessed December 1, 2013. http://www.bls.gov/opub/mlr/2013/03/art1full.pdf. The Heritage Foundation. 2015. "2015 Index of Economic Freedom." Accessed March 11, 2015. http://www.heritage.org/index/ranking. Garling, Caleb. “S.F. plane crash: Reporting, emotions on social media,” The San Francisco Chronicle. July 7, 2013. http://www.sfgate.com/news/article/S-F-plane-crash-Reporting-emotions-on-social-4651639.php. Irvine, Jessica. “Social Networking Sites are Factories of Modern Ideas.” The Sydney Morning Herald. November 25, 2011.http://www.smh.com.au/federal-politics/society-and-culture/social-networking-sites-are-factories-ofmodern-ideas-20111124-1nwy3.html#ixzz2YZhPYeME. Pew Research Center. 2015. "Social Networking Fact Sheet." Accessed March 11, 2015. http://www.pewinternet.org/ fact-sheets/social-networking-fact-sheet/. The World Bank Group. 2015. "World Data Bank." Accessed March 30, 2014. http://databank.worldbank.org/data/. Choice in a World of Scarcity Bureau of Labor Statistics, U.S. Department of Labor. 2015. “Median Weekly
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Index INDEX A AARP, 433 absolute advantage, 444, 460 Accounting profit, 157 accounting profit, 181, 242 acquisition, 256, 270 actual rate of return, 407, 423 Adam Smith, 12, 39, 245 additional external cost, 291 additional external costs, 278 Adverse selection, 391 adverse selection, 396 Affirmative action, 342 affirmative action, 347 Affordable Care Act (ACA), 381, 394, 433 Age Discrimination in Employment Act, 342 Agriculture and Food Research Initiative (AFRI), 309 Aid to Families with Dependent Children (AFDC), 361 Alfred Marshall, 73 Allocative efficiency, 36, 229 allocative efficiency, 42, 206, 231, 292 American Federation of State, 335 Anthony Downs, 431 Anti-dumping laws, 478 anti-dumping laws, 488 Antitrust Division of the Justice Department, 245 antitrust laws, 257, 270 asymmetric information, 383, 396 average cost, 166 average cost curve, 264 average profit, 170, 181 Average total cost, 169 average total cost, 181 average variable cost, 169, 181 B bar graph, 502 Barriers to entry, 216 barriers to entry, 231 Behavioral economics, 147 behavioral economics, 150 behavioral economists, 385 bilateral monopoly, 337, 347 biodiversity, 288, 291 Bipartisan Campaign Reform Act (BCRA), 431 bond, 404, 423 bond yield, 409, 423 bondholder, 404, 423 bonds, 409, 523 break even point, 196, 209 budget constraint, 29, 42, 135, 141, 512 budget constraint (or budget line), 135, 150 budget constraint line, 357 budget line, 511 bundling, 262, 270 C Capital, 159 capital gain, 405, 423 cartel, 244, 250 Celler-Kefauver Act, 258 certificate of deposit (CD), 408, 423 ceteris paribus, 52, 64, 75, 86 checking account, 407, 423 circular flow diagram, 16, 23 Civil Rights Act of 1964, 340 Civil Rights Act of 1991, 342 Clayton Antitrust Act, 258 Clean Air Act, 280, 283, 285 Clean Water Act, 280 coinsurance, 390, 396 collateral, 387, 396 collective bargaining, 330, 347 collusion, 244, 250 command economy, 18, 23 command-and-control regulation, 280, 291 common market, 488 common markets, 483 common resources, 312 comparative advantage, 13, 37, 42, 445, 475 competition, 303 competitive market, 341 complements, 55
, 75 Compound interest, 420 compound interest, 423 concentration ratio, 258, 270 constant cost industry, 205 constant returns to scale, 176, 181 Constant unitary elasticity, 114 569 constant unitary elasticity, 127 consumer equilibrium, 140, 150 consumer surplus, 71, 75, 471 consumption, 134 consumption budget constraint, 35 copayment, 390, 396 copyright, 218, 231 core competency, 13 corporate bond, 404, 423 corporate governance, 406, 423 corporation, 404, 423 cosigner, 387, 396 cost, 35 cost-plus regulation, 266, 270 County and Municipal Employees (AFSCME), 335 coupon rate, 409, 423 cross-price elasticity of demand, 123, 127 CTC, 362 D David Ricardo, 444 deadweight loss, 72, 75 debit card, 408, 423 decreasing cost industry, 205 deductible, 396 deductibles, 390 demand, 47, 75, 304 demand and supply diagram, 71 demand and supply models, 98 demand curve, 47, 52, 75, 111, 113, 237, 241, 469 demand curves, 261 demand schedule, 47, 75 democracy, 436 Deposit insurance, 389 deregulation, 219, 231, 267 derived demand, 323 differentiated product, 250 differentiated products, 237, 243 diminishing marginal productivity, 168, 181 diminishing marginal utility, 136, 150, 510 Discrimination, 338 discrimination, 347 diseconomies of scale, 177, 181 disruptive market change, 486, 488 diversification, 414, 423 570 Index dividend, 405, 423 division of labor, 12, 23, 455 Dodd-Frank Act, 268 Dow Jones Index, 412 dumping, 458, 488 Dumping, 478 duopoly, 246, 250 E earned income tax credit (EITC), 361, 375 economic efficiency, 39 Economic profit, 157 economic profit, 181 economic surplus, 72, 75 economic union, 488 economic unions, 483 Economics, 10 economics, 23 economies of scale, 13, 23, 181, 181, 456 Economies of scale, 174, 217 ecotourism, 285 effective income tax, 371, 375 efficiency, 71 elastic demand, 109, 127 elastic supply, 109, 127 Elasticity, 108 elasticity, 127 elasticity of savings, 124, 127 Entrepreneurship, 159 entry, 204, 209 Equal Employment Opportunity Commission (EEOC), 342 Equal Pay Act of 1963, 342 equilibrium, 51, 71, 75, 86, 242, 383, 467 equilibrium price, 51, 75,
384 equilibrium quantity, 51, 66, 75 equity, 414, 423 estate tax, 372, 375 European Union, 483 European Union (EU), 70 excess demand, 51, 75 excess supply, 51, 75 exclusive dealing, 262, 270 exit, 204, 209 expected rate of return, 407, 423 Explicit costs, 157 explicit costs, 181 export, 452 Exports, 21 exports, 23 external benefits (or positive externalities), 314 external costs, 278 externality, 277, 291 F face value, 409, 423 Factor payments, 165 factors of production, 57, 75 factors of production (or inputs), 181 Federal Deposit Insurance Corporation (FDIC), 408 Federal Reserve Economic Data (FRED), 402 Federal Trade Commission, 245 Federal Trade Commission (FTC), 387 fee-for-service, 391, 396 financial capital, 93 financial capital market, 387 financial capital markets, 383 Financial capital markets, 402 financial intermediary, 407, 423 firm, 37, 59, 156, 181, 242, 248, 303, 403, 437 firms, 236 first rule of labor markets, 320, 347 fiscal policy, 15, 23 fixed, 160 fixed cost, 181 Fixed inputs, 160 fixed inputs, 181 fossil fuels, 288 four-firm concentration ratio, 270 Francine Blau, 338 free rider, 310, 310, 314 free trade, 479 free trade agreement, 488 free trade agreements, 483 function, 493 fungible, 147, 150 G gain from trade, 448, 460 game theory, 245, 250 Gary Becker, 341 General Agreement on Tariffs and Trade (GATT), 482, 488 George Psacharopoulos, 305 globalization, 21, 23, 261, 370, 444, 475 good, 48 goods and services market, 16, 23 This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Great Depression, 444 Great Recession, 363 Gross domestic product (GDP), 21 gross domestic product (GDP), 23, 289 growth rate, 497 H Health Care for America Now (HCAN), 433 health maintenance organization (HMO), 391, 396 Herfindahl-Hirschman Index (HHI), 259, 270 high yield, 409 high yield bonds, 423 High-income countries, 470 high-income countries, 479 I immigrants, 343 imperfect information, 383, 396 imperfectly competitive, 236, 250 Implicit costs, 157 implicit costs, 181 import quotas, 4
66, 488 Imports, 21 imports, 23, 452 Income, 354, 371 income, 375 income effect, 143, 150, 513 income elasticity of demand, 123 income inequality, 375 increasing cost industry, 205 index fund, 414, 423 indifference curve, 509 inelastic demand, 109, 127 inelastic supply, 109, 127 inequality, 354 infant industry argument, 476 inferior good, 54, 75, 123, 142 Infinite elasticity, 113 infinite elasticity, 127 initial public offering (IPO), 405, 423 inputs, 57, 75 Insurance, 388 insurance, 396 intellectual property, 219, 231, 314 Intellectual property, 306 Interest and dividends, 165 interest rate, 93, 103, 523 Index 571 international externalities, 288, 291 international trade, 457, 486 International Trade Commission (ITC), 438 intertemporal choices, 41 intertemporal decision making, 95 intra-industry trade, 455, 460 invisible hand, 40, 42 J Joan Robinson, 438 John Maynard Keynes, 16 junk bonds, 409, 423 K key input, 118 kinked demand curve, 248, 250 L Labor, 159 labor market, 16, 23, 86, 386 labor markets, 365 labor-leisure budget constraint, 357 labor-leisure diagram, 515 Laurence Kahn, 338 law of demand, 47, 75, 94 Law of Diminishing Marginal Product, 162 law of diminishing marginal utility, 32, 42 law of diminishing returns, 35, 42 law of supply, 48, 75 legal monopoly, 217, 231 leviathan effect, 177 line graphs, 498 liquidity, 407, 423 living wage, 90 lobbyists, 433 Logrolling, 435 logrolling, 440 long run, 160, 181 long-run average cost (LRAC) curve, 175, 181 long-run equilibrium, 205, 209 Lorenz curve, 366, 375 loss aversion, 147 low-income countries, 470, 479 M Macroeconomics, 14 macroeconomics, 23 marginal analysis, 32, 42 marginal benefits, 286 marginal cost, 166, 181, 194, 224, 266 Marginal cost, 169 marginal cost curve, 286 marginal cost curves, 281 marginal cost of labor, 328 marginal product, 161, 181 Marginal profit, 226 marginal profit, 231 marginal rate of substitution, 510 marginal revenue, 191, 209, 224 marginal utility, 136, 150, 510 Marginal utility per dollar, 138 marginal utility per dollar, 150 market, 19,
23, 261, 392 market economy, 19, 23, 37, 303 market failure, 280, 291 market price, 237 market share, 258, 270 market structure, 188, 209 marketable permit program, 283, 291 maturity date, 409, 424 maximizing utility, 139 median, 498 median voter theory, 436, 440 Medicaid, 356, 363, 375 Medicare, 433 merger, 256, 270 Michael S. Clune, 344 Microeconomics, 14 microeconomics, 23 Midpoint Formula, 111 Midpoint Method, 109, 112 minimum resale price maintenance agreement, 262, 270 minimum wage, 90, 103, 475 model, 16, 23 Mollie Orshansky, 354 monetary policy, 15, 23 money-back guarantee, 385, 396 monopolistic competition, 236, 250 monopoly, 216, 231, 263 monopsony, 328, 347 Moody’s, 410 Moral hazard, 390 moral hazard, 396 municipal bond, 404 municipal bonds, 424 mutual funds, 414, 424 N National Academy of Engineers, 309 National Academy of Scientists, 309 National Association of Insurance Commissioners, 393 National Education Association, 335 National Institutes of Health, 309 national interest argument, 481, 488 National Labor-Management Relations Act, 336 National Venture Capital Association, 403 natural monopoly, 217, 231, 244, 264 Natural Resources (Land and Raw Materials), 159 near-poor, 360, 375 negative externalities, 433 negative externality, 277, 291 negative slope, 495 non-rival, 310 nonexcludable, 310, 314 nonrivalrous, 314 Nontariff barriers, 467 nontariff barriers, 488 normal good, 54, 75 normal goods, 123, 142 normative statement, 42 normative statements, 39 North American Free Trade Agreement (NAFTA), 473, 483 O occupational license, 396 occupational licenses, 386 oligopoly, 236, 250 Oligopoly, 244 opportunity cost, 29, 42, 445, 453, 473 opportunity set, 42, 513 Organization of Petroleum Exporting Countries (OPEC), 247 output, 247 Oxfam International, 470 P partnership, 405, 424 572 Index patent, 218, 231, 306 Patient Protection and Affordable Care Act (PPACA), 381, 394 Pension insurance, 389 perfect competition, 188, 209 perfect elasticity, 113, 127 perfect inelasticity, 113, 127 perfectly competitive firm, 188, 189, 205, 220 Perfectly Competitive Labor Market, 322 perfectly competitive labor market, 347
perfectly elastic, 237 Pew Research Center for People and the Press, 63 pie chart, 501 pie graph, 501 pollution charge, 281, 291 pork-barrel spending, 434, 440 positive externalities, 303, 314 positive externality, 277, 291 positive slope, 495 positive statement, 42 positive statements, 39 poverty, 354, 375 poverty line, 354, 375 poverty rate, 354, 375 poverty trap, 357, 375 predatory pricing, 219, 231, 263 premium, 396 premiums, 388 present discounted value (PDV), 523 present value, 409, 424 price, 31, 47, 52, 75, 248 price cap regulation, 266, 270 price ceiling, 68, 72, 75 price control, 72, 75 price controls, 68, 100, 434 Price elasticity, 108 price elasticity, 127 price elasticity of demand, 108, 127 price elasticity of supply, 108, 127 price floor, 68, 73, 76 price taker, 188, 209 price takers, 236 prisoner’s dilemma, 245, 250 private benefits, 303, 314 private company, 405, 424 private enterprise, 19, 23, 181 private insurance, 389 Private markets, 277 private rates of return, 305, 314 producer surplus, 72, 76, 471 product differentiation, 237, 250 production, 156, 181 production function, 160, 181 production possibilities frontier (PPF), 33, 42 production possibility frontier (PPF), 289, 446 production technologies, 172, 181 Productive efficiency, 36, 206 productive efficiency, 42, 242, 292 productivity, 450 Profit, 165 profit, 189 profit margin, 171, 194 profit-maximizing, 239 profits, 247 progressive tax system, 371, 375 property rights, 284, 291 protectionism, 466, 470, 488 Protectionism, 473 public company, 405, 424 public good, 309, 314 Public policy, 372 Q quantity demanded, 47, 76 Quantity demanded, 244 quantity supplied, 48, 76 quintile, 375 quintiles, 365 quotas, 436 R race to the bottom, 479, 488 rational ignorance, 430, 440 Raw materials prices, 165 Redistribution, 371 redistribution, 375 regulatory capture, 268, 270, 433 Rent, 165 restrictive practices, 262, 270 Retirement insurance, 389 revenue, 157, 181 Risk, 407 risk, 424 risk group, 390, 396 S safety net, 360, 375 This OpenStax book is available for free at http://cnx.org/content/
col12170/1.7 salary, 84 Sarbanes-Oxley Act, 268 savings account, 408, 424 Scarcity, 10 scarcity, 23, 39 service, 48 service contract, 386, 396 Service Employees International Union, 335 shareholders, 404, 424 shares, 404, 424 Sherman Antitrust Act, 258 shift in demand, 55, 76 shift in supply, 57, 76 short run, 160, 182 short-run average cost (SRAC) curve, 182 short-run average cost (SRAC) curves, 175 shortage, 51, 76 shutdown point, 209 Simple interest, 419 simple interest, 424 slope, 34, 495 social benefits, 303, 314 social costs, 277, 291 social rate of return, 305, 314 social surplus, 72, 76 sole proprietorship, 405, 424 Special interest groups, 433 special interest groups, 440 Special Supplemental Food Program for Women, Infants and Children (WIC), 363 specialization, 13, 23, 448 spillover, 291 spillovers, 277 splitting up the value chain, 456, 460 Standard & Poor’s 500 index, 412 Stock, 404 stock, 418, 424 stocks, 523 straight-line demand curve, 111 subsidies, 470 substitute, 55, 76 substitution effect, 143, 150, 513 sunk costs, 33, 42, 170 Supplemental Nutrition Assistance Program (SNAP), 362, 375 supply, 48, 76 supply curve, 49, 52, 76, 469 supply curves, 261 Index 573 voting cycle, 437, 440 W wage, 84 wage elasticity of labor supply, 124, 127 Wages and salaries, 165 Walter McMahon, 305 warranty, 386, 396 Wealth, 371 wealth, 375 Workman’s compensation insurance, 389 World Trade Organization (WTO), 458, 467, 478, 482, 488 Z Zero elasticity, 113 zero inelasticity, 127 supply schedule, 49, 76 surplus, 51, 76 T tariff, 432 Tariffs, 458 tariffs, 460, 466 tax incidence, 120, 127 Technology, 159 technology, 303 Temporary Assistance for Needy Families (TANF), 361 theory, 16, 23 thick market, 384 thin market, 384 time series, 500 total cost, 182 total costs, 190 total product, 182 total revenue, 190, 224 total surplus, 72, 76 total utility, 135, 150 trade secrets, 219, 231 trademark, 218, 231 tradeoffs, 39 traditional economy, 18, 23 Treasury bond, 404
