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on a N maturity bond observed at time t, i1,t is the interest rate on a 1 maturity bond observed at time t, ie 1,t+1 is the expected interest rate on a one maturity bond one period into the future, and so on. tpt is a term premium that accounts for the risk involved in holding longer maturity bonds. See also the discussion in Chapter 34. For example, suppose that you are considering a Treasury security with a one year maturity. If the current interest rate on a Treasury security with a three month maturity 935 𝑃𝑃𝑡𝑡𝐵𝐵 𝐵𝐵𝑡𝑡 𝐷𝐷 𝑆𝑆 𝐵𝐵0,𝑡𝑡 𝐷𝐷′ 𝐵𝐵1,𝑡𝑡 𝑃𝑃0,𝑡𝑡𝐵𝐵 𝑃𝑃1,𝑡𝑡𝐵𝐵 1 4 is 4 percent, and the expected interest rates on three month maturity Treasury securities for the ensuing three quarters are 5, 6, and 7 percent, then the interest rate on the one year (4 + 5 + 6 + 7)) plus a term premium to Treasury security ought to be 5.5 percent (i.e. account for risk. The basic intuition behind the expectations hypothesis is straightforward. If you are looking to save and transfer resources across a long period of time, you can either buy a long maturity bond or a sequence of shorter maturity bonds. If your only objective is to transfer resources intertemporally, then you ought to be indiﬀerent between these two options once you adjust for diﬀerent levels of risk (i.e. add in a term premium). This ends up necessitating that the interest rate on the long maturity bond approximately equals the average of expected short maturity interest rates plus a term premium. Using similar theoretical considerations, we can think about yields on long maturity, risky, private sector debt (e.g. corporate bonds) as being related to yields on long term government bonds, plus a risk premium to compensate for default risk. Let ir N,t denote the yield on risky private sector debt with N periods to maturity. It ought to equal: ir N,t = iN
,t + rpt (37.2) In (37.2), rpt is a risk premium to compensate holders of such debt for default risk. (37.1) and (37.2) embody the idea that bond with diﬀerent characteristics are perfect substitutes once one controls for risk. As discussed in Chapter 34, the risk and term premia relate to how bond prices co-vary with the marginal utility of consumption, and it is not clear how or why quantitative easing could inﬂuence these terms. If this is the case, the only way for a central bank to inﬂuence ir N,t is to inﬂuence iN,t, and the way to inﬂuence iN,t is to aﬀect the current and expected future path of short term riskless yields. Since quantitative easing does not impact the current or expected future path of short term yields, it is not clear why it (in either form of purchasing long term government debt or risky private sector debt) should impact ir N,t. For these reasons, the analysis portrayed graphically in Figure 37.22 might be too simple. In particular, if long term and risky bonds are perfectly substitutable with short term riskless bonds (once one controls for risk), the demand for such bonds ought to be perfectly elastic (i.e. horizontal), and large-scale purchases of such bonds ought not to impact their prices or yields. The Fed purchasing such debt would temporarily increase the demand for such debt, driving prices up and yields down. But if long term and risky debt are perfectly substitutable with short term riskless debt (once one controls for risk), this would cause private sector demand to shift way from these securities and into shorter term, riskless debt until yields are equalized. Based on the logic of the expectations hypothesis, in addition to its quantitative easing 936 programs the Fed also resorted to another form of unconventional monetary policy, what is called forward guidance. Whereas quantitative easing will be ineﬀective under the expectations hypothesis, forward guidance relies on the expectations hypothesis characterizing reality. Under forward guidance, the Fed tries to communicate to the private sector its intended path of short term interest rates, particularly the path after the period where the ZLB binds is over. For example, suppose that the current and expected future short term interest rates are both at 0, but expected short term interest rates in the two periods after that are 1
and 2 percent, respectively. Under the expectations hypothesis, the current interest rate on a (0 + 0 + 1 + 2)) (plus a four period maturity debt instrument ought to be 0.75 percent (i.e. 1 4 term premium). If the Fed can convince the private sector that it will lower the short term interest rates to 0 and 1 percent three and four periods out into the future, respectively, then it could, in principle, lower the current four period interest rate to 0.25 percent (i.e. (0 + 0 + 0 + 1)) (plus a term premium which would be unaﬀected). Lower long term rates 1 4 would ﬁlter through to yields on risky private sector debt, as in (37.2). The Fed began using forward guidance at the same time it started quantitative easing. In December 2008, for example, the Fed issued a press release stating that “The Committee anticipates that weak economic conditions are likely to warrant exceptionally low levels of the Federal Funds rate for some time.” In March 2009, the Fed changed language from “some time” to “an extended period.” In subsequent meetings the Fed gave speciﬁcity to how long it anticipated the Fed Funds rate being low – for example, in January of 2012 the Fed said that it anticipated it would remain low until “late 2014.”3 In a sense, the Fed was hedging its bets. It wanted to lower interest rates on risky and longer term debt. It tried both quantitative easing and forward guidance to do so, both of which should not be able to work at the same time. Nevertheless, the evidence suggests that both the Fed’s QE programs and its forward guidance attempts did work to lower longer term, riskier interest rates (see, e.g. Campbell, Evans, Fisher, and Justiniano 2012 for evidence on forward guidance and Krishnamurthy and Vissing-Jorgensen (2011) for the eﬀects of quantitative easing). In terms of the AD-AS model, we can think about the unconventional monetary policies of quantitative easing and forward guidance as both having the same objective – lowering credit spreads, ft. In this sense, these policies were designed to work similarly to the Fed’s extraordinary lending policies. The objective was to lower ft, and hence the relevant interest rate for investment spending, without adjusting the short term Federal Funds Rate. If successful,
the decrease in ft would shift the IS curve to the right with a resulting rightward shift of the AD curve. The desired eﬀects of these interventions are shown in Figure 37.23 3The wording in this paragraph follows the ﬁrst paragraph in Campbell, Evans, Fisher, and Justiniano (2012). 937 below. We label the hypothetical new curves with lower ft as being dated in 2012, because this was the last period in which a new quantitative easing program was announced. Figure 37.23: AD-AS Eﬀects of Federal Reserve Quantitative Easing and Forward Guidance 37.5 Lingering Questions The Great Depression was a formative experience for the generation of economists that lived through it and shaped macroeconomic thought and policy for decades. In a similar way, the Great Recession has been a formative experience for macroeconomists in the last decade. Among other things, the Great Recession has emphasized the importance of linkages between the ﬁnancial system and the macroeconomy. Prior to the crisis, most macroeconomics textbooks abstracted from things like credit spreads altogether. The Great Recession has forced us to reconsider this and other aspects of economists’ modeling frameworks. 938 𝐴𝐴𝐴𝐴 𝐿𝐿𝐿𝐿 𝑟𝑟09=𝑟𝑟12′=−𝜋𝜋𝑡𝑡+1𝑒𝑒 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑃𝑃𝑡𝑡 2009 equilibrium Desired effects of quantitative easing and forward guidance to ↓𝑓𝑓𝑡𝑡 𝑃𝑃12′ 𝐼𝐼𝐴𝐴12′ 𝐴𝐴𝐴𝐴12′ 𝑌𝑌12′ 𝐼𝐼𝐴𝐴09 𝐴𝐴𝐴𝐴09 𝑃𝑃09 𝑌𝑌09 The Great Recession has also spurred a whole new set of questions on which macroeconomists now focus. Why and how did