, 424 Tying sales, 262 tying sales, 270 Tyler Cowen, 313 U U.S. Census Bureau, 54 U.S. Department of the Treasury, 404 underground economies, 20 underground economy, 23 Unemployment insurance, 389 Unitary elasticities, 109 unitary elasticity, 127 usury laws, 98, 103 utility, 32, 42, 135, 509 utility maximizing, 385 utility-maximizing, 430 utility-maximizing choice, 142 V value chain, 456, 460 variable, 160, 494 variable cost, 182 Variable costs, 167 Variable inputs, 160 variable inputs, 182 Venture capital, 403 venture capital, 424tions Many economic models start from the assumption that the economic actors being studied are rationally pursuing some goal. We brieﬂy discussed such an assumption when investigating the notion of ﬁrms maximizing proﬁts. Example 1.1 shows how that model can be used to make testable predictions. Other examples we will encounter in this book include consumers maximizing their own well-being (utility), ﬁrms minimizing costs, and government regulators attempting to maximize public welfare. Although, as we will show, all these assumptions are unrealistic, and all have won widespread acceptance as good starting places for developing economic models. There seem to be two reasons for this acceptance. First, the optimization assumptions are useful for generating precise, solvable models, primarily because such models can draw on a variety of mathematical techniques suitable for optimization problems. Many of these techniques, together with the logic behind them, are reviewed in Chapter 2. A second reason for the popularity of optimization models concerns their apparent empirical validity. As some of our Extensions show, such models seem to be fairly good at explaining reality. In all, then, optimization models have come to occupy a prominent position in modern economic theory. EXAMPLE 1.1 Proﬁt Maximization The proﬁt-maximization hypothesis provides a good illustration of how optimization assumptions can be used to generate empirically testable propositions about economic behavior. Suppose that a ﬁrm can sell all the output that it wishes at a price of p per unit and that the total costs of production, C, depend on the amount produced, q. Then proﬁts are given by profits p pq C q : Þ ð " ¼ ¼ (1:1) 8 Part 1: Introduction Maximization of proﬁts consists of ﬁnd
ing that value of q which maximizes the proﬁt expression in Equation 1.1. This is a simple problem in calculus. Differentiation of Equation 1.1 and setting that derivative equal to 0 give the following ﬁrst-order condition for a maximum: dp dq or p C 0 q ð : Þ ¼ (1:2) In words, the proﬁt-maximizing output level (q%) is found by selecting that output level for which price is equal to marginal cost, C 0ð. This result should be familiar to you from your introductory Þ economics course. Notice that in this derivation the price for the ﬁrm’s output is treated as a constant because the ﬁrm is a price-taker. That is, price is an exogenous variable in this model. q Equation 1.2 is only the ﬁrst-order condition for a maximum. Taking account of the secondorder condition can help us to derive a testable implication of this model. The second-order condition for a maximum is that at q% it must be the case that d 2p dq 2 ¼ " C 00 q ð Þ < 0 or C 00 > 0: q% ð Þ (1:3) That is, marginal cost must be increasing at q% for this to be a true point of maximum proﬁts. Our model can now be used to ‘‘predict’’ how a ﬁrm will react to a change in price. To do so, we differentiate Equation 1.2 with respect to price (p), assuming that the ﬁrm continues to choose a proﬁt-maximizing level of q: p d ½ " C 0 q% ð dp 0 ’ Þ ¼ 1 C 00 q% ð Þ ( " ¼ dq% dp ¼ 0: (1:4) Rearranging terms a bit gives dq% dp ¼ 1 q%Þ C 00ð Here the ﬁnal inequality again reﬂects the fact that marginal cost must be increasing at q% if this point is to be a true maximum. This then is one of the testable propositions of the proﬁtmaximization hypothesis—if other things do not change, a price-taking
ﬁrm should respond to an increase in price by increasing output. On the other hand, if ﬁrms respond to increases in price by reducing output, there must be something wrong with our model. > 0: (1:5) Although this is a simple model, it reﬂects the way we will proceed throughout much of this book. Speciﬁcally, the fact that the primary implication of the model is derived by calculus, and consists of showing what sign a derivative should have, is the kind of result we will see many times. Notice that in this model there is only one endogenous variable—q, the quantity the ﬁrm chooses to produce. There is also only one exogenous variable—p, the price of the product, which the ﬁrm takes as a given. Our model makes a speciﬁc prediction about how changes in this exogenous variable affect the ﬁrm’s output choice. QUERY: In general terms, how would the implications of this model be changed if the price a ﬁrm obtains for its output were a function of how much it sold? That is, how would the model work if the price-taking assumption were abandoned? Positive-normative distinction A ﬁnal feature of most economic models is the attempt to differentiate carefully between ‘‘positive’’ and ‘‘normative’’ questions. Thus far we have been concerned primarily with positive economic theories. Such theories take the real world as an object to be studied, attempting to explain those economic phenomena that are observed. Positive economics seeks to determine how resources are in fact allocated in an economy. A somewhat different use of economic theory is normative analysis, taking a deﬁnite stance about what should be done. Under the heading of normative analysis, economists have a great deal to Chapter 1: Economic Models 9 say about how resources should be allocated. For example, an economist engaged in positive analysis might investigate how prices are determined in the U.S. health-care economy. The economist also might want to measure the costs and beneﬁts of devoting even more resources to health care by, for example, offering government-subsidized health insurance. But when he or she speciﬁcally advocates that such an insurance plan should be adopted, the analysis becomes normative. Some economists believe that the only proper economic analysis is positive analysis
. Drawing an analogy with the physical sciences, they argue that ‘‘scientiﬁc’’ economics should concern itself only with the description (and possibly prediction) of real-world economic events. To take political positions and to plead for special interests are considered to be outside the competence of an economist acting as such. Of course, an economist, like any other citizen, is free to express his or her views on political matters. But when doing so he or she is acting as a citizen, not an economist. For other economists, however, the positive-normative distinction seems artiﬁcial. They believe that the study of economics necessarily involves the researchers’ own views about ethics, morality, and fairness. According to these economists, searching for scientiﬁc ‘‘objectivity’’ in such circumstances is hopeless. Despite some ambiguity, this book tries to adopt a positivist tone, leaving normative concerns for you to decide for yourself. Development of the Economic Theory of Value Because economic activity has been a central feature of all societies, it is surprising that these activities were not studied in any detail until fairly recently. For the most part, economic phenomena were treated as a basic aspect of human behavior that was not sufﬁciently interesting to deserve speciﬁc attention. It is, of course, true that individuals have always studied economic activities with a view toward making some kind of personal gain. Roman traders were not above making proﬁts on their transactions. But investigations into the basic nature of these activities did not begin in any depth until the eighteenth century.3 Because this book is about economic theory as it stands today, rather than the history of economic thought, our discussion of the evolution of economic theory will be brief. Only one area of economic study will be examined in its historical setting: the theory of value. Early economic thoughts on value The theory of value, not surprisingly, concerns the determinants of the ‘‘value’’ of a commodity. This subject is at the center of modern microeconomic theory and is closely intertwined with the fundamental economic problem of allocating scarce resources to alternative uses. The logical place to start is with a deﬁnition of the word ‘‘value.’’ Unfortunately, the meaning of this term has not been consistent throughout the development of the subject. Today we regard value as being synonymous with the price of a commodity.4
Earlier philosopher-economists, however, made a distinction between the market price of a commodity and its value. The term value was then thought of as being, in some sense, synonymous with ‘‘importance,’’ ‘‘essentiality,’’ or (at times) ‘‘godliness.’’ Because ‘‘price’’ and ‘‘value’’ were separate concepts, they could differ, and most early economic 3For a detailed treatment of early economic thought, see the classic work by J. A. Schumpeter, History of Economic Analysis (New York: Oxford University Press, 1954), pt. II, chaps. 1–3. 4This is not completely true when ‘‘externalities’’ are involved, and a distinction must be made between private and social value (see Chapter 19). 10 Part 1: Introduction discussions centered on these divergences. For example, St. Thomas Aquinas believed value to be divinely determined. Because prices were set by humans, it was possible for the price of a commodity to differ from its value. A person accused of charging a price in excess of a good’s value was guilty of charging an ‘‘unjust’’ price. St. Thomas believed that, in most cases, the ‘‘just’’ rate of interest was zero. Any lender who demanded a payment for the use of money was charging an unjust price and could be—and sometimes was—prosecuted by church ofﬁcials. The founding of modern economics During the latter part of the eighteenth century, philosophers began to take a more scientiﬁc approach to economic questions. The 1776 publication of The Wealth of Nations by Adam Smith (1723–1790) is generally considered the beginning of modern economics. In his vast, all-encompassing work, Smith laid the foundation for thinking about market forces in an ordered and systematic way. Still, Smith and his immediate successors, such as David Ricardo (1772–1823), continued to distinguish between value and price. To Smith, for example, the value of a commodity meant its ‘‘value in use,’’ whereas the price represented its ‘‘value in exchange.’’ The distinction between these two concepts was illustrated by the famous water–diamond paradox.
Water, which obviously has great value in use, has little value in exchange (it has a low price); diamonds are of little practical use but have a great value in exchange. The paradox with which early economists struggled derives from the observation that some useful items have low prices whereas certain nonessential items have high prices. Labor theory of exchange value Neither Smith nor Ricardo ever satisfactorily resolved the water–diamond paradox. The concept of value in use was left for philosophers to debate, while economists turned their attention to explaining the determinants of value in exchange (i.e., to explaining relative prices). One obvious possible explanation is that exchange values of goods are determined by what it costs to produce them. Costs of production are primarily inﬂuenced by labor costs—at least this was so in the time of Smith and Ricardo—and therefore it was a short step to embrace a labor theory of value. For example, to paraphrase an example from Smith, if catching a deer takes twice the number of labor hours as catching a beaver, then one deer should exchange for two beavers. In other words, the price of a deer should be twice that of a beaver. Similarly, diamonds are relatively costly because their production requires substantial labor input, whereas water is freely available. To students with even a passing knowledge of what we now call the law of supply and demand, Smith’s and Ricardo’s explanation must seem incomplete. Did they not recognize the effects of demand on price? The answer to this question is both yes and no. They did observe periods of rapidly rising and falling relative prices and attributed such changes to demand shifts. However, they regarded these changes as abnormalities that produced only a temporary divergence of market price from labor value. Because they had not really developed a theory of value in use, they were unwilling to assign demand any more than a transient role in determining relative prices. Rather, long-run exchange values were assumed to be determined solely by labor costs of production. The marginalist revolution Between 1850 and 1880, economists became increasingly aware that to construct an adequate alternative to the labor theory of value, they had to devise a theory of value in use. During the 1870s, several economists discovered that it is not the total usefulness of a commodity that helps to determine its exchange value, but rather the usefulness of the last unit consumed. For example, water is certainly useful—it is necessary for all life. Chapter 1: Economic Models 11 FIGURE 1.2 The Marshallian Supply–Demand Cross Marshall
theorized that demand and supply interact to determine the equilibrium price (p%) and the quantity (q%) that will be traded in the market. He concluded that it is not possible to say that either demand or supply alone determines price or therefore that either costs or usefulness to buyers alone determines exchange value. Price D p* S S D q* Quantity per period However, because water is relatively plentiful, consuming one more pint (ceteris paribus) has a relatively low value to people. These ‘‘marginalists’’ redeﬁned the concept of value in use from an idea of overall usefulness to one of marginal, or incremental, usefulness— the usefulness of an additional unit of a commodity. The concept of the demand for an incremental unit of output was now contrasted with Smith’s and Ricardo’s analysis of production costs to derive a comprehensive picture of price determination.5 Marshallian supply–demand synthesis The clearest statement of these marginal principles was presented by the English economist Alfred Marshall (1842–1924) in his Principles of Economics, published in 1890. Marshall showed that demand and supply simultaneously operate to determine price. As Marshall noted, just as you cannot tell which blade of a scissors does the cutting, so too you cannot say that either demand or supply alone determines price. That analysis is illustrated by the famous Marshallian cross shown in Figure 1.2. In the diagram the quantity of a good purchased per period is shown on the horizontal axis, and its price appears on the vertical axis. The curve DD represents the quantity of the good demanded per period at each possible price. The curve is negatively sloped to reﬂect the marginalist principle that as quantity increases, people are willing to pay less for the last unit purchased. It is the value of this last unit that sets the price for all units purchased. The curve SS shows how (marginal) production costs increase as more output is produced. This reﬂects the increasing cost of producing one more unit as total output expands. In other words, the upward slope of the SS curve reﬂects increasing marginal costs, just as the downward slope of the DD curve reﬂects decreasing marginal value. The two curves intersect at p%, q%. This is an equilibrium point—both buyers and sellers are content with the quantity being traded and the price at which it is traded. If one of the curves should shift, the equilibrium point would shift to a new location.
Thus, price and quantity are simultaneously determined by the joint operation of supply and demand. 5Ricardo had earlier provided an important ﬁrst step in marginal analysis in his discussion of rent. Ricardo theorized that as the production of corn increased, land of inferior quality would be used and this would cause the price of corn to increase. In his argument Ricardo recognized that it is the marginal cost—the cost of producing an additional unit—that is relevant to pricing. Notice that Ricardo implicitly held other inputs constant when discussing decreasing land productivity; that is, he used one version of the ceteris paribus assumption. 12 Part 1: Introduction EXAMPLE 1.2 Supply–Demand Equilibrium Although graphical presentations are adequate for some purposes, economists often use algebraic representations of their models both to clarify their arguments and to make them more precise. As an elementary example, suppose we wished to study the market for peanuts and, based on the statistical analysis of historical data, concluded that the quantity of peanuts demanded each week (q, measured in bushels) depended on the price of peanuts (p, measured in dollars per bushel) according to the equation: quantity demanded 1,000 100p: qD ¼ ¼ Because this equation for qD contains only the single independent variable p, we are implicitly holding constant all other factors that might affect the demand for peanuts. Equation 1.6 indicates that, if other things do not change, at a price of $5 per bushel people will demand 500 bushels of peanuts, whereas at a price of $4 per bushel they will demand 600 bushels. The negative coefﬁcient for p in Equation 1.6 reﬂects the marginalist principle that a lower price will cause people to buy more peanuts. " (1:6) To complete this simple model of pricing, suppose that the quantity of peanuts supplied also depends on price: quantity supplied qS ¼ " 125 125p: ¼ Here the positive coefﬁcient of price also reﬂects the marginal principle that a higher price will call forth increased supply—primarily because (as we saw in Example 1.1) it permits ﬁrms to incur higher marginal costs of production without incurring losses on the additional units produced. þ (1:7) Equilibrium price determination. Therefore, Equations 1.6 and 1.7 reﬂect our model of price determination in the
market for peanuts. An equilibrium price can be found by setting quantity demanded equal to quantity supplied: or or thus, qD ¼ qS 1,000 100p " 125 ¼ " 125p þ 225p ¼ 1,125 (1:8) (1:9) (1:10) ¼ At a price of $5 per bushel, this market is in equilibrium: At this price people want to purchase 500 bushels, and that is exactly what peanut producers are willing to supply. This equilibrium is pictured graphically as the intersection of D and S in Figure 1.3. p% 5: (1:11) A more general model. To illustrate how this supply–demand model might be used, let’s adopt a more general notation. Suppose now that the demand and supply functions are given by qD ¼ a þ bp and qS ¼ c þ dp (1:12) where a and c are constants that can be used to shift the demand and supply curves, respectively, and b (<0) and d (>0) represent demanders’ and suppliers’ reactions to price. Equilibrium in this market requires Thus, equilibrium price is given by6 qD ¼ bp ¼ qS c þ or dp: a þ p1:13) (1:14) 6Equation 1.14 is sometimes called the ‘‘reduced form’’ for the supply–demand structural model of Equations 1.12 and 1.13. It shows that the equilibrium value for the endogenous variable p ultimately depends only on the exogenous factors in the model (a and c) and on the behavioral parameters b and d. A similar equation can be calculated for equilibrium quantity. Chapter 1: Economic Models 13 FIGURE 1.31Changing Supply–Demand Equilibria The initial supply–demand equilibrium is illustrated by the intersection of D and S (p% When demand shifts to qD 0 ¼ 100p denoted as D 0Þ ð, the equilibrium shifts to p% 1; 450 " ¼ 5, q% ¼ 500). ¼ 7, q% 750. ¼ Price ($) D′ 14.5 D 10 7 5 S 0 S D 500 750 1,000 D′ 1,450 Quantity per period (bushels) Notice that in our previous example a p% ¼ ¼ 1,000 125 125 100 ¼ 1,125 225 ¼
þ þ 1,000, b 100, c ¼ " 125, and d ¼ 125; therefore, ¼ " 5: (1:15) With this more general formulation, however, we can pose questions about how the equilibrium price might change if either the demand or supply curve shifted. For example, differentiation of Equation 1.14 shows that dp% da ¼ dp% dc ; < 0: b b (1:16) That is, an increase in demand (an increase in a) increases equilibrium price, whereas an increase in supply (an increase in c) reduces price. This is exactly what a graphical analysis of supply and demand curves would show. For example, Figure 1.3 shows that when the constant 7 term, a, in the demand equation increases to 1,450, equilibrium price increases to p% [ ¼ QUERY: How might you use Equation 1.16 to ‘‘predict’’ how each unit increase in the exogenous constant a affects the endogenous variable p%? Does this equation correctly predict the increase in p% when the constant a increases from 1,000 to 1,450? 125)/225]. (1,450 þ ¼ 14 Part 1: Introduction Paradox resolved Marshall’s model resolves the water–diamond paradox. Prices reﬂect both the marginal evaluation that demanders place on goods and the marginal costs of producing the goods. Viewed in this way, there is no paradox. Water is low in price because it has both a low marginal value and a low marginal cost of production. On the other hand, diamonds are high in price because they have both a high marginal value (because people are willing to pay quite a bit for one more) and a high marginal cost of production. This basic model of supply and demand lies behind much of the analysis presented in this book. General equilibrium models Although the Marshallian model is an extremely useful and versatile tool, it is a partial equilibrium model, looking at only one market at a time. For some questions, this narrowing of perspective gives valuable insights and analytical simplicity. For other, broader questions, such a narrow viewpoint may prevent the discovery of important relationships among markets. To answer more general questions we must have a model of the whole economy that suitably mirrors the connections among various markets and economic agents. The French economist Leon Walras (1831–1910), building on a long Continental tradition in such analysis, created the basis for modern investigations
into those broad questions. His method of representing the economy by a large number of simultaneous equations forms the basis for understanding the interrelationships implicit in general equilibrium analysis. Walras recognized that one cannot talk about a single market in isolation; what is needed is a model that permits the effects of a change in one market to be followed through other markets. For example, suppose that the demand for peanuts were to increase. This would cause the price of peanuts to increase. Marshallian analysis would seek to understand the size of this increase by looking at conditions of supply and demand in the peanut market. General equilibrium analysis would look not only at that market but also at repercussions in other markets. An increase in the price of peanuts would increase costs for peanut butter makers, which would, in turn, affect the supply curve for peanut butter. Similarly, the increasing price of peanuts might mean higher land prices for peanut farmers, which would affect the demand curves for all products that they buy. The demand curves for automobiles, furniture, and trips to Europe would all shift out, and that might create additional incomes for the providers of those products. Consequently, the effects of the initial increase in demand for peanuts eventually would spread throughout the economy. General equilibrium analysis attempts to develop models that permit us to examine such effects in a simpliﬁed setting. Several models of this type are described in Chapter 13. Production possibility frontier Here we brieﬂy introduce some general equilibrium ideas by using another graph you should remember from introductory economics—the production possibility frontier. This graph shows the various amounts of two goods that an economy can produce using its available resources during some period (say, one week). Because the production possibility frontier shows two goods, rather than the single good in Marshall’s model, it is used as a basic building block for general equilibrium models. Figure 1.4 shows the production possibility frontier for two goods: food and clothing. The graph illustrates the supply of these goods by showing the combinations that can be produced with this economy’s resources. For example, 10 pounds of food and 3 units of clothing could be produced, or 4 pounds of food and 12 units of clothing. Many other combinations of food and clothing could also be produced. The production possibility frontier shows all of them. Combinations of food and clothing outside the frontier cannot Chapter 1: Economic Models 15 FIGURE 1.4 Production Possibility Frontier The production possibility frontier shows the different combinations of two goods that can be produced from a certain amount of scarce resources. It also shows
the opportunity cost of producing more of one good as the amount of the other good that cannot then be produced. The opportunity cost at two different levels of clothing production can be seen by comparing points A and B. Quantity of food per week 10 9.5 4 2 Opportunity cost of clothing = pound of food 1 2 A Opportunity cost of clothing = 2 pounds of food B 0 3 4 12 13 Quantity of clothing per week be produced because not enough resources are available. The production possibility frontier reminds us of the basic economic fact that resources are scarce—there are not enough resources available to produce all we might want of every good. This scarcity means that we must choose how much of each good to produce. Figure 1.4 makes clear that each choice has its costs. For example, if this economy produces 10 pounds of food and 3 units of clothing at point A, producing 1 more unit of clothing would ‘‘cost’’ ½ pound of food—increasing the output of clothing by 1 unit means the production of food would have to decrease by ½ pound. Thus, the opportunity cost of 1 unit of clothing at point A is ½ pound of food. On the other hand, if the economy initially produces 4 pounds of food and 12 units of clothing at point B, it would cost 2 pounds of food to produce 1 more unit of clothing. The opportunity cost of 1 more unit of clothing at point B has increased to 2 pounds of food. Because more units of clothing are produced at point B than at point A, both Ricardo’s and Marshall’s ideas of increasing incremental costs suggest that the opportunity cost of an additional unit of clothing will be higher at point B than at point A. This effect is shown by Figure 1.4. The production possibility frontier provides two general equilibrium insights that are not clear in Marshall’s supply and demand model of a single market. First, the graph shows that producing more of one good means producing less of another good because resources are scarce. Economists often (perhaps too often!) use the expression ‘‘there is no such thing as a free lunch’’ to explain that every economic action has opportunity costs. Second, the production possibility frontier shows that opportunity costs depend on how much of each good is produced. The frontier is like a supply curve for two goods: It 16 Part 1: Introduction shows the opportunity cost of producing more of one good as the decrease in the amount of the second good. Therefore, the production possibility frontier is a particularly useful tool for
studying several markets at the same time. EXAMPLE 1.3 The Production Possibility Frontier and Economic Inefﬁciency General equilibrium models are good tools for evaluating the efﬁciency of various economic arrangements. As we will see in Chapter 13, such models have been used to assess a wide variety of policies such as trade agreements, tax structures, and environmental regulations. In this simple example, we explore the idea of efﬁciency in its most elementary form. Suppose that an economy produces two goods, x and y, using labor as the only input. The (where lx is the quantity of labor used in x labor available is ¼ 200. Construction of the production possibility frontier in this production function for good x is x production), and the production function for good y is y constrained by lx þ ly * economy is extremely simple:. Total 2l 0:5 y l 0:5 x ¼ lx þ ly ¼ x2 0:25y2 200 þ where the equality holds exactly if the economy is to be producing as much as possible (which, after all, is why it is called a ‘‘frontier’’). Equation 1.17 shows that the frontier here has the shape of a quarter ellipse—its concavity derives from the diminishing returns exhibited by each production function. * (1:17) Opportunity cost. Assuming this economy is on the frontier, the opportunity cost of good y in terms of good x can be derived by solving for y as y2 ¼ 800 " 4x2 or y ¼ And then differentiating this expression: p 800 4x2 " ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 800 0:5 4x2 ’ " ¼ ½ dy dx ¼ 800 0:5 ½ " 4x2 ’ 0:5 " 8x ð" Þ ¼ 4x " y : (1:18) (1:19) 4(10)/20 20, Suppose, for example, labor is equally allocated between the two goods. Then x and dy/dx 2. With this allocation of labor, each unit increase in x output ¼ " would require a reduction
in y of 2 units. This can be veriﬁed by considering a slightly different allocation, lx ¼ 19.9. Moving to this alternative allocation would have ¼ " 101 and ly ¼ 99. Now production is x 10.05 and y 10, y ¼ ¼ ¼ ¼ Dy Dx ¼ 19:9 ð 10:05 ð " " 20 Þ 10 Þ 0:1 " 0:05 ¼ " 2, ¼ which is precisely what was derived from the calculus approach. Concavity. Equation 1.19 clearly illustrates the concavity of the production possibility frontier. The slope of the frontier becomes steeper (more negative) as x output increases and y output decreases. For example, if labor is allocated so that lx ¼ 144 and ly ¼ 12 and y ¼ " ¼ " opportunity cost of one more unit of x increases from 2 to 3.2 units of y. ¼ 3.2. With expanded x production, the 56, then outputs are x 15 and so dy /dx 4(12)/15 + Inefﬁciency. If an economy operates inside its production possibility frontier, it is operating inefﬁciently. Moving outward to the frontier could increase the output of both goods. In this book we will explore many reasons for such inefﬁciency. These usually derive from a failure of some market to perform correctly. For the purposes of this illustration, let’s assume that the labor market in this economy does not work well and that 20 workers are permanently unemployed. Now the production possibility frontier becomes 0:25y2 x2 þ ¼ 180, (1:20) Chapter 1: Economic Models 17 and the output combinations we described previously are no longer feasible. For example, if x ¼ 10, then y output is now y 17.9. The loss of approximately 2.1 units of y is a measure of the cost of the labor market inefﬁciency. Alternatively, if the labor supply of 180 were allocated 19, and evenly between the production of the two goods, then we would have x the inefﬁciency would show up in both goods’ production—more of both goods could be produced if the labor market inefﬁciency were resolved. 9.5 and y + + + QUERY: How would the inefﬁ
ciency cost of labor market imperfections be measured solely in terms of x production in this model? How would it be measured solely in terms of y production? What would you need to know to assign a single number to the efﬁciency cost of the imperfection when labor is equally allocated to the two goods? Welfare economics In addition to using economic models to examine positive questions about how the economy operates, the tools used in general equilibrium analysis have also been applied to the study of normative questions about the welfare properties of various economic arrangements. Although such questions were a major focus of the great eighteenth- and nineteenthcentury economists (e.g., Smith, Ricardo, Marx, and Marshall), perhaps the most signiﬁcant advances in their study were made by the British economist Francis Y. Edgeworth (1848–1926) and the Italian economist Vilfredo Pareto (1848–1923) in the early years of the twentieth century. These economists helped to provide a precise deﬁnition for the concept of ‘‘economic efﬁciency’’ and to demonstrate the conditions under which markets will be able to achieve that goal. By clarifying the relationship between the allocation pricing of resources, they provided some support for the idea, ﬁrst enunciated by Adam Smith, that properly functioning markets provide an ‘‘invisible hand’’ that helps allocate resources efﬁciently. Later sections of this book focus on some of these welfare issues. Modern Developments Research activity in economics expanded rapidly in the years following World War II. A major purpose of this book is to summarize much of this research. By illustrating how economists have tried to develop models to explain increasingly complex aspects of economic behavior, this book seeks to help you recognize some of the remaining unanswered questions. The mathematical foundations of economic models A major postwar development in microeconomic theory was the clariﬁcation and formalization of the basic assumptions that are made about individuals and ﬁrms. The ﬁrst landmark in this development was the 1947 publication of Paul Samuelson’s Foundations of Economic Analysis, in which the author (the ﬁrst American Nobel Prize winner in economics) laid out a number of models of optimizing behavior.7 Samuelson demonstrated the importance of basing behavioral models on well-speciﬁed mathematical postulates so that various optimization
techniques from mathematics could be applied. The power of his approach made it inescapably clear that mathematics had become an integral part of modern economics. In Chapter 2 of this book we review some of the mathematical concepts most often used in microeconomics. 7Paul A. Samuelson, Foundations of Economic Analysis (Cambridge, MA: Harvard University Press, 1947). 18 Part 1: Introduction New tools for studying markets A second feature that has been incorporated into this book is the presentation of a number of new tools for explaining market equilibria. These include techniques for describing pricing in single markets, such as increasingly sophisticated models of monopolistic pricing or models of the strategic relationships among ﬁrms that use game theory. They also include general equilibrium tools for simultaneously exploring relationships among many markets. As we shall see, all these new techniques help to provide a more complete and realistic picture of how markets operate. The economics of uncertainty and information A ﬁnal major theoretical advance during the postwar period was the incorporation of uncertainty and imperfect information into economic models. Some of the basic assumptions used to study behavior in uncertain situations were originally developed in the 1940s in connection with the theory of games. Later developments showed how these ideas could be used to explain why individuals tend to be adverse to risk and how they might gather information to reduce the uncertainties they face. In this book, problems of uncertainty and information enter the analysis on many occasions. Computers and empirical analysis One ﬁnal aspect of the postwar development of microeconomics should be mentioned— the increasing use of computers to analyze economic data and build economic models. As computers have become able to handle larger amounts of information and carry out complex mathematical manipulations, economists’ ability to test their theories has dramatically improved. Whereas previous generations had to be content with rudimentary tabular or graphical analyses of real-world data, today’s economists have available a wide variety of sophisticated techniques together with extensive microeconomic data with which to test their models. To examine these techniques and some of their limitations would be beyond the scope and purpose of this book. However, the Extensions at the end of most chapters are intended to help you start reading about some of these applications. SUMMARY This chapter provided background on how economists approach the study of the allocation of resources. Much of the material discussed here should be familiar to you from introductory economics. In many respects, the study of economics represents acquiring increasingly sophisticated tools for addressing the same basic problems. The purpose of this book (and, indeed