house prices get so high in the ﬁrst place, and why did they come crashing down starting in 2006? What is the appropriate level of regulation of ﬁnancial institutions, and should this regulation extend beyond traditional banks? What is the appropriate level of government intervention in response to crises? Does the implicit promise of being bailed out incentivize excessive risk-taking behavior? Why has the economic recovery from the crisis been relatively weak – i.e. why has real GDP not returned to a pre-recession trend (see, e.g., Figure 37.16)? What have been the eﬀects of the Fed’s unconventional monetary policy actions? In the years since the crisis, why has inﬂation been persistently below where the Fed would like it to be (2 percent) in spite of all the Fed’s actions? What are the downward pressures on interest rates throughout the world, and what are the implications for monetary policy going forward? We close this chapter by noting that these questions are not likely to go away anytime soon. Getting compelling answers to important macroeconomic questions is always diﬃcult. To ascertain what the eﬀects of diﬀerent policies are, we need to know what would have happened in historical episodes in the absence of those policies. This counterfactual history is never (or at least rarely) observed, so we are forced to rely upon simpliﬁed modeling frameworks to arrive at answers, and the answers these models give us always depend on the underlying assumptions in the models. Sooner rather than later, the economy is likely to experience another cyclical downturn. With short term interest rates still at very low levels, what room will the Fed and other central banks around the world have to maneuver? Will unconventional monetary policies like forward guidance and quantitative easing be the main lines of defense against recessions in the future? Only time will tell. It is an exciting time to study macroeconomics! 37.6 Summary • The tell-tale sign of a ﬁnancial crisis is sharp increase in credit spreads • Financial crises are typically preceded by asset price booms and then busts. In the Great Depression, it was a general stock market boom that ended in 1929 and set oﬀ the Depression. In the Great Recession, it was a boom in housing prices that ended in 2006. • Following asset price booms and busts there is typically a “run” on
ﬁnancial intermediaries as investors become worried about the value of the investments a ﬁnancial institution holds. In the Great Depression, this run was a run on deposits by house- 939 holds against traditional banks. In the Great Recession, the run was somewhat more complicated. It involved institutions running on other institutions, with short term credit markets drying up and institutions being forced to liquidate assets. • In the terms of the AD-AS model, we can think about the Great Recession as featuring several adverse shocks to the IS curve. The ﬁrst was a direct eﬀect of home price declines and was not large. The second was due to an increase in credit spreads in 2007 and into 2008. The Fed responded to both of these shocks by aggressively lowering interest rates, which meant that by the end of 2008 the Federal Funds rate was at its zero lower bound (ZLB). • The Great Recession entered its most virulent stage at the end of 2008 and into the ﬁrst half of 2009 when the ﬁnancial crisis intensiﬁed. The macroeconomic eﬀects of the negative IS shock were exacerbated by the binding ZLB. • There were many unconventional policy actions taken to combat the Great Recession. This diﬀers somewhat from the Great Depression, where many economists feel that the Fed let things get out of hand and should have done a better job as a lender of last resort. These unconventional policy actions included emergency Federal Reserve lending to ﬁnancial institutions, ﬁscal stimulus, and unconventional monetary policies in the form of quantitative easing and forward guidance. 940 Bibliography Abel, Andrew, Ben Bernanke, and Dean Croushore. 2017. Macroeconomics, 9th Edition. Pearson. Acemoglu, Daron, Simon Johnson, and James A. Robinson. 2001. “The Colonial Origins of Comparative Development: An Empirical Investigation.” American Economic Review 91 (5):1369–1401. URL http://www.aeaweb.org/articles?id=10.1257/aer.91.5. 1369. Aguiar, Mark and Erik Hurst. 2005. “Consumption vs. Expenditure.” Journal of Political Economy 113 (5):919–948. Akerlof, George. 1970. “The Market for ‘Lemons’: Quality Uncertain
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..................................... 33 1.5 CPI............................................. 35 1.6 Labor Market Variables................................. 38 4.1 Real GDP per Worker in the US 1950-2011..................... 58 4.2 Capital per worker in the US 1950-2011........................ 60 4.3 Capital to Output Ratio in the U.S. 1950–2011................... 61 4.4 Labor Share in the US 1950-2011............................ 62....................... 64 4.5 Return on capital in the US 1950-2011. 4.6 Wages in the US 1950-2011............................... 65 4.7 Relationship Between Human Capital and Income per Person.......... 69 5.1 Plot of Central Equation of Solow Model...................... 83 5.2 Convergence to Steady State from kt < k∗...................... 84 5.3 Convergence to Steady State from kt > k∗...................... 85 5.4 Convergence to Steady State.........
..................... 86 5.5 Alternative Plot of Central Equation of Solow Model............... 87 5.6 Exogenous Increase in s, s1 > s0............................ 89 5.7 Dynamic Responses to Increase in s......................... 90 5.8 Dynamic Path of Output Growth........................... 92 5.9 Exogenous Increase in A, A1 > A0........................... 93 5.10 Dynamic Responses to Increase in A......................... 94 5.11 Dynamic Path of Output Growth........................... 95 5.12 s and c∗: The Golden Rule............................... 97 5.13 The Golden Rule Saving Rate............................. 99 5.14 Eﬀects of ↑ s Above and Below the Golden Rule.................. 100 6.1 Plot of Central Equation of Augmented Solow Model............... 113 948 Increase in s........................................ 116 6.2 6.3 Dynamic Responses to Increase in s.......................
.. 117 6.4 Dynamic Responses to Increase in s, Per Worker Variables............ 118 6.5 Dynamic Path of Output Per Worker Growth.................... 119 6.6 Increase in A....................................... 120 6.7 Dynamic Responses to Increase in A......................... 121........... 122 6.8 Dynamic Responses to Increase in A, Per Worker Variables 6.9 Dynamic Responses to Increase in s, Quantitative Exercise........... 124 6.10 Dynamic Responses to Increase in s, Quantitative Exercise, Per Worker Variables125 6.11 Dynamic Responses to Increase in s, Quantitative Exercise Per Worker Output Growth........................................... 126 6.12 The Golden Rule Saving Rate............................. 127 7.1 Country 1 Initially Endowed With More Capital than Country 2........ 134 7.2 Paths of Capital and Output Growth for Countries 1 and 2........... 135 Initial GDP Per Capita in 1950 and Cumulative Growth From 1950–2010... 136 7.3 Initial GDP Per Capita in 1950 and Cumulative Growth From 1950–2010 7.4 OECD Countries..................................... 137 7.5 Real GDP Per Capita Relative to the United States................ 138 7.6 Scatter Plot: TFP and GDP Per Worker in 2011.................. 142
8.1 Plot of the Central Equation of the OLG Model.................. 159 8.2 Dynamics in the OLG Model.............................. 160 8.3 Quantitative Convergence in the OLG Model.................... 161 Implementing Taxes to Hit the Golden Rule 8.4.................... 168 Implementing Taxes to Hit the Golden Rule: Starting from Dynamic Ineﬃciency170 8.5 9.1 Utility and Marginal Utility.............................. 184 9.2 Budget Line........................................ 189 Indiﬀerence Curves.................................... 191 9.3 9.4 An Optimal Consumption Bundle........................... 192 Increase in Yt 9.5....................................... 193 Increase in Yt+1...................................... 194 9.6 Increase in rt and Pivot of the Budget Line..................... 196 9.7 9.8.......................... 196 Increase in rt: Initially a Borrower Increase in rt: Initially a Saver.