, of most upper-level books on economics) is to provide you with more of these tools. As a starting place, this chapter reminded you of the following points: • Economics is the study of how scarce resources are allocated among alternative uses. Economists seek to develop simple models to help understand that process. these models have a mathematical basis Many of because the use of mathematics offers a precise shorthand for stating the models and exploring their consequences. • The most commonly used economic model is the supply–demand model ﬁrst thoroughly developed by Alfred Marshall in the latter part of the nineteenth century. This model shows how observed prices can be taken to represent an equilibrium balancing of the production costs incurred by ﬁrms and the willingness of demanders to pay for those costs. • Marshall’s model of equilibrium is only ‘‘partial’’—that is, it looks only at one market at a time. To look at many markets together requires an expanded set of general equilibrium tools. • Testing the validity of an economic model is perhaps the most difﬁcult task economists face. Occasionally, a model’s validity can be appraised by asking whether it is based on ‘‘reasonable’’ assumptions. More often, however, models are judged by how well they can explain economic events in the real world. SUGGESTIONS FOR FURTHER READING Chapter 1: Economic Models 19 On Methodology Blaug, Mark, and John Pencavel. The Methodology of Economics: Or How Economists Explain, 2nd ed. Cambridge: Cambridge University Press, 1992. A revised and expanded version of a classic study on economic methodology. Ties the discussion to more general issues in the philosophy of science. Boland, Lawrence E. Journal of Economic Literature (June 1979): 503–22. ‘‘A Critique of Friedman’s Critics.’’ Good summary of criticisms of positive approaches to economics and of the role of empirical veriﬁcation of assumptions. Friedman, Milton. ‘‘The Methodology of Positive Economics.’’ In Essays in Positive Economics, pp. 3–43. Chicago: University of Chicago Press, 1953. Basic statement of Friedman’s positivist views. Harrod, Roy F. ‘‘Scope and Method in Economics.’’ Economic Journal 48 (1938): 383–412. Classic statement of appropriate role for economic modeling
. Hausman, David M., and Michael S. McPherson. Economic Analysis, Moral Philosophy, and Public Policy, 2nd ed. Cambridge, UK: Cambridge University Press, 2006. The authors stress their belief that consideration of issues in moral philosophy can improve economic analysis. McCloskey, Donald N. If You’re So Smart: The Narrative of Economic Expertise. Chicago: University of Chicago Press, 1990. Discussion of McCloskey’s view that economic persuasion depends on rhetoric as much as on science. For an interchange on this topic, see also the articles in the Journal of Economic Literature, June 1995. Sen, Amartya. On Ethics and Economics. Oxford, UK: Blackwell Reprints, 1989. The author seeks to bridge the gap between economics and ethical studies. This is a reprint of a classic study on this topic. Primary Sources on the History of Economics Edgeworth, F. Y. Mathematical Psychics. London: Kegan Paul, 1881. Initial investigations of welfare economics, including rudimentary notions of economic efﬁciency and the contract curve. Marshall, A. Principles of Economics, 8th ed. London: Macmillan & Co., 1920. Complete summary of neoclassical view. A long-running, popular text. Detailed mathematical appendix. Marx, K. Capital. New York: Modern Library, 1906. Full development of labor theory of value. Discussion of ‘‘transformation problem’’ provides a (perhaps faulty) start for general equilibrium analysis. Presents fundamental criticisms of institution of private property. Ricardo, D. Principles of Political Economy and Taxation. London: J. M. Dent & Sons, 1911. Very analytical, tightly written work. Pioneer in developing careful analysis of policy questions, especially trade-related issues. Discusses ﬁrst basic notions of marginalism. Smith, A. The Wealth of Nations. New York: Modern Library, 1937. First great economics classic. Long and detailed, but Smith had the ﬁrst word on practically every economic matter. This edition has helpful marginal notes. Walras, L. Elements of Pure Economics. Translated by W. Jaffe. Homewood, IL: Richard D. Irwin, 1954. Beginnings of general equilibrium theory. Rather difﬁcult reading. Secondary Sources on the History of Economics Backhouse, Roger E. The Ordinary Business of Life: The History of Economics from the
Ancient World to the 21st Century. Princeton, NJ: Princeton University Press, 2002. An iconoclastic history. Good (although brief) on the earliest economic ideas, but some blind spots on recent uses of mathematics and econometrics. Blaug, Mark. Economic Theory in Retrospect, 5th ed. Cambridge, UK: Cambridge University Press, 1997. Complete ‘‘Readers’ Guides’’ to the classics in each chapter. stressing analytical summary issues. Excellent Heilbroner, Robert L. The Worldly Philosophers, 7th ed. New York: Simon & Schuster, 1999. Fascinating, easy-to-read biographies of leading economists. Chapters on Utopian Socialists and Thorstein Veblen highly recommended. Keynes, John M. Essays in Biography. New York: W. W. Norton, 1963. (Lloyd George, Winston Essays on many famous persons Churchill, Leon Trotsky) and on several economists (Malthus, Marshall, Edgeworth, F. P. Ramsey, and Jevons). Shows the true gift of Keynes as a writer. Schumpeter, J. A. History of Economic Analysis. New York: Oxford University Press, 1954. Encyclopedic treatment. Covers all the famous and many notso-famous economists. Also brieﬂy summarizes concurrent developments in other branches of the social sciences. This page intentionally left blank CHAPTER TWO Mathematics for Microeconomics Microeconomic models are constructed using a wide variety of mathematical techniques. In this chapter we provide a brief summary of some of the most important techniques that you will encounter in this book. A major portion of the chapter concerns mathematical procedures for ﬁnding the optimal value of some function. Because we will frequently adopt the assumption that an economic actor seeks to maximize or minimize some function, we will encounter these procedures (most of which are based on calculus) many times. After our detailed discussion of the calculus of optimization, we turn to four topics that are covered more brieﬂy. First, we look at a few special types of functions that arise in economics. Knowledge of properties of these functions can often be helpful in solving problems. Next, we provide a brief summary of integral calculus. Although integration is used in this book far less frequently than is differentiation, we will nevertheless encounter situations where we will want to use integrals to measure areas that are important to economic theory or to add up outcomes
that occur over time or across many individuals. One particular use of integration is to examine problems in which the objective is to maximize a stream of outcomes over time. Our third added topic focuses on techniques to be used for such problems in dynamic optimization. Finally, Chapter 2 concludes with a brief summary of mathematical statistics, which will be particularly useful in our study of economic behavior in uncertain situations. Maximization of a Function of One Variable We can motivate our study of optimization with a simple example. Suppose that a manager of a ﬁrm desires to maximize1 the proﬁts received from selling a particular good. Suppose also that the proﬁts (p) received depend only on the quantity (q) of the good sold. Mathematically, p q ð Figure 2.1 shows a possible relationship between p and q. Clearly, to achieve maximum proﬁts, the manager should produce output q$, which yields proﬁts p$. If a graph such as that of Figure 2.1 were available, this would seem to be a simple matter to be accomplished with a ruler. (2:1) : Þ ¼ f 1Here we will generally explore maximization problems. A virtually identical approach would be taken to study minimization problems because maximization of f (x) is equivalent to minimizing f (x). % 21 22 Part 1: Introduction FIGURE 2.1 Hypothetical Relationship between Quantity Produced and Proﬁts If a manager wishes to produce the level of output that maximizes proﬁts, then q$ should be produced. Notice that at q$, dp/dq 0. ¼ π π* π2 π3 π1 π = f(q) q1 q2 q* q3 Quantity Suppose, however, as is more likely, the manager does not have such an accurate picture of the market. He or she may then try varying q to see where a maximum proﬁt is obtained. For example, by starting at q1, proﬁts from sales would be p1. Next, the manager may try output q2, observing that proﬁts have increased to p2. The commonsense idea that proﬁts have increased in response to an increase in q can be stated formally as p2 % q2 % p1 q1 > 0 or Dp Dq > 0, (2:2) where the D notation is
used to mean ‘‘the change in’’ p or q. As long as Dp/Dq is positive, proﬁts are increasing and the manager will continue to increase output. For increases in output to the right of q$, however, Dp/Dq will be negative, and the manager will realize that a mistake has been made. Derivatives As you probably know, the limit of Dp/Dq for small changes in q is called the derivative of the function, p f (q), and is denoted by dp/dq or df/dq or f 0(q). More formally, the derivative of a function p ¼ ¼ dp dq ¼ f (q) at the point q1 is deﬁned as q1Þ q1 þ ð ð f f : df dq ¼ lim 0 h! h Þ % h (2:3) Notice that the value of this ratio obviously depends on the point q1 that is chosen. The derivative of a function may not always exist or it may be undeﬁned at certain points. Most of the functions studied in this book are fully differentiable, however. Value of the derivative at a point A notational convention should be mentioned: Sometimes we wish to note explicitly the point at which the derivative is to be evaluated. For example, the evaluation of the derivative at the point q q1 could be denoted by ¼ : q1 q ¼ dp dq!!!! (2:4) Chapter 2: Mathematics for Microeconomics 23 At other times, we are interested in the value of dp/dq for all possible values of q, and no explicit mention of a particular point of evaluation is made. In the example of Figure 2.1, whereas dp dq q1 q ¼!!!! > 0, < 0. dp dq q ¼ What is the value of dp/dq at q$? It would seem to be 0 because the value is positive for values of q less than q$ and negative for values of q greater than q$. The derivative is the slope of the curve in question; this slope is positive to the left of q$ and negative to the right of q$. At the point q$, the slope of f (q) is 0.!!!! q3 First-order condition for a maximum This result is general. For
a function of one variable to attain its maximum value at some point, the derivative at that point (if it exists) must be 0. Hence if a manager could estimate the function f (q) from some sort of real-world data, it would theoretically be possible to ﬁnd the point where df/dq 0. At this optimal point (say, q$), ¼ 0: ¼ df dq q q$ ¼!!!! (2:5) Second-order conditions An unsuspecting manager could be tricked, however, by a naive application of this ﬁrst-derivative rule alone. For example, suppose that the proﬁt function looks like that shown in either Figure 2.2a or 2.2b. If the proﬁt function is that shown in Figure 0, will choose point q$a. This point in 2.2a, the manager, by producing where dp/dq fact yields minimum, not maximum, proﬁts for the manager. Similarly, if the proﬁt the manager will choose point q$b, which, function is that shown in Figure 2.2b, although it yields a proﬁt greater than that for any output lower than q$b, is certainly inferior to any output greater than q$b. These situations illustrate the mathematical fact that dp/dq 0 is a necessary condition for a maximum, but not a sufﬁcient condition. To ensure that the chosen point is indeed a maximum point, a second condition must be imposed. ¼ ¼ Intuitively, this additional condition is clear: The proﬁt available by producing either a bit more or a bit less than q$ must be smaller than that available from q$. If this is not true, the manager can do better than q$. Mathematically, this means that dp/dq must be greater than 0 for q < q$ and must be less than 0 for q > q$. Therefore, at q$, dp/dq must be decreasing. Another way of saying this is that the derivative of dp/dq must be negative at q$. Second derivatives The derivative of a derivative is called a second derivative and is denoted by d2p dq2 or d2f dq2 or f 00 : q Þ ð 24 Part 1: Introduction FIGURE 2.2 Two Proﬁt Functions That Give Mis
leading Results If the First Derivative Rule Is Applied Uncritically In (a), the application of the ﬁrst derivative rule would result in point q$a being chosen. This point is in fact a point of minimum proﬁts. Similarly, in (b), output level q$b would be recommended by the ﬁrst derivative rule, but this point is inferior to all outputs greater than q$b. This demonstrates graphically that ﬁnding a point at which the derivative is equal to 0 is a necessary, but not a sufﬁcient, condition for a function to attain its maximum value. π πa (a) π πb qa Quantity qb Quantity (b) The additional condition for q$ to represent a (local) maximum is therefore d2p dq2 q$ q ¼!!!! f 00 q ð Þ ¼ q$ q ¼!!!! < 0, (2:6) ¼ Hence although Equation 2.5 (dp/dq where the notation is again a reminder that this second derivative is to be evaluated at q$. 0) is a necessary condition for a maximum, that equation must be combined with Equation 2.6 (d2p/dq2 < 0) to ensure that the point is a local maximum for the function. Therefore, Equations 2.5 and 2.6 together are sufﬁcient conditions for such a maximum. Of course, it is possible that by a series of trials the manager may be able to decide on q$ by relying on market information rather than on mathematical reasoning (remember Friedman’s pool-player analogy). In this book we shall be less interested in how the point is discovered than in its properties and how the point changes when conditions change. A mathematical development will be helpful in answering these questions. Rules for ﬁnding derivatives Here are a few familiar rules for taking derivatives of a function of a single variable. We will use these at many places in this book. 1. If a is a constant, then 2. If a is a constant, then 3. If a is a constant, then da dx ¼ 0: d ½ af x Þ( ð dx ¼ af 0 x ð : Þ dxa dx ¼ axa 1: % Chapter 2: Mathematics for Microeconomics 25 d ln x
dx ¼ 1 x where ln signiﬁes the logarithm to the base e ( dax axlna for any constant a dx ¼ A particular case of this rule is dex/dx ex. ¼ ¼ 2.71828). Now suppose that f (x) and g (x) are two functions of x and that f 0(x) and g 0(x) exist. Then dx : g x Þ ð dx =g x Þ ð dx x ð Þ( x ð Þ( d d f f ½ ½ Þ( f 0 x ð Þ þ g0 x ð. Þ ¼ x g0 g0 Þ x ð x ð Þ, provided that g (x) Finally, if y 0. f (x) and x 6¼ dy dz ¼ dy dx ) dx dz ¼ ¼ df dx ) dg dz. g (z) and if both f 0(x) and g 0(z) exist, then ¼ d This result is called the chain rule. It provides a convenient way to study how one variable (z) affects another variable ( y) solely through its inﬂuence on some intermediate variable (x). Some examples are deax dx ¼ ax ln d ð dx x2 ln ð dx ax ð Þ dx ¼ ln d ax ½ ð d ax ð Þ x2 ln ð ½ Þ( x2 d ) ax ð Þ dx ¼ x2 ð Þ dx ¼ 1 ax ) 1 x2 ) deax ax d ð Þ( 1 x 2 x aeax. eax 2x Þ d ¼ ¼ ¼ ¼ ¼ Þ( Þ. 5. 6. 7. 8. 9. 10. 11. 12. EXAMPLE 2.1 Proﬁt Maximization Suppose that the relationship between proﬁts (p) and quantity produced (q) is given by A graph of this function would resemble the parabola shown in Figure 2.1. The value of q that maximizes proﬁts can be found by differentiation: p q ð Þ ¼ 1,000q 5q2: % (2:7) thus dp dq ¼ 1
,000 10q 0, ¼ % q$ ¼ 100: (2:8) (2:9) 100, Equation 2.7 shows that proﬁts are 50,000—the largest value possible. If, for example, 50, proﬁts would be 37,500. At q 200, proﬁts are precisely 0. ¼ That q 100 is a ‘‘global’’ maximum can be shown by noting that the second derivative of 10 (see Equation 2.8). Hence the rate of increase in proﬁts is always % 100 this rate of increase is still positive, but beyond that point it becomes 100 is the only local maximum value for the function p. With ¼ the proﬁt function is decreasing—up to q negative. In this example, q more complex functions, however, there may be several such maxima. ¼ ¼ At q the ﬁrm opted to produce q ¼ ¼ QUERY: Suppose that a ﬁrm’s output (q) is determined by the amount of labor (l) it hires according lp. Suppose also that the ﬁrm can hire all the labor it wants at $10 per unit and to the function q sells its output at $50 per unit. Therefore, proﬁts are a function of l given by p 10l. How much labor should this ﬁrm hire to maximize proﬁts, and what will those proﬁts be? Þ ¼ 100 lp l ð ¼ % 2 ﬃﬃ ﬃﬃ 26 Part 1: Introduction Functions of Several Variables Economic problems seldom involve functions of only a single variable. Most goals of interest to economic agents depend on several variables, and trade-offs must be made among these variables. For example, the utility an individual receives from activities as a consumer depends on the amount of each good consumed. For a ﬁrm’s production function, the amount produced depends on the quantity of labor, capital, and land devoted to production. In these circumstances, this dependence of one variable ( y) on a series of other variables (x1, x2, …, xn) is denoted by y f : x1, x2,..., xnÞ ð ¼ (2:10) Partial derivatives