............................ 198 9.9 9.10 Increase in Qt....................................... 203 949 9.11 Increase in Qt+1...................................... 205 9.12 Expected Marginal Utility and Marginal Utility of Expected Consumption.. 211 9.13 An Increase in Uncertainty............................... 213 9.14 Borrowing Constraint: St ≥ 0.............................. 217 9.15 Borrowing Constraint: rb............................. 218 t........................... 219 9.16 A Binding Borrowing Constraint.................. 220 9.17 A Binding Borrowing Constraint, Increase in Yt................. 221 9.18 A Binding Borrowing Constraint, Increase in Yt+1 > rs t 10.1 Consumption and Income: The Life Cycle...................... 240 10.2 The Stock of Savings over The Life Cycle...................... 241 11.1 Expenditure and Income................................ 251 11.2 Derivation of the IS Curve........
....................... 253 11.3 IS Curve Shift: ↑ Yt+1.................................. 254 11.4 The Y s Curve....................................... 255 11.5 Equilibrium........................................ 256 11.6 Supply Shock: Increase in Yt.............................. 257 11.7 Demand Shock: Increase in Yt+1............................ 258 11.8 Real Interest Rates and Expected Output Growth................. 260 12.1 The Production Function................................ 271 12.2 Labor Demand...................................... 275 12.3 Investment Demand................................... 277 12.4 The Consumption-Leisure Budget Line........................ 285 12.5 Optimal Consumption-Leisure Choice........................ 286............... 287 12.6 Optimal Consumption-Leisure Choice, Increase in wt 12.7 Labor Supply......
................................. 289 13.1 Budget Line........................................ 297 13.2 Increase in Gt....................................... 300 13.3 Increase in Gt+1...................................... 302 16.1 Perfect Competition Versus Monopoly........................ 348 16.2 Continuum Versus an Interval............................. 351 16.3 Labor Share in US.................................... 358 16.4 Proﬁts a Function of Mt................................ 364 16.5 Startup and working age population growth rates................. 367 950 17.1 Total Hires and separations in the US 2000-2016.................. 370 17.2 Net job creation in the US 2000-2016......................... 371........................... 372 17.3 Separations in the US 2000-2016......................... 373 17.4 Job ﬁnding
rate in the US 2000-2016. 17.5 Job ﬁnding rate in the US 2000-2016......................... 373 17.6 Uniform Probability Distribution........................... 376 17.7 Expected Utility of Reservation Wage........................ 378 17.8 Eﬀects of Mean Preserving Spread.......................... 381 17.9 The top panel shows the transition dynamics starting at u0 = 0.047. The bottom panel shows the transition dynamics associated with moving to a lower separation rate....................................... 385 18.1 Desired Expenditure and Income........................... 414 18.2 Desired Expenditure and Income........................... 415......... 416 18.3 Desired Expenditure and Income: Expenditure Equals Income 18.4 The IS Curve: Derivation................................ 417 18.5 The Y s Curve: Derivation............................... 418 18.6 IS − Y s Equilibrium................................... 420 18.7 Money Demand...................................... 421 18.8
Money Demand, Alternative Graphical Representations.............. 422 18.9 Money Supply....................................... 422 18.10Equilibrium in the Money Market........................... 423 19.1 IS − Y s Equilibrium................................... 427 19.2 Equilibrium in the Money Market........................... 428 19.3 Increase in At....................................... 430 19.4 Increase in At: The Money Market.......................... 431 19.5 Increase in At+1...................................... 433 19.6 Increase in At+1: The Money Market......................... 434 19.7 Increase in Gt....................................... 436 19.8 Increase in Gt: The Money Market.......................... 437 19.9 Increase in Mt....................................... 441 19.10Increase in πe t+1..........
............................ 442 20.1 Cyclical Component of Real GDP........................... 449 20.2 Increase in At and At+1................................. 454 20.3 Cyclical Components of Real GDP and TFP.................... 456 951 21.1 Diﬀerent Measures of the Money Supply....................... 462 21.2 M2 and the Monetary Base............................... 464 21.3 M2 Divided by the Monetary Base.......................... 465 21.4 Scatter Plot: Money Growth and Inﬂation...................... 468 21.5 Smoothed Money Growth and Inﬂation....................... 469 21.6 Velocity.......................................... 470 21.7 Nominal Federal Funds Rate.............................. 471 21.8 Smoothed Inﬂation and Interest Rates........................ 473 21.9 Cylical Components of Real GDP and the Money Supply............ 475 22.1 Cyclical Components of GDP and Utilization
-Adjusted TFP.......... 483 22.2 Cyclical Components of GDP and Baa Credit Spread............... 486 23.1 IS Curve: Open vs. Closed Economy......................... 497 23.2 IS Curve: Small Open Economy............................ 498 23.3 Shift of IS Curve due to ↑ Gt: Open vs. Closed Economy............. 499 23.4 IS − Y s Equilibrium................................... 501 23.5 Equilibrium in the Money Market........................... 502 23.6 Equilibrium Real Exchange Rate........................... 503 23.7 Eﬀects of a Positive IS Shock............................. 504 23.8 Eﬀect of Positive IS Shock on the Price Level.................... 506 23.9 Eﬀects of Positive IS Shock on the Real Exchange Rate.............. 506 23.10Eﬀects of an Increase in At............................... 508 23.11Eﬀect of Increase in At on the Price Level...................... 509 23.12Eﬀects of Increase in At Shock on the Real Exchange Rate............ 510 23.
13Eﬀects of Increase in Qt................................. 511 23.14Eﬀect of Increase in Qt on the Price Level...................... 512 23.15Eﬀects of Increase in Qt Shock on the Real Exchange Rate............ 512................................. 513 23.16Eﬀect of Increase in Mt 24.1 The LM Curve: Derivation............................... 523 24.2 The LM Curve: Increase in Mt............................ 523 24.3 The LM Curve: Increase in Pt............................. 524 24.4 The LM Curve: Increase in πe t+1............................ 525 24.5 The IS Curve: Derivation................................ 526 24.6 The AD Curve: Derivation............................... 528 24.7 Shift of the AD Curve: Increase in Mt........................ 529 952 24.8 Shift of the AD Curve: IS Shock........................... 530 25.1 Derivation of the AS Curve........
....................... 536................. 537 25.2 Shift of the Neoclassical AS Curve: Increase in At 25.3 Shift of the Neoclassical AS Curve: Increase in θt................. 538 25.4 The Simple Sticky Price AS Curve: Derivation................... 541 25.5 Shift of the Simple Sticky Price AS Curve: Increase in ¯Pt............ 542 25.6 The Partial Sticky Price AS Curve: Derivation................... 545 25.7 The Partial Sticky Price AS Curve: Role of γ.................... 547 25.8 Shift of the Partial Sticky Price AS Curve: ↑ ¯Pt.................. 549 25.9 Shift of the Partial Sticky Price AS Curve: ↑ At.................. 550 25.10Shift of the Partial Sticky Price AS Curve: ↑ θt................... 551 26.1 Neoclassical IS-LM-AD-AS Equilibrium....................... 555 26.2 Neoclassical Model: Increase in Mt.......................... 557.......................... 558 26.3 Neoclassical Model: Increase in At 26.4 Neoclassical Model: Positive IS Shock........................ 560 26.5 Equilibrium in the Simple Sticky Price
Model.................... 562 26.6 Simple Sticky Price: Eﬀect of Increase in Mt.................... 564 26.7 Simple Sticky Price: Eﬀect of Positive IS Shock.................. 566.................... 568 26.8 Simple Sticky Price: Eﬀect of Increase in At 26.9 Simple Sticky Price: Eﬀect of Increase in θt..................... 569 26.10Simple Sticky Price: Eﬀect of Increase in ¯Pt..................... 570................... 573 26.11Equilibrium in the Partial Sticky Price Model 26.12Partial Sticky Price: Eﬀect of Increase in Mt.................... 575 26.13Partial Sticky Price: Eﬀect of Positive IS Shock.................. 577.................... 579 26.14Partial Sticky Price: Eﬀect of Increase in At 26.15Partial Sticky Price: Eﬀect of Increase in θt..................... 581 26.16Partial Sticky Price: Eﬀect of Increase in ¯Pt..................... 582 < Y f 27.1 Sticky Price Model: Y sr............................. 590 0,t 0,t < Y f
27.2 Sticky Price Model: Y sr 0,t, Short Run to Medium Run Price Adjustment. 592 0,t 27.3 Simple Sticky Price Model: Increase in Mt, Short Run to Medium Run.... 594 27.4 Short Run and Medium Responses: Increase in Mt................ 595 27.5 Simple Sticky Price Model: Positive IS Shock, Short Run to Medium Run.. 596 27.6 Short Run and Medium Responses: Positive IS Shock............... 597 27.7 Sticky Price Model: Increase in At, Short Run to Medium Run......... 598 953 27.8 Short Run and Medium Responses: Increase in At................. 599 27.9 Partial Sticky Price Model: Y0,t < Y f 0,t......................... 601 27.10Partial Sticky Price Model: Y0,t < Y f 0,t, Short Run to Medium Run Price Adjustment........................................ 603 27.11Partial Sticky Price: Increase in Mt, Short Run to Medium Run........ 605................ 606 27.12Short Run and Medium Responses: Increase in Mt 27.13Short Run and Medium Responses: Increase in Mt Comparing Simple Sticky Price to Partial Sticky Price Model.......................... 607 27.14Partial Sticky Price: Positive IS Shock, Short Run to Medium Run....... 608 27.15Short Run and Medium Responses: Positive IS Shock............... 609 27.16Partial Sticky Price: Eﬀect of Increase in At, Dynamics..........
.... 610 27.17Short Run and Medium Responses: Increase in At................. 611 27.18Inﬂation and the Output Gap............................. 613 27.19Inﬂation and the Output Gap............................. 614............................ 615 27.20Average Inﬂation Expectations 27.21Sticky Price Model: Anticipated Increase in Mt, Reﬂected in πe t......... 618 27.22Sticky Price Model: Unanticipated Disinﬂation................... 620 27.23Sticky Price Model: Anticipated Disinﬂation.................... 622 28.1 Equilibrium in the Partial Sticky Price Model................... 628 28.2 Optimal Monetary Response to Positive IS Shock................. 629................... 631 28.3 Optimal Monetary Response to Increase in At 28.4 Optimal Monetary Response to Increase in ¯Pt................... 633 28.5 A Strict Inﬂation (Price Level) Target and the Eﬀective AD Curve....... 635 28.6 A Strict Inﬂation (Price Level) Target: Response to Positive IS Shock..... 636 28.7 A Strict Inﬂation (Price Level) Target: Response to ↑ At or ↓ θt......... 637 28.8 A Strict In�
��ation (Price Level) Target: Response to ↑ ¯Pt............. 638 28.9 The Natural Rate of Interest.............................. 640 28.10IS Shocks and the Natural Rate of Interest..................... 641 28.11Y f t Shocks and the Natural Rate of Interest..................... 642 28.12Using Fiscal Policy to Combat an IS Shock..................... 644 28.13Actual and Monetary Policy Rule Implied Fed Funds Rate............ 646 29.1 The LM Curve and the ZLB.............................. 650 29.2 AD Derivation with a Non-Binding ZLB....................... 651 29.3 AD Derivation with a Binding ZLB.......................... 652 29.4 AD Derivation with Both a Binding and Non-Binding ZLB........... 653 954 29.5 Changes in the Money Supply and a Binding ZLB................. 654 29.6 Sticky Price Model: Negative IS Shock with Binding ZLB............ 656 29.7 Sticky Price Model: Positive Supply Shock with Binding ZLB.......... 658 29.8 Medium Run Supply-Side Dynamics at the ZLB, Sticky Price Model...... 660 29.9 Medium Run Supply-Side Dynamics at the ZLB, Deﬂationary Expectations, Sticky Price Model.................
................... 662 29.10Sticky Price Model: Fiscal Expansion with Binding ZLB............. 664 29.11Fiscal Expansion to Exit the ZLB........................... 665 29.12Engineering Higher Expected Inﬂation to Exit the ZLB.............. 666 30.1 The AD Curve: Open vs. Closed Economy..................... 673.............. 675 30.2 Equilibrium in the Open Economy Sticky Price Model 30.3 Increase in Mt: Open vs. Closed Economy..................... 677 30.4 Positive IS Shock: Open vs. Closed Economy.................... 679.......... 682 30.5 Equilibrium in the Small Open Economy Sticky Price Model 30.6 Increase in Mt: Small Open vs. Closed Economy.................. 683 30.7 Eﬀect of an Increase in rF............................... 686 t............................... 688 30.8 Eﬀect of an Increase in Qt 30.9 A Positive IS Shock and a Fixed Exchange Rate Regime............. 691 30.10An Increase in rF in a Fixed Exchange Rate Regime............... 693 t 31.1 Financial Intermediation................................ 701
31.2 Traditional Banking System.............................. 720 31.3 Shadow Banking System................................ 721 32.1 The M1 Money Multiplier and Its Components................... 729 32.2 Money Stock Measures: Great Depression...................... 750 32.3 Federal Reserve Assets................................. 754 32.4 Money Stock Measures: Great Recession....................... 756 34.1 Increase in Current Enowment............................. 783 34.2 Increase in Future Endowment............................. 783 34.3 Bond Prices and p1: Income Always High...................... 789 34.4 Yields and Risk Premium: Income Always High.................. 790 34.5 Bond Prices and p3: Income Always Low...................... 791 34.6 Yields and Risk Premium: Income Always Low................... 792 34.7 Bond Prices and p1: No Default Risk......................... 793 34.8 Yields: No Default Risk...........................