We are interested in the point at which y reaches a maximum and in the trade-offs that must be made to reach that point. It is again convenient to picture the agent as changing the variables at his or her disposal (the x’s) to locate a maximum. Unfortunately, for a function of several variables, the idea of the derivative is not well deﬁned. Just as the steepness of ascent when climbing a mountain depends on which direction you go, so does the slope (or derivative) of the function depend on the direction in which it is taken. Usually, the only directional slopes of interest are those that are obtained by increasing one of the x’s while holding all the other variables constant (the analogy for mountain climbing might be to measure slopes only in a north–south or east–west direction). These directional slopes are called partial derivatives. The partial derivative of y with respect to (i.e., in the direction of) x1 is denoted by @y @x1 or @f @x1 or fx1 or f1: It is understood that in calculating this derivative all the other x’s are held constant. Again it should be emphasized that the numerical value of this slope depends on the value of x1 and on the (preassigned and constant) values of x2, …, xn. A somewhat more formal deﬁnition of the partial derivative is @f @x1!!!! x2,...,xn ¼ lim 0 h! f x1 þ ð h, x2,..., xnÞ % h f x1, x2,..., xnÞ ð, (2:11) where the notation is intended to indicate that x2, …, xn are all held constant at the preassigned values x2,..., xn so the effect of changing x1 only can be studied. Partial derivatives with respect to the other variables (x2, …, xn) would be calculated in a similar way. Calculating partial derivatives It is easy to calculate partial derivatives. The calculation proceeds as for the usual derivative by treating x2, …, xn as constants (which indeed they are in the deﬁnition of a partial derivative). Consider the following examples. 1. If y f x1, x2Þ ¼ ð ¼ ax2 1 þ and 2, then cx
2 bx1x2 þ @f @x1 ¼ f1 ¼ 2ax1 þ bx2 @f @x2 ¼ f2 ¼ bx1 þ 2cx2: Chapter 2: Mathematics for Microeconomics 27 Notice that @f/@x1 is in general a function of both x1 and x2; therefore, its value will depend on the particular values assigned to these variables. It also depends on the parameters a, b, and c, which do not change as x1 and x2 change. 2. If y = f x1, x2Þ ¼ ð eax1þ bx2, then and 3. If y f (x1, x2) a ln x1 þ ¼ ¼ and @f @x1 ¼ f1 ¼ aeax1þ bx2 @f @x2 ¼ f2 ¼ beax1þ bx2 : b ln x2, then @f @x1 ¼ f1 ¼ a x1 @f @x2 ¼ f2 ¼ b x2 : Notice here that the treatment of x2 as a constant in the derivation of @f/@x1 causes the term b ln x2 to disappear on differentiation because it does not change when x1 changes. In this case, unlike our previous examples, the size of the effect of x1 on y is independent of the value of x2. In other cases, the effect of x1 on y will depend on the level of x2. Partial derivatives and the ceteris paribus assumption In Chapter 1, we described the way in which economists use the ceteris paribus assumption in their models to hold constant a variety of outside inﬂuences so the particular relationship being studied can be explored in a simpliﬁed setting. Partial derivatives are a precise mathematical way of representing this approach; that is, they show how changes in one variable affect some outcome when other inﬂuences are held constant—exactly what economists need for their models. For example, Marshall’s demand curve shows the relationship between price ( p) and quantity (q) demanded when other factors are held constant. Using partial derivatives, we could represent the slope of this curve by @q/@p to indicate the ceteris paribus assumptions that are in effect. The fundamental law of demand
—that price and quantity move in opposite directions when other factors do not change—is therefore reﬂected by the mathematical statement @q/@p < 0. Again, the use of a partial derivative serves as a reminder of the ceteris paribus assumptions that surround the law of demand. Partial derivatives and units of measurement In mathematics relatively little attention is paid to how variables are measured. In fact, most often no explicit mention is made of the issue. However, the variables used in economics usually refer to real-world magnitudes; therefore, we must be concerned with how they are measured. Perhaps the most important consequence of choosing units of measurement is that the partial derivatives often used to summarize economic behavior will reﬂect these units. For example, if q represents the quantity of gasoline demanded by all U.S. consumers during a given year (measured in billions of gallons) and p represents the price in dollars per gallon, then @q/@p will measure the change in demand (in billions of gallons per year) for a dollar per gallon change in price. The numerical size of this 28 Part 1: Introduction derivative depends on how q and p are measured. A decision to measure consumption in millions of gallons per year would multiply the size of the derivative by 1,000, whereas a decision to measure price in cents per gallon would reduce it by a factor of 100. The dependence of the numerical size of partial derivatives on the chosen units of measurement poses problems for economists. Although many economic theories make predictions about the sign (direction) of partial derivatives, any predictions about the numerical magnitude of such derivatives would be contingent on how authors chose to measure their variables. Making comparisons among studies could prove practically impossible, especially given the wide variety of measuring systems in use around the world. For this reason, economists have chosen to adopt a different, unit-free way to measure quantitative impacts. Elasticity—A general deﬁnition Economists use elasticities to summarize virtually all the quantitative impacts that are of interest to them. Because such measures focus on the proportional effect of a change in one variable on another, they are unit-free—the units ‘‘cancel out’’ when the elasticity is calculated. For example, suppose that y is a function of x (which we can denote by y(x)). Then the elasticity of y with respect to x (which we will denote by ey, x) is deﬁned as ey,x ¼
Dy y Dx x Dy Dx ) x y ¼ dy x ð Þ dx ) x y ¼ (2:12) If the variable y depends on several variables in addition to x (as will often be the case), the derivative in Equation 2.12 would be replaced by a partial derivative. In either case, notice how the units in which y and x are measured cancel out in the deﬁnition of elasticity; the result is a ﬁgure that is a pure number with no dimensions. This makes it possible for economists to compare elasticities across different countries or across rather different goods. You should already be familiar with the price elasticities of demand and supply usually encountered in a ﬁrst economics course. Throughout this book you will encounter many more such concepts. EXAMPLE 2.2 Elasticity and Functional Form The deﬁnition in Equation 2.12 makes clear that elasticity should be evaluated at a speciﬁc point on a function. In general the value of this parameter would be expected to vary across different ranges of the function. This observation is most clearly shown in the case where y is a linear function of x of the form In this case, y a þ ¼ bx þ other terms: ey,x ¼ dy dx ) bx,... þ ) a þ (2:13) which makes clear that ey,x is not constant. Hence for linear functions it is especially important to note the point at which elasticity is to be computed. If the functional relationship between y and x is of the exponential form then the elasticity is a constant, independent of where it is measured: axb; y ¼ ey,x ¼ dy dx ) x y ¼ 1 abxb % x axb ¼ ) b: Chapter 2: Mathematics for Microeconomics 29 A logarithmic transformation of this equation also provides a convenient alternative deﬁnition of elasticity. Because we have ln y ln a þ ¼ b ln x, ey,x ¼ b ¼ d ln y d ln x : (2:14) Hence elasticities can be calculated through ‘‘logarithmic differentiation.’’ As we shall see, this is frequently the easiest way to proceed in making such calculations. QUERY: Are there any functional forms in addition to the exponential that have a constant
elasticity, at least over some range? Second-order partial derivatives The partial derivative of a partial derivative is directly analogous to the second derivative of a function of one variable and is called a second-order partial derivative. This may be written as or more simply as @ @f =@xiÞ ð @xj @2f @xj@xi ¼ f ij: (2:15) For the examples discussed previously: cx2 2 ax2 1. 1 þ bx1x2 þ f x1, x2Þ ¼ ð 2a 2c b b y ¼ f11 ¼ f12 ¼ f21 ¼ f22 ¼ y ¼ f11 ¼ f12 ¼ f21 ¼ f22 ¼ y ¼ f11 ¼ % f12 ¼ 0 f21 ¼ 0 f22 ¼ % 2. 3. eax1þ bx2 bx2 x1; x2Þ ¼ f ð a2eax1þ abeax1þ abeax1þ b2eax1þ bx2 bx2 bx2 b lnx2 a lnx1 þ ax% 1 2 2 bx% 2 30 Part 1: Introduction Young’s theorem These examples illustrate the mathematical result that, under general conditions, the order in which partial differentiation is conducted to evaluate second-order partial derivatives does not matter. That is, f ij ¼ f ji (2:16) for any pair of variables xi, xj. This result is sometimes called Young’s theorem. For an intuitive explanation of the theorem, we can return to our mountain-climbing analogy. In this example, the theorem states that the gain in elevation a hiker experiences depends on the directions and distances traveled, but not on the order in which these occur. That is, the gain in altitude is independent of the actual path taken as long as the hiker proceeds from one set of map coordinates to another. He or she may, for example, go one mile north, then one mile east or proceed in the opposite order by ﬁrst going one mile east, then one mile north. In either case, the gain in elevation is the same because in both cases the hiker is moving from one speciﬁc place to another. In later chapters we will make good use of this
result because it provides a convenient way of showing some of the predictions that economic models make about behavior.2 Uses of second-order partials Second-order partial derivatives will play an important role in many of the economic theories that are developed throughout this book. Probably the most important examples relate to the ‘‘own’’ second-order partial, fii. This function shows how the marginal inﬂuence of xi on y (i.e., @y/@xi) changes as the value of xi increases. A negative value for fii is the mathematical way of indicating the economic idea of diminishing marginal effectiveness. Similarly, the cross-partial fij indicates how the marginal effectiveness of xi changes as xj increases. The sign of this effect could be either positive or negative. Young’s theorem indicates that, in general, such cross-effects are symmetric. More generally, the second-order partial derivatives of a function provide information about the curvature of the function. Later in this chapter we will see how such information plays an important role in determining whether various second-order conditions for a maximum are satisﬁed. They also play an important role in determining the signs of many important derivatives in economic theory. The chain rule with many variables Calculating partial derivatives can be rather complicated in cases where some variables depend on other variables. As we will see, in many economic problems it can be hard to tell exactly how to proceed in differentiating complex functions. In this section we illustrate a few simple cases that should help you to get the general idea. We start with looking at how the ‘‘chain rule’’ discussed earlier in a single variable context can be generalized to many variables. Speciﬁcally, suppose that y is a function of three variables, f (x1, x2, x3). Suppose further that each of these x’s is itself a function of a single pay rameter, say a. Hence we can write y f [x1(a), x2(a), x3(a)]. Now we can ask how a change in a affects the value of y, using the chain rule: ¼ ¼ dy da ¼ @f @x1 ) dx1 da þ @f @x2 ) dx2 da þ @f @x3 ) dx3 da (2:17) 2Young’s theorem implies that the matrix of the second-order partial
derivatives of a function is symmetric. This symmetry offers a number of economic insights. For a brief introduction to the matrix concepts used in economics, see the Extensions to this chapter. Chapter 2: Mathematics for Microeconomics 31 In words, changes in a affect each of the x’s, and then these changes in the x’s affect the ﬁnal value of y. Of course, some of the terms in this expression may be zero. That would be the case if one of the x’s is not affected by a or if a particular x had no effect on y (in which case it should not be in the function). But this version of the chain rule shows that a can inﬂuence y through many routes.3 In our economic models we will want to be sure that all those routes are taken into account. EXAMPLE 2.3 Using the Chain Rule As a simple (and probably unappetizing) example, suppose that each week a pizza fanatic consumes three kinds of pizza, denoted by x1, x2, and x3. Type 1 pizza is a simple cheese pizza costing p per pie. Type 2 pizza adds two toppings and costs 2p. Type 3 pizza is the house special, which includes ﬁve toppings and costs 3p. To ensure a (modestly) diversiﬁed menu, this fanatic decides to allocate $30 each week to each type of pizza. Here we wish to examine how the total number of pizzas purchased is affected by the underlying price p. Notice that this problem includes a single exogenous variable, p, which is set by the pizza shop. The quantities of each pizza purchased (and total purchases) are the endogenous variables in the model. Because of the way this fanatic budgets his pizza purchases, the quantity purchased of each 30/3p. Now total type depends only on the price p. Speciﬁcally, x1 ¼ pizza purchases (y) are given by x1ð, x2ð Þ Applying the chain rule from Equation 2.17 to this function yields: 30/p, x2 ¼, x3ð Þ x2ð x1ð Þ 30/2p, x3 ¼ x3ð p Þ Þ þ dy dp ¼ dx1 dp þ f2 ) dx2 dp þ f3 ) f1 ) dx3 dp ¼ %
2 30p% 2 15p% % % 2 10p% 2 55p% ¼ % (2:18) (2:19) We can interpret this with a numerical illustration. Suppose that initially p 5. With this price total pizza purchases will be 11 pies. Equation 2.19 implies that each unit price increase would reduce purchases by 2.2 ( 55/25) pies, but such a change is too large for calculus (which assumes small changes) to work correctly. Therefore, instead, let’s assume p increases by 5 cents 5.05. Equation 2.19 now predicts that total pizza purchases will decrease by 0.11 pies to p (0.05 55/25). If we calculate pie purchases directly we get x1 ¼ 1.98. Hence total pies purchased are 10.89—a reduction of 0.11 from the original level, just what was predicted by Equation 2.19. 2.97, x3 ¼ 5.94, x2 ¼ ¼ * ¼ ¼ QUERY: It should be obvious that a far easier way to solve this problem would be to deﬁne total pie purchases (y) directly as a function of p. Provide a proof using this approach, and then describe some reasons why this simpler approach may not always be possible to implement. One special case of this chain rule might be explicitly mentioned here. Suppose x3(a) ¼ a. That is, suppose that the parameter a enters directly into the determination of y f [x1(a), x2(a), a]. In this case the effect of a on y can be written as:4 ¼ dy da ¼ @f @x1 ) dx1 da þ @f @x2 ) dx2 da þ @f @a (2:20) 3If the x’s in Equation 2.17 depended on several parameters, all the derivatives in the equation would be partial derivatives to indicate that the chain rule looks at the effect of only one parameter at a time, holding the others constant. 4The expression in Equation 2.20 is sometimes called the total derivative or full derivative of the function f, although this usage is not consistent across various ﬁelds of applied mathematics. 32 Part 1: Introduction This shows that the effect of a on y can be decomposed into two different kinds of effects: (1) a direct effect (which is given by fa); and
(2) an indirect effect that operates only through the ways in which a affects the x’s. In many economic problems, analyzing these two effects separately can provide a number of important insights. Implicit functions If the value of a function is held constant, an implicit relationship is created among the independent variables that enter into the function. That is, the independent variables can no longer take on any values, but must instead take on only that set of values that result in the function’s retaining the required value. Examining these implicit relationships can often provide another analytical tool for drawing conclusions from economic models. ¼ Probably the most useful result provided by this approach is in the ability to quantify the trade-offs inherent in most economic models. Here we will look at a simple case. f (x1, x2). If we hold the value of y constant, we have created Consider the function y an implicit relationship between the x’s showing how changes in them must be related to keep the value of the function constant. In fact, under fairly general conditions5 (the most 0) it can be shown that holding y constant allows the creimportant of which is that f2 6¼ ation of an implicit function of the form x2 ¼ g (x1). Although computing this function may sometimes be difﬁcult, the derivative of the function g is related in a speciﬁc way to the partial derivatives of the original function f. To show this, ﬁrst set the original function equal to a constant (say, zero) and write the function as x1ÞÞ ð x1, x2Þ ¼ ð Using the chain rule to differentiate this relationship with respect to x1 yields: (2:21) x1, g ¼ ¼ 0 y ð f f f1 þ f2 ) Rearranging terms gives the ﬁnal result that ¼ 0 dg x1Þ ð dx1 dg x1Þ ð dx1 ¼ dx2 dx1 ¼ % f1 f2 : (2:22) (2:23) Thus, we have shown6 that the partial derivatives of the function f can be used to derive an explicit expression for the trade-offs between x1 and x2. The next example shows how this can make computations much easier in certain situations. EXAMPLE 2.4 A Production Possibility Frontier—Again
In Example 1.3 we examined a production possibility frontier for two goods of the form Because this function is set equal to a constant, we can study the relationship between the variables by using the implicit function result: 0:25y2 x2 þ ¼ 200: (2:24) 5For a detailed discussion of this implicit function theorem and of how it can be extended to many variables, see Carl P. Simon and Lawrence Blume, Mathematics for Economists (New York: W.W. Norton, 1994), chapter 15. 6An alternative approach to proving this result uses the total differential of f: dy ing terms gives the same result (assuming one can make the mathematically questionable move of dividing by dx1). f2 dx2. Setting dy 0 and rearrang- f1 dx1 þ ¼ ¼ Chapter 2: Mathematics for Microeconomics 33 dy dx ¼ f x % f y ¼ 2x % 0:5y ¼ 4x % y, (2:25) which is precisely the result we obtained earlier, with considerably less work. QUERY: Why does the trade-off between x and y here depend only on the ratio of x to y and not on the size of the labor force as reﬂected by the 200 constant? Maximization of Functions of Several Variables Using partial derivatives allows us to ﬁnd the maximum value for a function of several variables. To understand the mathematics used in solving this problem, an analogy to the one-variable case is helpful. In this one-variable case, we can picture an agent varying x by a small amount, dx, and observing the change in y, dy. This change is given by f 0 ¼ dy dx: Þ x ð The identity in Equation 2.26 records the fact that the change in y is equal to the change in x times the slope of the function. This formula is equivalent to the point-slope formula used for linear equations in basic algebra. As before, the necessary condition for a maximum is that dy 0 for small changes in x around the optimal point. Otherwise, y could be increased by suitable changes in x. But because dx does not necessarily equal 0 in 0. This is another 0 must imply that at the desired point, f 0(x) Equation 2.26, dy way of obtaining the ﬁrst-order condition for a maximum that we already derived. (2:26) ¼ ¼ ¼ Using