...... 794 955 34.9 Yields: No Default Risk................................. 796 34.10Yields: 10 Yr Treasury Note and Three Month Treasury Bill........... 798 34.11Representative Yield Curves.............................. 799 34.12Yield Curves Prior to Recent Recessions....................... 799 34.13The Yield Curve and the Term Premium...................... 817 34.14The Markets for Short and Long Term Government Bonds............ 824 34.15The Market for Long Term Risky Bonds....................... 825 34.16Conventional Monetary Policy: Open Market Purchase.............. 827 34.17Conventional Monetary Policy: Impact on Market for Risky Long Term Debt 828 34.18The Zero Lower Bound and the Market for Short Term Riskless Bonds.... 829 34.19Unconventional Monetary Policy: Forward Guidance............... 831 34.20Forward Guidance and the Market for Risky Long Term Debt.......... 832 34.21Unconventional Monetary Policy: Quantitative Easing, Government Securities 833 34.22Quantitative Easing (Government Securities) and Market for Risky Long Term Debt............................................ 834 34.23Quantitative Easing (Risky Long Term Debt).................... 835 35.1 The Equity Premium..
................................ 839 35.2 Total Stock Market Price Index............................ 839 35.3 Simulated Stock Price: No Bubbles.......................... 869............................... 870 35.4 Simulated Bubble Process 35.5 Actual and Fundamental Stock Price......................... 871 35.6 PE Ratio and Subsequent 20 Year Realized Return: S&P 500.......... 872 35.7 Price-Dividend Ratio and Subsequent Realized Return: Model with No Bubble873........ 874 35.8 Price-Dividend Ratio and Subsequent Realized Return: Bubbles 36.1 Empirical Measure of ft................................. 885 36.2 Equilibrium in the Partial Sticky Price Model with Financial Frictions..... 886 36.3 Increase in Credit Spread................................ 891 36.4 The IS Curve with the Financial Accelerator.................... 894 36.5 The AD Curve with the Financial Accelerator................... 896 36.6 Shift of the IS Curve with the Financial Accelerator................ 897 36.7 IS-LM-AD-AS Curves with Financial Accelerator................. 898 36.8 Supply Shock with
Financial Accelerator...................... 899 36.9 Demand Shock with Financial Accelerator, Simple Sticky Price Model..... 900 37.1 S&P 500 Stock Market Index Around Great Depression............. 904 956................. 904 37.2 Real Home Price Index Around Great Recession 37.3 Interest Rate Spread Around Great Depression.................. 906 37.4 Interest Rate Spread Around Great Recession................... 907 37.5 Industrial Production Index Around Great Depression.............. 908 37.6 Industrial Production Index Around Great Recession............... 908 37.7 Repo Haircuts During the Crisis............................ 912 37.8 AA-AAA Spread..................................... 914 37.9 The TED Spread During the Financial Crisis.................... 915 37.102007 Decline in House Prices.............................. 917 37.112007-2008 Financial Crisis............................... 918 37.12Federal Funds Rate Around Great Recession.................... 919 37.13The ZLB and the Intensiﬁcation of the Financial Crisis in 2008-2009...... 920 37.14Intensiﬁcation of the Financial Crisis in 2008-2009 with and without the ZLB 921 37.15Price Level and In
ﬂation Rate Around Great Recession.............. 922 37.16Real GDP and Hypothetical Trend.......................... 923 37.17Emergency Fed Lending................................. 925 37.18AD-AS Eﬀects of Federal Reserve Lending Programs............... 927 37.19AD-AS Eﬀects of Fiscal Stimulus........................... 929 37.20Demand and Supply for a Bond, Segmented Markets............... 932 37.21Unconventional Asset Holdings by Fed........................ 933 37.22Desired Eﬀects of Quantitative Easing Programs.................. 934 37.23AD-AS Eﬀects of Federal Reserve Quantitative Easing and Forward Guidance 937 A.1 Yt = ln Xt and dYt.................................... 971 dXt A.2 Y = X 2........................................... 977 A.3 Y = ln X − 2X....................................... 978 C.1 The Y s Curve: Derivation with Intertemporal Labor Supply........... 1000 C.2 IS − Y s Equilibrium with Upward
-Sloping Y s Curve............... 1002 C.3 Equilibrium in the Money Market........................... 1003 C.4 Increase in At with Upward-Sloping Y s Curve................... 1004 C.5 Increase in At: the Money Market........................... 1005 C.6 Positive IS Shock with Upward-Sloping Y s Curve................. 1006 C.7 Positive IS Shock: the Money Market........................ 1007 D.1 The Sticky Wage AS Curve: Derivation....................... 1010..................... 1012 D.2 The Sticky Wage AS Curve: Increase in At 957 D.3 The Sticky Wage AS Curve: Increase in ¯Wt..................... 1013 D.4 Sticky Wage IS-LM-AD-AS Equilibrium....................... 1016 D.5 Eﬀects of Increase in Mt................................ 1017 D.6 Eﬀects of Positive IS Shock............................... 1020 D.7 Eﬀects of Increase in At................................. 1021 D.8 Eﬀects of
Increase in ¯Wt................................ 1023 D.9 Eﬀects of Increase in At on Sticky Wage AS and Neoclassical AS........ 1025 D.10 Eﬀects of Increase in At on Sticky Wage AS and Neoclassical AS, Slope of AD 1026 D.11 Sticky Wage Model: Y0,t > Y f............................. 1029 0,t D.12 Sticky Wage Model: Y0,t > Y f........... 1030 0,t, Dynamic Wage Adjustment D.13 Sticky Wage Model: Increase in Mt, Dynamics................... 1032 D.14 Sticky Wage Model: Positive IS Shock, Dynamics................. 1034 D.15 Sticky Wage Model: Increase in At, Dynamics................... 1036 E.1 The MP Curve...................................... 1039 E.2 The AD Curve with the MP Curve: Derivation................... 1041 E.3 Shift of the AD Curve: Positive IS Shock...................... 1042....................... 1043 E.4 Shift of the AD Curve: Reduction in ¯rt E.5 The AD Curve with the MP Curve: Role of φπ................... 1044 E.6 Supply Side of the Model.......................
......... 1046 E.7 The IS-MP-AD-AS Equilibrium............................ 1048 E.8 Positive IS Shock..................................... 1049 E.9 Increase in ¯rt....................................... 1050 E.10 Increase in At....................................... 1051 958 List of Tables 4.1 GDP Per Capita for Selected Countries 4.2 Growth Miracles and Growth Disasters....................... 67....................... 68 9.1 Income and Substitution Eﬀects of Higher rt.................... 199 19.1 Qualitative Eﬀects of Exogenous Shocks on Endogenous Variables....... 443 20.1 Correlations Among Variables in the Data and in the Model........... 450 21.1 Dynamic Correlations between M2 and Output................... 476 23.1 Qualitative Eﬀects of Exogenous Shocks on Endogenous Variables in Open Economy Model..................................... 514 24.1 LM Curve Shifts..................................... 525 24.2 IS Curve Shifts...
................................... 526 24.3 AD Curve Shifts..................................... 530 25.1 Neoclassical AS Curve Shifts.............................. 538 25.2 Simple Sticky Price AS Curve Shifts......................... 543 25.3 Partial Sticky Price AS Curve Shifts......................... 552 26.1 Qualitative Eﬀects of Exogenous Shocks on Endogenous Variables in the Neoclassical Model.................................... 561 26.2 Qualitative Eﬀects of Exogenous Shocks on Endogenous Variables in the Simple Sticky Price Model.................................... 571 26.3 Qualitative Eﬀects of Exogenous Shocks on Endogenous Variables in the Partial Sticky Price Model.................................... 583 26.4 Comparing the Sticky Price and Neoclassical Models............... 584 27.1 Qualitative Eﬀects of Exogenous Shocks on Endogenous Variables in the Sticky Price Model, Transition from Short Run to Medium Run............. 600 27.2 Qualitative Eﬀects of Exogenous Shocks on Endogenous Variables in the Partial Sticky Price Model, Transition from
Short Run to Medium Run........ 612 959 28.1 Optimal Monetary Policy Reaction to Diﬀerent Shocks.............. 634 30.1 Comparing the Open and Closed Economy Variants of the Sticky Price Model 681 30.2 Comparing the Small Open and Open Economy Variants of the Sticky Price Model........................................... 685........................ 704 31.1 Potential Buyer and Seller Valuations 31.2 T-Account for Hypothetical Bank........................... 709 31.3 T-Account for Homeowner............................... 711 31.4 T-Account for Hypothetical Bank........................... 712 31.5 T-Account for Hypothetical Bank After Loan Default............... 712 31.6 T-Account for Hypothetical Bank........................... 713 31.7 T-Account for Hypothetical Bank........................... 713 31.8 T-Account for Hypothetical Bank........................... 714 31.9 T-Account for Hypothetical Bank........................... 715 31.10T-Account for Hypothetical Bank..................
......... 715 31.11T-Account for Hypothetical Bank........................... 716 31.12T-Account for Hypothetical Shadow Bank...................... 722 31.13T-Account for Hypothetical Shadow Bank...................... 723 32.1 Balance Sheet for Banking System as a Whole................... 729 32.2 Balance Sheet for Central Bank............................ 730 32.3 Balance Sheet for a Particular Bank......................... 730 32.4 Open Market Purchase from Bank A......................... 731 32.5 Balance Sheet for Central Bank After Open Market Purchase.......... 731 32.6 Bank A Makes a Loan.................................. 731 32.7 Bank A’s Loan is Deposited Elsewhere........................ 732 32.8 Bank B Gets a Deposit................................. 732 32.9 Bank B Makes a Loan.................................. 732 32.10Bank B’s Loan Gets Deposited Elsewhere...................... 732 32.11Bank C Gets a Deposit..
............................... 733 32.12Bank C Makes a Loan.................................. 733 32.13Funds are Withdrawn from Bank C.......................... 733 32.14Bank D Gets a Deposit................................. 734 32.15Deposit Creation..................................... 734 32.16Balance Sheet for Banking System as a Whole After Open Market Purchase. 736 32.17Balance Sheet for Banking System as a Whole................... 736 960 32.18Balance Sheet for Non-Bank Public.......................... 737 32.19Balance Sheet for Central Bank............................ 737 32.20Balance Sheet for Government............................. 738 32.21Initial Balance Sheet for Bank A........................... 738 32.22Open Market Purchase from Bank A......................... 739 32.23Balance Sheet for Central Bank After Open Market Purchase.......... 739 32.24Bank A Makes a Loan.................................. 739 32.
25Non-Bank Public gets a Loan............................. 739 32.26Non-Bank Public Withdraws Cash.......................... 740 32.27Bank A Handles Cash Withdrawal.......................... 740 32.28Bank A’s Loan is Deposited Elsewhere........................ 741 32.29Bank B Gets a Deposit................................. 741 32.30Bank B Makes a Loan.................................. 741 32.31Non-Bank Public Gets a Loan............................. 742 32.32Non-Bank Public Withdraws Cash.......................... 742 32.33Bank B Handles Withdrawal.............................. 742 32.34Bank B’s Loan is Deposited Elsewhere........................ 743 32.35Bank C Receives Deposit................................ 743 32.36Bank C Makes Loan................................... 743 32.37Non-Bank Public Gets a Loan.......
...................... 743 32.38Non-Bank Public Withdraws Cash.......................... 744 32.39Bank C Handles Withdrawal.............................. 744 32.40Deposits are Withdrawn from Bank C........................ 744 32.41Bank D Receives a Deposit............................... 744 32.42Bank D Makes a Loan.................................. 745 32.43Non-Bank Public Gets a Loan............................. 745 32.44Non-Bank Public Withdraws Cash.......................... 745 32.45Funds are Withdrawn from Bank D.......................... 745 32.46Deposits are Withdrawn from Bank D........................ 746 32.47Deposit, Loan, and Cash Changes........................... 746 32.48Stylized Great Depression Banking System Balance Sheet............ 751 32.49Stylized Great Depression Non-Bank Public Balance Sheet............ 751 32.50Stylized Great Depression Fed Balance Sheet....................
751 32.51Stylized Great Depression Non-Bank Public Balance Sheet: Cash Withdrawal 751 32.52Stylized Great Depression Banking System Balance Sheet: Cash Withdrawal. 752 32.53Stylized Great Depression Banking System Balance Sheet: Reducing Loans.. 752 961 32.54Stylized Great Depression Non-Bank Public Balance Sheet: Reduction in Loans753 32.55Stylized Great Depression Fed Balance Sheet: After Cash Withdrawal and Loan Reduction...................................... 753 32.56Stylized Great Recession Fed Balance Sheet: Before the Crisis.......... 755 32.57Stylized Great Recession Banking System Balance Sheet: Before the Crisis.. 755 32.58Stylized Great Recession Fed Balance Sheet: After the Crisis.......... 755 32.59Stylized Great Recession Banking System Balance Sheet: After the Crisis... 756 33.1 T-Account for Bank in Bad Equilibrium....................... 767 33.2 T-Account for Bank in Bad Equilibrium....................... 767 33.3 T-Account for Hypothetical Bank........................... 770 33.4 T-Account for Hypothetical Bank........................... 770 34.1 The Nature of Uncertainty and Risk Premia.................... 795 35.1 Uncertainty and the Equity Premium........................ 845 35.2 Stock and Bond Pricing: Three Periods....................... 860 37.1 T-Account for Bear Stearns 37.