this analogy, let’s look at the decisions made by an economic agent who must choose the levels of several variables. Suppose that this agent wishes to ﬁnd a set of x’s f (x1, x2, …, xn). The agent might consider changing that will maximize the value of y only one of the x’s, say x1, while holding all the others constant. The change in y (i.e., dy) that would result from this change in x1 is given by ¼ @f @x1 ¼ dy dx1 ¼ This says that the change in y is equal to the change in x1 times the slope measured in the x1 direction. Using the mountain analogy again, the gain in altitude a climber heading north would achieve is given by the distance northward traveled times the slope of the mountain measured in a northward direction. f 1dx1: (2:27) First-order conditions for a maximum For a speciﬁc point to provide a (local) maximum value to the function f it must be the case that no small movement in any direction can increase its value. Hence all the directional terms similar to Equation 2.27 must not increase y, and the only way this can happen is if all the directional (partial) derivatives are zero (remember, the term dx1 in Equation 2.27 could be either positive or negative). That is, a necessary condition for a point to be a local maximum is that at this point: f2 ¼ ) ) ) ¼ fn ¼ f1 ¼ (2:28) 0 Technically, a point at which Equation 2.25 holds is called a critical point of the function. It is not necessarily a maximum point unless certain second-order conditions (to be 34 Part 1: Introduction discussed later) hold. In most of our economic examples, however, these conditions will hold; thus, Equation 2.28 will allow us to ﬁnd a maximum. The necessary ‘‘ﬁrst-order’’ conditions for a maximum described by Equation 2.28 also have an important economic interpretation. They say that for a function to reach its maximal value, any input to the function must be increased up to the point at which its marginal (or incremental) value to the function is zero. If, say, f1 were positive at a point, this could not be a true
maximum because an increase in x1 (holding all other variables constant) would, by Equation 2.27, increase f. EXAMPLE 2.5 Finding a Maximum Suppose that y is a function of x1 and x2 given by or y ¼ %ð x1 % 2 1 Þ x2 % 2 2 Þ þ % ð 10 (2:29) y 2x1 % For example, y might represent an individual’s health (measured on a scale of 0 to 10), and x1 and x2 might be daily dosages of two health-enhancing drugs. We wish to ﬁnd values for x1 and x2 that make y as large as possible. Taking the partial derivatives of y with respect to x1 and x2 and applying the necessary conditions given by Equation 2.28 yields 4x2 þ ¼ % 5: x2 2 þ x2 1 þ or @y @x1 ¼ % @y @x2 ¼ % 2x1 þ 2 ¼ 0, 2x2 þ 4 ¼ 0 x$1 ¼ x$2 ¼ 1, 2: (2:30) Therefore, the function is at a critical point when x1 ¼ 10 is the best health status possible. A bit of experimentation provides convincing evidence that this is the greatest value y can have. For example, if x1 ¼ 1, then x2 ¼ 9. Values of x1 and x2 larger than 1 and 2, respectively, reduce y because the negative y quadratic terms in Equation 2.29 become large. Consequently, the point found by applying the necessary conditions is in fact a local (and global) maximum.7 1, x2 ¼ 0, then y 5, or if x1 ¼ 2. At that point, y ¼ x2 ¼ ¼ ¼ QUERY: Suppose y took on a ﬁxed value (say, 5). What would the relationship implied between x1 and x2 look like? How about for y 10? (These graphs are contour lines of the 7? Or y function and will be examined in more detail in several later chapters. See also Problem 2.1.) ¼ ¼ Second-order conditions Again, however, the conditions of Equation 2.28 are not sufﬁcient to ensure a maximum. This can be illustrated by returning to an already overworked analogy: All
hilltops are (more or less) ﬂat, but not every ﬂat place is a hilltop. A second-order condition is needed to ensure that the point found by applying Equation 2.28 is a local maximum. Intuitively, for a local maximum, y should be decreasing for any small changes in the x’s away from the critical point. As in the single variable case, this involves looking at the curvature of the function 7More formally, the point x1 ¼ our discussion later in this chapter). 1, x2 ¼ 2 is a global maximum because the function described by Equation 2.29 is concave (see Chapter 2: Mathematics for Microeconomics 35 around the critical point to be sure that the value of the function really does decrease for movements in every direction. To do this we must look at the second partial derivatives of the function. A ﬁrst condition (that draws in obvious ways from the single variable case) is that the own second partial derivative for any variable ( fii) must be negative. If we conﬁne our attention only to movements in a single direction, a true maximum must be characterized by a pattern in which the slope of the function goes from positive (up), to zero (ﬂat), to negative (down). That is what the mathematical condition fii < 0 means. Unfortunately, the conditions that assure the value of f decreases for movements in any arbitrary direction involve all the second partial derivatives. A two-variable example is discussed later in this chapter, but the general case is best discussed with matrix algebra (see the Extensions to this chapter). For economic theory, however, the fact that the own second partial derivatives must be negative for a maximum is often the most important fact. The Envelope Theorem One major application of the idea of implicit functions, which will be used many times in this book, is called the envelope theorem; it concerns how the optimal value for a particular function changes when a parameter of the function changes. Because many of the economic problems we will be studying concern the effects of changing a parameter (e.g., the effects that changing the market price of a commodity will have on an individual’s purchases), this is a type of calculation we will frequently make. The envelope theorem often provides a nice shortcut to solving the problem. A speciﬁc example Perhaps the easiest way to understand the envelope theorem is through an example. Suppose y is a
function of a single variable (x) and a parameter (a) given by y x2 ax: (2:31) ¼ % þ For different values of the parameter a, this function represents a family of inverted parabolas. If a is assigned a speciﬁc value, Equation 2.31 is a function of x only, and the 1 2 and, for value of x that maximizes y can be calculated. For example, if a 1 these values of x and a, y 1 and y$ 4 (its maximal value). Similarly, if a 1. Hence an increase of 1 in the value of the parameter a has increased the maximum ¼ value of y by 3 4. In Table 2.1, integral values of a between 0 and 6 are used to calculate the optimal values for x and the associated values of the objective, y. Notice that as a increases, the maximal value for y also increases. This is also illustrated in Figure 2.3, which shows that the relationship between a and y$ is quadratic. Now we wish to calculate explicitly how y$ changes as the parameter a changes. 1, then x$ 2, then x$ ¼ ¼ ¼ ¼ ¼ TABLE 2.1 OPTIMAL VALUES OF y AND x FOR ALTERNATIVE VALUES OF a IN y Value of a Value of x$ x2 ax þ ¼ % Value of y 25 4 9 36 Part 1: Introduction FIGURE 2.3 Illustration of the Envelope Theorem The envelope theorem states that the slope of the relationship between y$ (the maximum value of y) and the parameter a can be found by calculating the slope of the auxiliary relationship found by substituting the respective optimal values for x into the objective function and calculating @y/@a. y* = f(a) y* 10 direct, time-consuming approach The envelope theorem states that there are two equivalent ways we can make this calculation. First, we can calculate the slope of the function in Figure 2.3 directly. To do so, we must solve Equation 2.32 for the optimal value of x for any value of a: hence dy dx ¼ % 2x a þ ¼ 0; x$ ¼ a 2 : Substituting this value of x$ in Equation 2.32 gives y$ ¼ %ð 2 2 Þ x$ a 2 # $ a2 4 þ ¼ % ¼ % �
� þ x$ a ð a a 2 # $ a2 4 þ a2 2 ¼ (2:32), and this is precisely the relationship shown in Figure 2.3. From the previous equation, it is easy to see that Chapter 2: Mathematics for Microeconomics 37 dy$ da ¼ 2a 4 ¼ a 2 (2:33) and, for example, at a increasing a is to increase y$ by the same amount. Near a increase y$ by three times this change. Table 2.1 illustrates this result. 1. That is, near a 2, dy$/da ¼ ¼ ¼ 2 the marginal impact of ¼ 6, any small increase in a will The envelope shortcut Arriving at this conclusion was a bit complicated. We had to ﬁnd the optimal value of x for each value of a and then substitute this value for x$ into the equation for y. In more general cases this may be burdensome because it requires repeatedly maximizing the objective function. The envelope theorem, providing an alternative approach, states that for small changes in a, dy$/da can be computed by holding x at its optimal value and simply calculating @y/@a from the objective function directly. Proceeding in this way gives dy$ da ¼ @y @a @ ð% x2 @a þ ax Þ ¼ x a x$ð ¼ Þ x$ a Þ ð ¼ x a x$ð ¼ Þ (2:34)!!!!!!!! The notation here is a reminder that the partial derivative used in the envelope theorem must be evaluated at the value of x, which is optimal for the particular parameter value for a. In Equation 2.32 we showed that, for any value of a, x$(a) a/2. Substitution into Equation 2.34 now yields: ¼ dy$ da ¼ x$ a ð Þ ¼ a 2 (2:35) This is precisely the result obtained earlier. The reason that the two approaches yield identical results is illustrated in Figure 2.3. The tangents shown in the ﬁgure report values of y for a ﬁxed x$. The tangents’ slopes are @y/@a. Clearly, at y$ this slope gives the value we seek. This result is general, and we will use it at several places in this book to
simplify our analysis. To summarize, the envelope theorem states that the change in the optimal value of a function with respect to a parameter of that function can be found by partially differentiating the objective function while holding x at its optimal value. That is, dy$ da ¼ @y x @a f x$, a Þg ð ¼ (2:36) where the notation again provides a reminder that @y/@a must be computed at that value of x that is optimal for the speciﬁc value of the parameter a being examined. Many-variable case An analogous envelope theorem holds for the case where y is a function of several variables. Suppose that y depends on a set of x’s (x1, …, xn) and on a particular parameter of interest, say, a: Finding an optimal value for y would consist of solving n ﬁrst-order equations of the form y f x1,..., xn, a Þ ð ¼ (2:37) @y @xi ¼ 0 i ð ¼ 1,..., n, Þ (2:38) x$1, x$2,..., x$nÞ and a solution to this process would yield optimal values for these x’s ð that would implicitly depend on the parameter a. Assuming the second-order conditions 38 Part 1: Introduction are met, the implicit function theorem would apply in this case and ensure that we could solve each x$i as a function of the parameter a: x$1ð, a Þ, a x$2ð Þ... a x$nð Substituting these functions into our original objective (Equation 2.37) yields an expression in which the optimal value of y (say, y$) depends on the parameter a both directly and indirectly through the effect of a on the x$’s: x$1 ¼ x$2 ¼ x$n ¼ (2:39) : Þ : ( ¼ Totally differentiating this expression with respect to a yields a,..., x$nð Þ a, x$2ð a x$1ð Þ, a Þ y$ f ½ dy$ da ¼ @f @x1 ) dx1 da þ @f @x2 ) dx2 da þ ) ) ) þ @f @xn )
dxn da þ @f @a : (2:40) But because of the ﬁrst-order conditions, all these terms except the last are equal to 0 if the x’s are at their optimal values. Hence again we have the envelope result: where this derivative is to be evaluated at the optimal values for the x’s. dy$ da ¼ @f @a, EXAMPLE 2.6 The Envelope Theorem: Health Status Revisited Earlier, in Example 2.5, we examined the maximum values for the health status function and found that and y ¼ %ð x1 % 2 1 Þ x2 % 2 2 Þ þ % ð 10 x$1 ¼ x$2 ¼ 1, 2, (2:41) (2:42) (2:43) ¼ Suppose now we use the arbitrary parameter a instead of the constant 10 in Equation 2.42. Here a might represent a measure of the best possible health for a person, but this value would obviously vary from person to person. Hence y$ 10: y f ¼ x1, x2, a ð Þ ¼ %ð x1 % 1 2 Þ x2 % % ð 2 2 Þ a þ (2:44) In this case the optimal values for x1 and x2 do not depend on a (they are always x$1 ¼ x$2 ¼ 2); therefore, at those optimal values we have 1, and y$ a ¼ dy$ da ¼ 1: (2:45) (2:46) Chapter 2: Mathematics for Microeconomics 39 People with ‘‘naturally better health’’ will have concomitantly higher values for y$, providing they choose x1 and x2 optimally. But this is precisely what the envelope theorem indicates because dy$ da ¼ @f @a ¼ 1 (2:47) from Equation 2.44. Increasing the parameter a simply increases the optimal value for y$ by an identical amount (again, assuming the dosages of x1 and x2 are correctly chosen). QUERY: Suppose we focused instead on the optimal dosage for x1 in Equation 2.42—that is, suppose we used a general parameter, say b, instead of 1. Explain in words and using mathematics why @y$/@b would necessarily be 0 in this case. Con
strained Maximization Thus far we have focused our attention on ﬁnding the maximum value of a function without restricting the choices of the x’s available. In most economic problems, however, not all values for the x’s are feasible. In many situations, for example, it is required that all the x’s be positive. This would be true for the problem faced by the manager choosing output to maximize proﬁts; a negative output would have no meaning. In other instances the x’s may be constrained by economic considerations. For example, in choosing the items to consume, an individual is not able to choose any quantities desired. Rather, choices are constrained by the amount of purchasing power available; that is, by this person’s budget constraint. Such constraints may lower the maximum value for the function being maximized. Because we are not able to choose freely among all the x’s, y may not be as large as it could be. The constraints would be ‘‘nonbinding’’ if we could obtain the same level of y with or without imposing the constraint. Lagrange multiplier method One method for solving constrained maximization problems is the Lagrange multiplier method, which involves a clever mathematical trick that also turns out to have a useful economic interpretation. The rationale of this method is simple, although no rigorous presentation will be attempted here.8 In a previous section, the necessary conditions for a local maximum were discussed. We showed that at the optimal point all the partial derivatives of f must be 0. Therefore, there are n equations ( fi ¼ 1, …, n) in n unknowns (the x’s). Generally, these equations can be solved for the optimal x’s. When the x’s are constrained, however, there is at least one additional equation (the constraint) but no additional variables. Therefore, the set of equations is overdetermined. The Lagrangian technique introduces an additional variable (the Lagrange multiplier), which not only helps to solve the 1 unknowns), but also has problem at hand (because there are now n an interpretation that is useful in a variety of economic circumstances. 1 equations in n 0 for i ¼ þ þ The formal problem More speciﬁcally, suppose that we wish to ﬁnd the values of x1, x2, …, xn that maximize y f, x1, x2,..., x
nÞ ð ¼ (2:48) 8For a detailed presentation, see A. K. Dixit, Optimization in Economic Theory, 2nd ed. (Oxford: Oxford University Press, 1990), chapter 2. 40 Part 1: Introduction subject to a constraint that permits only certain values of the x’s to be used. A general way of writing that constraint is where the function9 g represents the relationship that must hold among all the x’s. g x1, x2,..., xnÞ ¼ ð 0 (2:49) First-order conditions The Lagrange multiplier method starts with setting up the Lagrangian expression + f x1, x2,..., xn ð kg x1, x2,..., xn ; (2:50) ð ¼ Þ þ where l is an additional variable called the Lagrange multiplier. Later we will interpret this new variable. First, however, notice that when the constraint holds, + and f have the same value [because g (x1, x2, …, xn) 0]. Consequently, if we restrict our attention only to values of the x’s that satisfy the constraint, ﬁnding the constrained maximum value of f is equivalent to ﬁnding a critical value of +. Let’s proceed then to do so, treating l also as a variable (in addition to the x’s). From Equation 2.50, the conditions for a critical point are: ¼ Þ @+ @x1 ¼ @+ @x2 ¼... @+ @xn ¼ @+ @k ¼ f1 þ kg1 ¼ 0, f2 þ kg2 ¼ 0, fn þ kgn ¼ 0, x1, x2, g ð..., xnÞ ¼ 0: (2:51) þ þ The equations comprised by Equation 2.51 are then the conditions for a critical point for the function +. Notice that there are n 1 equations (one for each x and a ﬁnal one for l) in n 1 unknowns. The equations can generally be solved for x1, x2, …, xn, and l. Such a solution will have two properties: (1) The x’s will obey the constraint because the
last equation in 2.51 imposes that condition; and (2) among all those values of x’s that satisfy the constraint, those that also solve Equation 2.51 will make + (and hence f ) as large as possible (assuming second-order conditions are met). Therefore, the Lagrange multiplier method provides a way to ﬁnd a solution to the constrained maximization problem we posed at the outset.10 The solution to Equation 2.51 will usually differ from that in the unconstrained case (see Equations 2.28). Rather than proceeding to the point where the marginal contribution of each x is 0, Equation 2.51 requires us to stop short because of the constraint. Only if the constraint were ineffective (in which case, as we show below, l would be 0) would the constrained and unconstrained equations (and their respective solutions) agree. These revised marginal conditions have economic interpretations in many different situations. 10 0. In later chapters, we will usually follow this procedure in dealing with constraints. Often the 9As we pointed out earlier, any function of x1, x2,…, xn can be written in this implicit way. For example, the constraint x1 þ could be written 10 – x1 – x2 ¼ constraints we examine will be linear. 10Strictly speaking, these are the necessary conditions for an interior local maximum. In some economic problems, it is necessary to amend these conditions (in fairly obvious ways) to take account of the possibility that some of the x’s may be on the boundary of the region of permissible x’s. For example, if all the x’s are required to be non-negative, it may be that the conditions of Equation 2.51 will not hold exactly because these may require negative x’s. We look at this situation later in this chapter. x2 ¼ Chapter 2: Mathematics for Microeconomics 41 Interpretation of the Lagrange multiplier Thus far we have used the Lagrange multiplier (l) only as a mathematical ‘‘trick’’ to arrive at the solution we wanted. In fact, that variable also has an important economic interpretation, which will be central to our analysis at many points in this book. To develop this interpretation, rewrite the ﬁrst n equations of 2.51 as f 1 g1 ¼ % f 2 g2 ¼ ) ) ) ¼ % f n gn ¼