2 T-Account for Bear Stearns.............................. 910.............................. 913 A.1 Greek Letters....................................... 964 A.2 Approximation of ln(1 + α)............................... 968 A.3 Derivatives of Common Functions........................... 971 A.4 Derivative Rules for Composite Functions...................... 972 D.1 Sticky Wage AS Curve Shifts D.2 Qualitative Eﬀects of Exogenous Shocks on Endogenous Variables in the Sticky............................. 1014 Wage Model........................................ 1024 D.3 Comparing the Sticky Wage and Neoclassical Models............... 1027 D.4 Qualitative Eﬀects of Exogenous Shocks on Endogenous Variables in the Sticky Wage Model, Transition from Short Run to Medium Run............. 1037 962 Part VII Appendices 963 Appendix A Mathematical Appendix Modern economics makes use of mathematics. Mathematics is a convenient and clean tool to express ideas formally. Mathematics is well-suited for the rigorous comparison of concepts in a formal model to observed data on economic variables. This book makes use of a good deal of mathematics. Most of the mathematics that we use is high school level algebra and basic calculus
. This appendix reviews several mathematical concepts which will be used throughout the book. A.1 Variables and Parameters A variable is something which can be represented by a number that can change. In economic models, there are two types of variables. An exogenous variable is a variable whose value is determined “outside of the model.” Put diﬀerently, the value of an exogenous variable is taken as given when working through a model. An endogenous variable is a variable whose value is determined “inside of the model.” The values of endogenous variables are determined given the structure of the model, taking the value of exogenous variables as given. An example of an endogenous variable in economics is a price – it is determined by the forces of supply and demand. An example of an exogenous variable is the taste a consumer has for some good. We take the consumer’s preferences (i.e. its taste for a particular good) as given, and hence exogenous. Given tastes (as well as other factors), we determine endogenous variables in the context of a model. We will typically denote variables with Latin letters. Because macroeconomics is focused on observations of variables at a point in time, we will index variables by the period in which they are observed. In particular, let t be a period index (which could be years, quarters, months, etc.). Yt denotes the value of the variable Y observed in period t. We often take period t to denote the present period, so Yt−1 would denote the value of the variable Y observed one period ago, while Yt+1 would denote the value observed one period in the future. We will use the notation that ∆Yt = Yt − Yt−1 denotes the ﬁrst diﬀerence of a variable across adjacent periods of time. A parameter is a constant which governs mathematical relationships in a model. We will typically use either lowercase Greek letters or lowercase Latin letters (without time 964 subscripts) to denote parameters. Table A.1 provides several diﬀerent symbols for lowercase Greek letters and their pronunciation. Table A.1: Greek Letters Symbol Name alpha α beta β gamma γ delta δ epsilon theta θ kappa κ rho ρ sigma σ phi φ chi χ The equation below provides a very simple example of an economic model: Yt = α + βXt (A.1
) In (A.1), Xt is an exogenous variable (the variable X observed at date t) and Yt is an endogenous variable. α and β are parameters. Given a value of the exogenous variable Xt, and given values of α and β, you can determine the value of Yt. The parameter β measures how Yt changes as Xt changes. A.2 Exponents and Logs We will be making frequent use of exponents and natural logs. The following are a sequence of rules for exponents. Xt and Yt denote variables and α and β are constant parameters: 965 (A.2) X 1 t X 0 t X −1 t X −α t = Xt = 1 = 1 Xt = 1 X α t = X α+β = X αβ = X α−β t (XtYt)α The natural log, which we will denoted by ln or log, is the inverse operator for the exponential function, which we will denote by exp(Xt) or eXt. e is called “Euler’s number,” and is approximately equal to 2.718. Below are some properties of the natural log and exponential function: (A.3) ln (exp(Xt)) = Xt exp(ln Xt) = Xt ln X α t = α ln Xt ) = ln Xt − ln Yt ln(XtYt) = ln Xt + ln Yt ln (Xt Yt ln 1 = 0 ln 0 → −∞ exp(0) = 1 exp(−∞) → 0 A.3 Summations and Discounted Summations In some applications we will be interested in summations of variables across time. Suppose that we want to sum up the value of X in periods t, t + 1, and t + 2. Formally: 966 S = Xt + Xt+1 + Xt+2 (A.4) We can write this in short hand using the summation operator, denoted by Σ (uppercase Greek sigma): S = 2 ∑ j=0 Xt+j (A.5) j is an integer index. The bottom part of the summation operator denotes where we start the sum (in this case, at j = 0). Starting with j = 0, you plug this
in to Xt+j and you get Xt. Then you go to the next integer, j = 1. You get Xt+1 You add this to the previous element, so you have Xt + Xt+1. You keep doing this until you get to the number/symbol at the top of the summation operator, in this case 2. More generally, the sum of the variable X from periods t to t + T, where T > 0, is: S = T ∑ j=0 Xt+j = Xt + Xt+1 +... Xt+T (A.6) You can also use summation operators to sum backwards in time. To do this, instead of writing +j in the subscripts on X, simply write −j. For example: S = T ∑ j=0 Xt−j = Xt + Xt−1 +... Xt−T (A.7) The summation of a constant times a variable is equal to the constant times the summation of a variable: T ∑ j=0 αXt+j = α T ∑ j=0 Xt+j (A.8) Suppose that you want to take the summation of two (or more) diﬀerent variables across time. You can distribute the summation operator across the two variables. In particular: T ∑ j=0 We will often be interested in computing discounted sums. Suppose that 0 ≤ α < 1 is a (Xt+j + Yt+j) = Xt+j + T ∑ j=0 T ∑ j=0 (A.9) Yt+j parameter and that Xt+j = αjXt. Suppose we want to compute the sum: We can write this as: S = T ∑ j=0 Xt+j 967 (A.10) Because Xt now does not vary with j, we can factor it out of the summation operator: S = T ∑ j=0 αjXt (A.11) S = Xt T ∑ j=0 αj Deﬁne S′ as the sum of α raised to successively higher powers: S′ = T ∑ j=0 αj = α0 + α1 + α2 +... αT Multiply both sides of the sum by α: αS′ =
α1 + α2 +... αT +1 Then, subtracting (A.14) from (A.13), we have: Solving for S′: S′(1 − α) = 1 − αT (A.12) (A.13) (A.14) (A.15) S′ = 1 − αT 1 − α If T is suﬃciently large, or α suﬃciently close to zero, αT ≈ 0, and we can approximate (A.16) the sum as: S′ = 1 1 − α (A.17) A.4 Growth Rates The growth rate of a variable is deﬁned as its change between two periods of time divided by the value in the “base” period. This is a general expression for a percentage diﬀerence, the change in a variable divided by its base. Most often when using the term growth rate we will mean the percentage change across two adjacent periods of time, but one could deﬁne growth rates over longer time horizons. Formally, deﬁne the period-over-period growth rate of variable Xt as: gX t = Xt − Xt−1 Xt−1 = ∆Xt Xt−1 968 (A.18) One can re-arrange this to get: 1 + gX t = Xt Xt−1 (A.19) One typically refers to gX t as the “net growth rate” and 1 + gX t as the “gross growth rate.” The gross growth rate is just equal to the ratio of a variable across time. A useful fact is that the log of one plus a small number is approximately equal to the small number. In particular: ln(1 + α) ≈ α (A.20) Table A.2 shows the actual value of ln(1 + α) for diﬀerent values of α. One can see that the approximation is pretty good. It is best for values of α closest to zero. Table A.2: Approximation of ln(1 + α) α -0.10 -0.05 -0.01 0.01 0.03 0.05 0.10 ln(1 + α) -0.1054 -0.0513 -0.01