% k: (2:52) In other words, at the maximum point, the ratio of fi to gi is the same for every xi. The numerators in Equation 2.52 are the marginal contributions of each x to the function f. They show the marginal beneﬁt that one more unit of xi will have for the function that is being maximized (i.e., for f ). A complete interpretation of the denominators in Equation 2.52 is probably best left until we encounter these ratios in actual economic applications. There we will see that these usually have a ‘‘marginal cost’’ interpretation. That is, they reﬂect the added burden on the constraint of using slightly more xi. As a simple illustration, suppose the constraint required that total spending on x1 and x2 be given by a ﬁxed dollar amount, F. Hence the F (where pi is the per unit cost of xi). Using our presconstraint would be p1x1 þ ent terminology, this constraint would be written in implicit form as x1, x2Þ ¼ p2x2 ¼ p2x2 ¼ p1x1 % (2:53) % 0: F g ð In this situation then, gi ¼ % pi (2:54) gi does indeed reﬂect the per unit, marginal cost of using xi. Practiand the derivative cally all the optimization problems we will encounter in later chapters have a similar interpretation for the denominators in Equation 2.52. % Lagrange multiplier as a beneﬁt–cost ratio Now we can give Equation 2.52 an intuitive interpretation. The equation indicates that, at the optimal choices for the x’s, the ratio of the marginal beneﬁt of increasing xi to the marginal cost of increasing xi should be the same for every x. To see that this is an obvious condition for a maximum, suppose that it were not true: Suppose that the ‘‘beneﬁt– cost ratio’’ were higher for x1 than for x2. In this case, slightly more x1 should be used to achieve a maximum. Consider using more x1 but giving up just enough x2 to keep g (the constraint) constant. Hence the marginal cost of the additional x1 used would equal the cost saved by
using less x2. But because the beneﬁt–cost ratio (the amount of beneﬁt per unit of cost) is greater for x1 than for x2, the additional beneﬁts from using more x1 would exceed the loss in beneﬁts from using less x2. The use of more x1 and appropriately less x2 would then increase y because x1 provides more ‘‘bang for your buck.’’ Only if the marginal beneﬁt–marginal cost ratios are equal for all the x’s will there be a local maximum, one in which no small changes in the x’s can increase the objective. Concrete applications of this basic principle are developed in many places in this book. The result is fundamental for the microeconomic theory of optimizing behavior. The Lagrange multiplier (l) can also be interpreted in light of this discussion. l is the common beneﬁt–cost ratio for all the x’s. That is, marginal benefit of xi marginal cost of xi k ¼ (2:55) 42 Part 1: Introduction for every xi. If the constraint were relaxed slightly, it would not matter exactly which x is changed (indeed, all the x’s could be altered) because, at the margin, each promises the same ratio of beneﬁts to costs. The Lagrange multiplier then provides a measure of how such an overall relaxation of the constraint would affect the value of y. In essence, l assigns a ‘‘shadow price’’ to the constraint. A high l indicates that y could be increased substantially by relaxing the constraint because each x has a high beneﬁt–cost ratio. A low value of l, on the other hand, indicates that there is not much to be gained by relaxing the constraint. If the constraint is not binding, l will have a value of 0, thereby indicating that the constraint is not restricting the value of y. In such a case, ﬁnding the maximum value of y subject to the constraint would be identical to ﬁnding an unconstrained maximum. The shadow price of the constraint is 0. This interpretation of l can also be shown using the envelope theorem as described later in this chapter.11 Duality This discussion shows that there is a clear relationship between the problem of maximizing a function subject to constraints and the problem of assigning values
to constraints. This reﬂects what is called the mathematical principle of duality: Any constrained maximization problem has an associated dual problem in constrained minimization that focuses attention on the constraints in the original (primal) problem. For example, to jump a bit ahead of our story, economists assume that individuals maximize their utility, subject to a budget constraint. This is the consumer’s primal problem. The dual problem for the consumer is to minimize the expenditure needed to achieve a given level of utility. Or, a ﬁrm’s primal problem may be to minimize the total cost of inputs used to produce a given level of output, whereas the dual problem is to maximize output for a given total cost of inputs purchased. Many similar examples will be developed in later chapters. Each illustrates that there are always two ways to look at any constrained optimization problem. Sometimes taking a frontal attack by analyzing the primal problem can lead to greater insights. In other instances the ‘‘back door’’ approach of examining the dual problem may be more instructive. Whichever route is taken, the results will generally, although not always, be identical; thus, the choice made will mainly be a matter of convenience. EXAMPLE 2.7 Constrained Maximization: Health Status Yet Again Let’s return once more to our (perhaps tedious) health maximization problem. As before, the individual’s goal is to maximize 2x1 % but now assume that choices of x1 and x2 are constrained by the fact that he or she can only tolerate one drug dose per day. That is, 4x2 þ ¼ % 5, y x2 1 þ x2 2 þ or x1 þ x2 ¼ 1 1 x1 % x2 ¼ % 0: (2:56) 11The discussion in the text concerns problems involving a single constraint. In general, one can handle m constraints (m < n) by simply introducing m new variables (Lagrange multipliers) and proceeding in an analogous way to that discussed above. Chapter 2: Mathematics for Microeconomics 43 2) is no longer attainable because of the 1, x2 ¼ Notice that the original optimal point (x1 ¼ constraint on possible dosages: Other values must be found. To do so, we ﬁrst set up the Lagrangian expression: + 4x2 þ Differentiation of + with respect to x1, x2, and
l yields the following necessary condition for a constrained maximum: 2x1 % x1 % k 1 ð ¼ % (2:57) : Þ x2 % þ 5 x2 1 þ x2 2 þ 2x1 þ 2x2, 0, (2:58) @+ @x1 ¼ % @+ @x2 ¼ % @+ @k ¼ x1 % These equations must now be solved for the optimal values of x1, x2, and l. Using the ﬁrst and second equations gives x2 ¼ % 1 0: or 2x1 þ % 2 ¼ k 2x2 þ ¼ % 4 Substitution of this value for x1 into the constraint yields the solution: x1 ¼ x2 % 1: (2:59) x2 ¼ x1 ¼ In words, if this person can tolerate only one dose of drugs, he or she should opt for taking only the second drug. By using either of the ﬁrst two equations, it is easy to complete our solution by showing that (2:60) 1, 0: 2: k ¼ (2:61) This then is the solution to the constrained maximum problem. If x1 ¼ 1, then y takes on the value 8. Constraining the values of x1 and x2 to sum to 1 has reduced the maximum value of health status, y, from 10 to 8. 0, x2 ¼ QUERY: Suppose this individual could tolerate two doses per day. Would you expect y to increase? Would increases in tolerance beyond three doses per day have any effect on y? EXAMPLE 2.8 Optimal Fences and Constrained Maximization Suppose a farmer had a certain length of fence, P, and wished to enclose the largest possible rectangular area. What shape area should the farmer choose? This is clearly a problem in constrained maximization. To solve it, let x be the length of one side of the rectangle and y be the length of the other side. The problem then is to choose x and y so as to maximize the area of the x Æ y), subject to the constraint that the perimeter is ﬁxed at P ﬁeld (given by A 2y. 2x Setting up the Lagrangian expression gives 2x 2y, Þ % (2:62) 44 Part 1: Introduction where l is an unknown Lag
range multiplier. The ﬁrst-order conditions for a maximum are @+ @x ¼ @+ @y ¼ @+ @k ¼ y x P 2k 2k % % 2x % ¼ ¼ % 0; 0, 2y 0: ¼ (2:63) ¼ x/2 The three equations in 2.63 must be solved simultaneously for x, y, and l. The ﬁrst two equations say that y/2 l, showing that x must be equal to y (the ﬁeld should be square). They also imply that x and y should be chosen so that the ratio of marginal beneﬁts to marginal cost is the same for both variables. The beneﬁt (in terms of area) of one more unit of x is given by y (area is increased by 1 Æ y), and the marginal cost (in terms of perimeter) is 2 (the available perimeter is reduced by 2 for each unit that the length of side x is increased). The maximum conditions state that this ratio should be equal for each of the variables. ¼ Because we have shown that x ¼ y, we can use the constraint to show that and because y 2l2:64) (2:65) Interpretation of the Lagrange multiplier. If the farmer were interested in knowing how much more ﬁeld could be fenced by adding an extra yard of fence, the Lagrange multiplier suggests that he or she could ﬁnd out by dividing the present perimeter by 8. Some speciﬁc numbers might make this clear. Suppose that the ﬁeld currently has a perimeter of 400 yards. If the farmer has planned ‘‘optimally,’’ the ﬁeld will be a square with 100 yards ( P/4) on a side. The enclosed area will be 10,000 square yards. Suppose now that the perimeter (i.e., the available fence) were enlarged by one yard. Equation 2.65 would then ‘‘predict’’ that the total area would be increased by approximately 50 ( P/8) square yards. That this is indeed the case can be shown as follows: Because the perimeter is now 401 yards, each side of the square will be 401/4 yards. Therefore, the total area of the ﬁeld is (401/4)2
, which, according to the author’s calculator, works out to be 10,050.06 square yards. Hence the ‘‘prediction’’ of a 50-square-yard increase that is provided by the Lagrange multiplier proves to be remarkably close. As in all constrained maximization problems, here the Lagrange multiplier provides useful information about the implicit value of the constraint. ¼ ¼ Duality. The dual of this constrained maximization problem is that for a given area of a rectangular ﬁeld, to minimize the fence required to surround it. Mathematically, the problem is to minimize the farmer wishes subject to the constraint Setting up the Lagrangian expression 2x P ¼ þ 2y, A y: x ) ¼ (2:66) (2:67) þ (where the D denotes the dual concept) yields the following ﬁrst-order conditions for a minimum: þ ¼ % Þ ð ) +D 2x 2y kD A x y (2:68) Chapter 2: Mathematics for Microeconomics 45 @+D @x ¼ @+D @y ¼ @+D @kD ¼ kD kD, 0, A x y ) % ¼ 0: (2:69) Solving these equations as before yields the result ¼ Again, the ﬁeld should be square if the length of fence is to be minimized. The value of the Lagrange multiplier in this problem is ¼ ﬃﬃﬃﬃ x y Ap : (2:70) kD 2 y ¼ 2 x ¼ 2 Ap : ¼ (2:71) ﬃﬃﬃﬃ As before, this Lagrange multiplier indicates the relationship between the objective (minimizing fence) and the constraint (needing to surround the ﬁeld). If the ﬁeld were 10,000 square yards, as we saw before, 400 yards of fence would be needed. Increasing the ﬁeld by one square yard 2=100 would require about.02 more yards of fence. The reader may wish to ﬁre up 2 Þ ð his or her calculator to show this is indeed the case—a fence 100.005 yards on each side will exactly enclose 10,001 square yards. Here, as in most duality problems, the value
of the Lagrange multiplier in the dual is the reciprocal of the value for the Lagrange multiplier in the primal problem. Both provide the same information, although in a somewhat different form. % ﬃﬃﬃﬃ Ap ¼ QUERY: An implicit constraint here is that the farmer’s ﬁeld be rectangular. If this constraint were not imposed, what shape ﬁeld would enclose maximal area? How would you prove that? Envelope Theorem in Constrained Maximization Problems The envelope theorem, which we discussed previously in connection with unconstrained maximization problems, also has important applications in constrained maximization problems. Here we will provide only a brief presentation of the theorem. In later chapters we will look at a number of applications. Suppose we seek the maximum value of subject to the constraint y f, x1,..., xn; a Þ ð ¼ g x1,..., xn; a ð Þ ¼ 0, (2:72) (2:73) where we have made explicit the dependence of the functions f and g on some parameter a. As we have shown, one way to solve this problem is to set up the Lagrangian expression + f x1,..., xn; a kg x1,..., xn; a Þ ð Þ þ and solve the ﬁrst-order conditions (see Equations 2.51) for the optimal, constrained values x$1,..., x$n. Alternatively, it can be shown that ¼ ð (2:74) dy$ da ¼ @+ @a ð x$1,..., x$n; a : Þ (2:75) 46 Part 1: Introduction That is, the change in the maximal value of y that results when the parameter a changes (and all the x’s are recalculated to new optimal values) can be found by partially differentiating the Lagrangian expression (Equation 2.74) and evaluating the resultant partial derivative at the optimal point. Hence the Lagrangian expression plays the same role in applying the envelope theorem to constrained problems as does the objective function alone in unconstrained problems. As a simple exercise, the reader may wish to show that this result holds for the problem of fencing a rectangular ﬁeld described in Example 2
.7.12 A sketch of the proof of the envelope theorem in constrained problems is provided in Problem 2.12. Inequality Constraints In some economic problems the constraints need not hold exactly. For example, an individual’s budget constraint requires that he or she spend no more than a certain amount per period, but it is at least possible to spend less than this amount. Inequality constraints also arise in the values permitted for some variables in economic problems. Usually, for example, economic variables must be non-negative (although they can take on the value of zero). In this section we will show how the Lagrangian technique can be adapted to such circumstances. Although we will encounter only a few problems later in the text that require this mathematics, development here will illustrate a few general principles that are consistent with economic intuition. A two-variable example To avoid much cumbersome notation, we will explore inequality constraints only for the simple case involving two choice variables. The results derived are readily generalized. Suppose that we seek to maximize y f (x1, x2) subject to three inequality constraints: x1, x2Þ + 0; 0: and 0; (2:76) ¼ 1: g ð 2: x1 + 3: x2 + Hence we are allowing for the possibility that the constraint we introduced before need not hold exactly (a person need not spend all his or her income) and for the fact that both of the x’s must be non-negative (as in most economic problems). Slack variables One way to solve this optimization problem is to introduce three new variables (a, b, and c) that convert the inequality constraints in Equation 2.76 into equalities. To ensure that the inequalities continue to hold, we will square these new variables, ensuring that the resulting values are positive. Using this procedure, the inequality constraints become a2 1: g x1, x2Þ % ð b2 2: x1 % 0; ¼ c2 3: x2 % 0: ¼ and (2:77) ¼ 0; 12For the primal problem, the perimeter P is the parameter of principal interest. By solving for the optimal values of x and y P/8. Differentiation of the and substituting into the expression for the area (A) of the ﬁeld, it is easy to show that dA/dP Lagrangian expression (Equation 2.62) yields @+/@P
P/8. The envelope theorem in this case then offers further proof that the Lagrange multiplier can be used to assign an implicit value to the constraint. l and, at the optimal values of x and y, dA/dP @+/@P ¼ ¼ ¼ ¼ ¼ l Chapter 2: Mathematics for Microeconomics 47 Any solution that obeys these three equality constraints will also obey the inequality constraints. It will also turn out that the optimal values for a, b, and c will provide several insights into the nature of the solutions to a problem of this type. Solution using Lagrange multipliers By converting the original problem involving inequalities into one involving equalities, we are now in a position to use Lagrangian methods to solve it. Because there are three constraints, we must introduce three Lagrange multipliers: l1, l2, and l3. The full Lagrangian expression is + f x1, x2Þ þ k1½ g x1, x2Þ % ð a2 k2ð x1 % b2 ( þ We wish to ﬁnd the values of x1, x2, a, b, c, l1, l2, and l3 that constitute a critical Þ þ ¼ ð k3ð x2 % c2 : Þ (2:78) point for this expression. This will necessitate eight ﬁrst-order conditions: f1 þ f2 þ k1g1 þ k2 ¼ 0, k1g2 þ k3 ¼ 0, @+ @x1 ¼ @+ @x2 ¼ @+ @a ¼ % @+ @b ¼ % @+ @c ¼ % @+ @k1 ¼ @+ @k2 ¼ @+ @k3 ¼ g 0, 0, 2ak1 ¼ 2bk2 ¼ 2ck3 ¼ x1, x2Þ % ð b2 0, x1 % 0, ¼ x2 % c2 0, ¼ (2:79) a2 0, ¼ In many ways these conditions resemble those we derived earlier for the case of a single equality constraint (see Equation 2.51). For example, the ﬁnal three conditions merely repeat the three revised constraints. This ensures that any solution will obey these conditions. The ﬁrst two
equations also resemble the optimal conditions developed earlier. If l2 and l3 were 0, the conditions would in fact be identical. But the presence of the additional Lagrange multipliers in the expressions shows that the customary optimality conditions may not hold exactly here. Complementary slackness The three equations involving the variables a, b, and c provide the most important insights into the nature of solutions to problems involving inequality constraints. For example, the third line in Equation 2.79 implies that, in the optimal solution, either l1 or a must be 0.13 In the second case (a 0 holds exactly, and the calculated value of l1 indicates its relative importance to the objective function, f. On 0, and this shows that the availability of some slackthe other hand, if a ness in the constraint implies that its value to the objective is 0. In the consumer context, 0), the constraint g (x1, x2) 0, then l1 ¼ ¼ 6¼ ¼ 13We will not examine the degenerate case where both of these variables are 0. 48 Part 1: Introduction this means that if a person does not spend all his or her income, even more income would do nothing to raise his or her well-being. Similar complementary slackness relationships also hold for the choice variables x1 and x2. For example, the fourth line in Equation 2.79 requires that the optimal solution 0, then the optimal solution has x1 > 0, and this choice have either b or l2 be 0. If l2 ¼ 0. Alternatively, solutions l1g1 ¼ variable meets the precise beneﬁt–cost test that f1 þ 0, and also require that l2 > 0. Thus, such solutions do not where b involve any use of x1 because that variable does not meet the beneﬁt–cost test as shown l1g1 < 0. An identical result by the ﬁrst line of Equation 2.79, which implies that f1 þ holds for the choice variable x2. 0 have x1 ¼ ¼ These results, which are sometimes called Kuhn–Tucker conditions after their discoverers, show that the solutions to optimization problems involving inequality constraints will differ from similar problems involving equality constraints in rather simple ways. Hence we cannot go far wrong by working primarily with constraints involving equalities and assuming that we can rely on intuition to state what would happen if the problems involved
inequalities. That is the general approach we will take in this book.14 Second-Order Conditions and Curvature Thus far our discussion of optimization has focused primarily on necessary (ﬁrst-order) conditions for ﬁnding a maximum. That is indeed the practice we will follow throughout much of this book because, as we shall see, most economic problems involve functions for which the second-order conditions for a maximum are also satisﬁed. This is because these functions have the right curvature properties to ensure that the necessary conditions for an optimum are also sufﬁcient. In this section we provide a general treatment of these curvature conditions and their relationship to second-order conditions. The economic explanations for these curvature conditions will be discussed throughout the text. Functions of one variable First consider the case in which the objective, y, is a function of only a single variable, x. That is, A necessary condition for this function to attain its maximum value at some point is that f (x). y ¼ (2:80) dy dx ¼ f 0 x ð Þ ¼ 0 (2:81) at that point. To ensure that the point is indeed a maximum, we must have y decreasing for movements away from it. We already know (by Equation 2.81) that for small changes in x, the value of y does not change; what we need to check is whether y is increasing before that ‘‘plateau’’ is reached and decreasing thereafter. We have already derived an expression for the change in y (dy), which is given by the total differential dy f 0 x ð dx: Þ ¼ (2:82) 14The situation can become much more complex when calculus cannot be relied on to give a solution, perhaps because some of the functions in a problem are not differentiable. For a discussion, see Avinask K. Dixit, Optimization in Economic Theory, 2nd ed. (Oxford: Oxford University Press, 1990). Chapter 2: Mathematics for Microeconomics 49 What we now require is that dy be decreasing for small increases in the value of x. The differential of Equation 2.82 is given by dy d ð Þ ¼ d2y d ½ f 0 x ð dx dx Þ ( ) ¼ dx f 00 x ð dx Þ ) ¼ dx ¼ f 00 x ð dx2: �
� (2:83) But implies that d2y < 0 dx2 < 0; x Þ ð and because dx2 must be positive (because anything squared is positive), we have f 00 (2:84) f 00 x ð Þ < 0 (2:85) as the required second-order condition. In words, this condition requires that the function f have a concave shape at the critical point (contrast Figures 2.1 and 2.2). The curvature conditions we will encounter in this book represent generalizations of this simple idea. EXAMPLE 2.9 Proﬁt Maximization Again In Example 2.1 we considered the problem of ﬁnding the maximum of the function The ﬁrst-order condition for a maximum requires 1,000q p ¼ 5q2: % dp dq ¼ 1,000 10q 0 ¼ % or ¼ The second derivative of the function is given by q$ 100: d2p dq2 ¼ % 10 < 0; (2:86) (2:87) (2:88) (2:89) and hence the point q$ ¼ 100 obeys the sufﬁcient conditions for a local maximum. QUERY: Here the second derivative is negative not only at the optimal point; it is always negative. What does that imply about the optimal point? How should the fact that the second derivative is a constant be interpreted? Functions of two variables As a second case, we consider y as a function of two independent variables: : x1; x2Þ ð A necessary condition for such a function to attain its maximum value is that its partial derivatives, in both the x1 and the x2 directions, be 0. That is, (2:90) ¼ y f 50 Part 1: Introduction 0; f1 ¼ (2:91) @y @x1 ¼ @y @x2 ¼ f2 ¼ A point that satisﬁes these conditions will be a ‘‘ﬂat’’ spot on the function (a point where dy 0) and therefore will be a candidate for a maximum. To ensure that the point is a local maximum, y must diminish for movements in any direction away from the critical point: In pictorial terms there is only one way to leave a true mountaintop, and that is to go down
. ¼ 0: An intuitive argument Earlier we described why a simple generalization of the single variable case shows that both own second partial derivatives ( f11 and f22) must be negative for a local maximum. In our mountain analogy, if attention is conﬁned only to north–south or east–west movements, the slope of the mountain must be diminishing as we cross its summit—the slope must change from positive to negative. The particular complexity that arises in the twovariable case involves movements through the optimal point that are not solely in the x1 or x2 directions (say, movements from northeast to southwest). In such cases, the secondorder partial derivatives do not provide complete information about how the slope is changing near the critical point. Conditions must also be placed on the cross-partial def21) to ensure that dy is decreasing for movements through the critical rivative ( f12 ¼ point in any direction. As we shall see, those conditions amount to requiring that the own second-order partial derivatives be sufﬁciently negative so as to counterbalance any possible ‘‘perverse’’ cross-partial derivatives that may exist. Intuitively, if the mountain falls away steeply enough in the north–south and east–west directions, relatively minor failures to do so in other directions can be compensated for. A formal analysis We now proceed to make these points more formally. What we wish to discover are the conditions that must be placed on the second partial derivatives of the function f to ensure that d 2y is negative for movements in any direction through the critical point. Recall ﬁrst that the total differential of the function is given by The differential of that function is given by dy f 1dx1 þ ¼ f 2dx2: d2y f 11dx1 þ f 12dx2Þ dx1 þ ð f 21dx1 þ f 22dx2Þ dx2 ¼ ð or d2y f 11 dx2 1 þ Because by Young’s theorem, f12 ¼ ¼ f 12dx2dx1 þ f21, we can arrange terms to get f 21dx1dx2 þ f 22 dx2 2: d2y f 11 dx2 2f 12dx1dx2 þ f 22 dx2 2: 1 þ ¼ (2:92) (2:93) (2:94) (2:95) For Equation 2.95 to be
unambiguously negative for any change in the x’s (i.e., for any choices of dx1 and dx2), it is obviously necessary that f11 and f22 be negative. If, for example, dx2 ¼ 0, then and d2y < 0 implies d2y f 11 dx2 1 ¼ f11 < 0: (2:96) (2:97) Chapter 2: Mathematics for Microeconomics 51 An identical argument can be made for f22 by setting dx1 ¼ 0. If neither dx1 nor dx2 is 0, we then must consider the cross-partial, f12, in deciding whether d2y is unambiguously negative. Relatively simple algebra can be used to show that the required condition is15 f 11 f 22 % f 2 12 > 0: (2:98) Concave functions Intuitively, what Equation 2.98 requires is that the own second partial derivatives ( f11 and f22) be sufﬁciently negative so that their product (which is positive) will outweigh f21). Functions that any possible perverse effects from the cross-partial derivatives ( f12 ¼ obey such a condition are called concave functions. In three dimensions, such functions resemble inverted teacups (for an illustration, see Example 2.11). This image makes it clear that a ﬂat spot on such a function is indeed a true maximum because the function always slopes downward from such a spot. More generally, concave functions have the property that they always lie below any plane that is tangent to them—the plane deﬁned by the maximum value of the function is simply a special case of this property. EXAMPLE 2.10 Second-Order Conditions: Health Status for the Last Time In Example 2.4 we considered the health status function x2 x1; x2Þ ¼ % 1 þ ð The ﬁrst-order conditions for a maximum are ¼ y f 2x1 % x2 2 þ 4x2 þ 5: or f 1 ¼ % f 2 ¼ % 2x1 þ 2x2 þ 2 4 ¼ ¼ 0; 0 x$1 ¼ x$2 ¼ 1; 2: (2:99) (2:100) (2:101) The second-order partial derivatives for Equation 2.99 are 2; f 11 ¼ % 2; f 22 �
� % 0: f 12 ¼ These derivatives clearly obey Equations 2.97 and 2.98, so both necessary and sufﬁcient conditions for a local maximum are satisﬁed.16 (2:102) QUERY: Describe the concave shape of the health status function, and indicate why it has only a single global maximum value. 2=f11 to Equation 2.95 and factoring. But this approach is 15The proof proceeds by adding and subtracting the term f12 dx2 only applicable to this special case. A more easily generalized approach that uses matrix algebra recognizes that Equation 2.95 is a ‘‘Quadratic Form’’ in dx1 and dx2, and that Equations 2.97 and 2.98 amount to requiring that the Hessian matrix Þ ð f11 f21 f12 f22 & ’ be ‘‘negative deﬁnite.’’ In particular, Equation 2.98 requires that the determinant of this Hessian matrix be positive. For a discussion, see the Extensions to this chapter. 16Notice that Equation 2.102 obeys the sufﬁcient conditions not only at the critical point but also for all possible choices of x1 and x2. That is, the function is concave. In more complex examples this need not be the case: The second-order conditions need be satisﬁed only at the critical point for a local maximum to occur. 52 Part 1: Introduction Constrained maximization As another illustration of second-order conditions, consider the problem of choosing x1 and x2 to maximize subject to the linear constraint y f, x1, x2Þ ð ¼ c b1x1 % b2x2 ¼ 0 % (2:103) (2:104) (where c, b1, and b2 are constant parameters in the problem). This problem is of a type that will be frequently encountered in this book and is a special case of the constrained maximum problems that we examined earlier. There we showed that the ﬁrst-order conditions for a maximum may be derived by setting up the Lagrangian expression + f x1, x2 ð ¼ Þ þ k c ð b1x1 % % b2x2 : Þ Partial differentiation with respect to x1, x2, and
l yields the familiar results: f1 % f 2 % b1x1 % kb1 ¼ kb2 ¼ b2x2 ¼ 0, 0, 0: c % (2:105) (2:106) These equations can in general be solved for the optimal values of x1, x2, and l. To ensure that the point derived in that way is a local maximum, we must again examine movements away from the critical points by using the ‘‘second’’ total differential: d2y f 11 dx2 1 þ ¼ 2f 12dx1dx2 þ f 22 dx2 2: (2:107) In this case, however, not all possible small changes in the x’s are permissible. Only those values of x1 and x2 that continue to satisfy the constraint can be considered valid alternatives to the critical point. To examine such changes, we must calculate the total differential of the constraint: or b1dx1 % b2dx2 ¼ % 0 dx2 ¼ % b1 b2 dx1: (2:108) (2:109) This equation shows the relative changes in x1 and x2 that are allowable in considering movements from the critical point. To proceed further on this problem, we need to use the ﬁrst-order conditions. The ﬁrst two of these imply f 1 f 2 ¼ b1 b2 ; and combining this result with Equation 2.109 yields dx2 ¼ % f 1 f 2 dx1: (2:110) (2:111) We now substitute this expression for dx2 in Equation 2.107 to demonstrate the conditions that must hold for d 2y to be negative: Chapter 2: Mathematics for Microeconomics 53 d2y f 11 dx2 1 þ f 11 dx2 1 % ¼ ¼ f 1 f 2 2f 12dx1 % ( f 1 dx2 f 2 2f 12 1 þ þ f 22 % ( f 1 f 2 dx1 2 ) dx2 1: dx1 ) f 2 1 f 2 2 f 22 Combining terms and putting each over a common denominator gives 2f 12 f 1 f 2 þ Consequently, for d 2y < 0, it must be the case that 2 % ¼ ð f 11 f 2 d2y f 22 f 2 1Þ dx2 1 f 2 2 : f 11 f 2 2 % 2f 12
f 1 f 2 þ f 22 f 2 1 < 0: (2:112) (2:113) (2:114) Quasi-concave functions Although Equation 2.114 appears to be little more than an inordinately complex mass of mathematical symbols, in fact the condition is an important one. It characterizes a set of functions termed quasi-concave functions. These functions have the property that the set of all points for which such a function takes on a value greater than any speciﬁc constant is a convex set (i.e., any two points in the set can be joined by a line contained completely within the set). Many economic models are characterized by such functions and, as we will see in considerable detail in Chapter 3, in these cases the condition for quasi-concavity has a relatively simple economic interpretation. Problems 2.9 and 2.10 examine two speciﬁc quasi-concave functions that we will frequently encounter in this book. Example 2.11 shows the relationship between concave and quasi-concave functions. EXAMPLE 2.11 Concave and Quasi-Concave Functions The differences between concave and quasi-concave functions can be illustrated with the function17 x1; x2Þ ¼ ð ð where the x’s take on only positive values, and the parameter k can take on a variety of positive values. No matter what value k takes, this function is quasi-concave. One way to show this is to look (2:115) x1 ) ¼ y f k, x2Þ at the ‘‘level curves’’ of the function by setting y equal to a speciﬁc value, say c. In this case y c ¼ k x1x2Þ ¼ ð or x1x2 ¼ c1=k c 0: ¼ (2:116) But this is just the equation of a standard rectangular hyperbola. Clearly the set of points for which y takes on values larger than c is convex because it is bounded by this hyperbola. A more mathematical way to show quasi-concavity would apply Equation 2.114 to this function. Although the algebra of doing this is a bit messy, it may be worth the struggle. The various components of Equation 2.114 are: f 1 ¼
f 2 ¼ f 11 ¼ f 22 ¼ f 12 ¼ kxk kxk 1 1 xk 2, % 1xk 1, % 2 k ð k k ð k2xk k 1 % 1 xk xk 2 2, % Þ 2 1xk xk, 1 % 2 % Þ 1 1 xk 1 : % % 2 (2:117) 17This function is a special case of the Cobb–Douglas function. See also Problem 2.10 and the Extensions to this chapter for more details on this function. 54 Part 1: Introduction Thus, f 11 f 2 2 % 2 f 12 f 1 f 2 þ f 22 f 2 1 ¼ ¼ k3 k % ð k3 k ð þ 2k3x3k 1 1 2 x3k % 1 Þ x3k 1 1 Þ 2 x3k % 2 x3k 2 % 2 % 2 x3k 2 % % 2, 1 ð% Þ % 2 % 2k4x3k 1 2 x3k 2 2 % % (2:118) which is clearly negative, as is required for quasi-concavity. Whether the function f is concave depends on the value of k. If k < 0.5 the function is indeed concave. An intuitive way to see this is to consider only points where x1 ¼ x2. For these points, y k x2 1Þ ¼ ð ¼ x2k 1, (2:119) which, for k < 0.5, is concave. Alternatively, for k > 0.5, this function is convex. FIGURE 2.42Concave and Quasi-Concave Functions In all three cases these functions are quasi-concave. For a ﬁxed y, their level curves are convex. But only for k 1.0 clearly shows nonconcavity because the function is not below its tangent plane. 0.2 is the function strictly concave. The case k ¼ ¼ (a) k = 0.2 (b) k = 0.5 (c) k = 1.0 Chapter 2: Mathematics for Microeconomics 55 A more deﬁnitive proof makes use of the partial derivatives from Equation 2.117. In this case the condition for concavity can be expressed as f
11 f 22 % f 2 12 ¼ ¼ k2 1 k % ð x2k x2k 2 % 2 1 x2k x2k 2 1 1 % 2x2k 1 Þ k2 2 % k2 % ½ 1 2 % k ð x2k 2 % 2 % 2 1 Þ k4x2k 1 k4 ( % 2k % 1, 2 x2k 2 2 % % (2:120) Þ( and this expression is positive (as is required for concavity) for ð% ¼ þ ½ On the other hand, the function is convex for k > 0.5. 2k 1 Þ þ ð% > 0 or k < 0:5: ¼ ¼ ¼ 0.2, k A graphic illustration. Figure 2.4 provides three-dimensional illustrations of three speciﬁc examples of this function: for k 1. Notice that in all three cases the level 0.5, and k curves of the function have hyperbolic, convex shapes. That is, for any ﬁxed value of y the functions are similar. This shows the quasi-concavity of the function. The primary differences among the functions are illustrated by the way in which the value of y increases as both x’s 0.2), the increase in y slows as the x’s increase. This increase together. In Figure 2.4a (when k ¼ gives the function a rounded, teacup-like shape that indicates its concavity. For k 0.5, y appears to increase linearly with increases in both of the x’s. This is the borderline between concavity and convexity. Finally, when k 1 (as in Figure 2.4c), simultaneous increases in the values of both of the x’s increase y rapidly. The spine of the function looks convex to reﬂect such increasing returns. ¼ ¼ A careful look at Figure 2.4a suggests that any function that is concave will also be quasiconcave. You are asked to prove that this is indeed the case in Problem 2.8. This example shows that the converse of this statement is not true—quasi-concave functions need not necessarily be concave. Most functions we will encounter in this book will also illustrate this fact; most will be quasi-
concave but not necessarily concave. QUERY: Explain why the functions illustrated both in Figure 2.4a and 2.4c would have maximum values if the x’s were subject to a linear constraint, but only the graph in Figure 2.4a would have an unconstrained maximum. Homogeneous Functions Many of the functions that arise naturally out of economic theory have additional mathematical properties. One particularly important set of properties relates to how the functions behave when all (or most) of their arguments are increased proportionally. Such situations arise when we ask questions such as what would happen if all prices increased by 10 percent or how would a ﬁrm’s output change if it doubled all the inputs that it uses. Thinking about these questions leads naturally to the concept of homogeneous functions. Speciﬁcally, a function f (x1, x2, …, xn) is said to be homogeneous of degree k if..., txnÞ ¼ :..., xnÞ tx1, tx2, x1, x2, (2:121) t k f ð ð f ¼ The most important examples of homogeneous functions are those for which k 1 or 0. In words, when a function is homogeneous of degree one, a doubling of all its k arguments doubles the value of the function itself. For functions that are homogeneous of degree zero, a doubling of all its arguments leaves the value of the function unchanged. Functions may also be homogeneous for changes in only certain subsets of their arguments—that is, a doubling of some of the x’s may double the value of the function if ¼ 56 Part 1: Introduction the other arguments of the function are held constant. Usually, however, homogeneity applies to changes in all the arguments in a function. Homogeneity and derivatives If a function is homogeneous of degree k and can be differentiated, the partial derivatives of the function will be homogeneous of degree k 1. A proof of this follows directly from the deﬁnition of homogeneity. For example, differentiating Equation 2.121 with respect to its ﬁrst argument gives % @f tx1,..., txnÞ ð @x1 t ) ¼ t k @f x1,..., xnÞ ð @x1 or f 1ð tx1,..
., txnÞ ¼ which shows that f1 meets the deﬁnition for homogeneity of degree k 1. Because marginal ideas are so prevalent in microeconomic theory, this property shows that some imthe portant properties of marginal effects can be inferred from the properties of underlying function itself., x1,..., xnÞ (2:122) % % t k 1f 1ð Euler’s theorem Another useful feature of homogeneous functions can be shown by differentiating the deﬁnition for homogeneity with respect to the proportionality factor, t. In this case, we differentiate the right side of Equation 2.121 ﬁrst, then the left side: tx1,..., txnÞ þ ) ) ) þ xn f nð : tx1,..., txnÞ If we let t ¼ ktk % 1f 1ð x1 f 1ð x1,..., xnÞ ¼ 1, this equation becomes x1 f1ð x1,..., xnÞ ¼ ð kf x1,..., xnÞ þ ) ) ) þ This equation is termed Euler’s theorem (after the mathematician who also discovered the constant e) for homogeneous functions. It shows that, for a homogeneous function, there is a deﬁnite relationship between the values of the function and the values of its partial derivatives. Several important economic relationships among functions are based on this observation. : x1,..., xnÞ xn fnð (2:123) Homothetic functions A homothetic function is one that is formed by taking a monotonic transformation of a homogeneous function.18 Monotonic transformations, by deﬁnition, preserve the order of the relationship between the arguments of a function and the value of that function. If certain sets of x’s yield larger values for f, they will also yield larger values for a monotonic transformation of f. Because monotonic transformations may take many forms, however, they would not be expected to preserve an exact mathematical relationship such as that embodied in homogeneous functions. Consider, for example, the function y ¼ f (x1, x2) x1x2. Clearly, this function is homogeneous of degree 2—a doubling

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