01 0.0100 0.0296 0.0488 0.0953 Since growth rates are typically small numbers (i.e. a 2 percent growth rate is 0.02), we can use (A.20) to approximate the growth rate of a variable as the log ﬁrst diﬀerence: gX t ≈ ln Xt − ln Xt−1 = ∆ ln Xt (A.21) The approximation is suﬃciently good that we will treat the log ﬁrst diﬀerence as equal to the growth rate. This approximation has several useful insights. First, this makes it clear why we often like to plot macroeconomic variables in logs rather than levels. Plotting in logs means that we can interpret diﬀerences across time as approximate percentage diﬀerences, and the slope of a trending series plotted in the log is approximately the average growth rate. Second, we can apply this approximation more generally, treating the log diﬀerence between any two variables (not necessarily the same variable observed at diﬀerent points in time) as the approximate percentage diﬀerence. Third, we can use this insight to think about growth rates of functions of variables. 969 As an example of the latter, suppose that Yt = XtZt. Taking logs, one gets ln Yt = ln Xt + ln Zt. Then taking ﬁrst diﬀerences, one gets ∆ ln Yt = ∆ ln Xt + ∆ ln Zt. Since log ﬁrst diﬀerences are approximately equal to growth rates, this tells us that the growth rate of a product of variables is approximately equal to the sum of the growth rates. Similarly, the growth rate of a quotient of variables is approximately the diﬀerence in the growth rates of the variables. A.5 Systems of Equations In economics one often ﬁnds that the variables of interest are related to each other in a way that can be expressed as a system of equations. As a simple example of a system of equations, suppose that we have demand and supply curves for some good: Qt = Xt − aPt Qt = bPt (A.22) Here, Qt is the quantity of the good and Pt is the price. The ﬁrst equation
is the demand function (decreasing in price) and the second is the supply function (increasing in price). Xt is an exogenous variable representing tastes for the good, and a and b are positive parameters. Pt and Qt are the endogenous variables, and Xt is an exogenous variable. Sometimes one will see endogenous variables referred to as “unknowns” (the variables we are attempting to solve for) and exogenous variables as “knowns” (the variables whose values are taken as given). Here we have two equations in two unknowns. Since we are working with a linear system of equations (Qt and Pt enter both demand supply functions in a linear fashion – e.g. no exponents and no multiplication/division), there being the same number of equations as unknowns will ordinarily mean that there is a unique solution for the unknowns. If the system of equations were non-linear, the analysis is often more complicated and a solution may or may not exist. We can solve this system of equations by plugging the demand function into the supply function, which eliminates Qt and leaves one equation in one unknown (Pt). Doing so yields: Simplifying and solving for Pt yields: bPt = Xt − aPt Pt = Xt a + b 970 (A.23) (A.24) Now that we have solved for Pt in terms of just the exogenous variable, Xt, and the parameters a and b, we can solve for Qt. Simply plug this expression for Pt into either the demand or supply function. Doing so for the supply function yields: Xt Qt = b a + b One has solved a system of equations when one can express each endogenous variable as a function of exogenous variables and parameters only. We have done so here. Economically, we see that both price and quantity are increasing in the exogenous variable Xt (which governs tastes). If one were to draw graphs, an increase in Xt would shift the demand curve to the right and would result in both a higher price and a higher quantity. This is what we observe here mathematically. (A.25) In a two equation linear system, it is fairly straightforward to solve for the endogenous variables by hand, as we have done here. In a system of equations with many more variables this process can become unwieldy. The mathematical ﬁeld of linear algebra oﬀers some tools that can help deal with larger systems of equations. A
.6 Calculus Suppose that Yt is a continuous (i.e. no discrete breaks) function of Xt that has no kinks, given by Yt = f (Xt). f (⋅) is a function which “maps” a value of Xt into Yt. The derivative is a measure of how the value of the function changes as Xt changes. It is important to note the distinction between the derivative (which is itself a function) and the derivative evaluated at a point (which is a number). We will use the following notation to denote a derivative: dYt dXt = f ′(Xt) (A.26) In words, the left hand side says “the change in Yt for a change in Xt.” The notation on the right hand side, f ′(Xt), is notation for denoting the derivative of f with respect to Xt. The second derivative is just the derivative of the derivative – it is a measure of how the change in the function changes as Xt changes. Formally: d2Yt dX 2 t = f ′′(Xt) (A.27) You can calculate many higher order derivatives – e.g. the third derivative is the derivative of the second derivative, and so on. Below are some derivatives of particular functions: 971 Table A.3: Derivatives of Common Functions f (Xt) α αXt X α t ln Xt exp(Xt) f ′(Xt) 0 α αX α−1 t 1 Xt exp(Xt) The last line here is not a typo – the exponential function has the special property that it is its own derivative. Note that the derivative is itself a function. Consider the function Yt = ln Xt. The upper panel of Figure A.1 plots Yt as a function of Xt for a range of values of Xt. The lower panel plots the derivative of Yt with respect to Xt, which for for this function is simply equal to 1. Xt The derivative at a point is the value of the derivative evaluated at a particular value of Xt. For example, at Xt = 0.5, the derivative is 2; at Xt = 2, the derivative is 1/2. Figure A.1: Yt = ln Xt and dYt dXt Now, suppose that you
have two separate functions, h(Xt) and g(Xt). Suppose that f (Xt) is some composite function of these two functions. Table A.4 below gives several rules for dealing with derivatives of composite functions: 972 0123Xt-3-2-1012Yt0123Xt0246810dYtdXt Table A.4: Derivative Rules for Composite Functions f (Xt) f ′(Xt) h(Xt) + g(Xt) h′(Xt) + g′(Xt) h(Xt)g(Xt) h(Xt)g′(Xt) + g(Xt)h′(Xt) h(Xt) g(Xt) g(Xt)h′(Xt) − h(Xt)g′(Xt) g(Xt)2 h(g(Xt)) h′(g(Xt))g′(Xt) The ﬁrst row just says that the derivative of a sum of functions is the sum of the derivatives. The second row gives what is called the “product rule.” In words, the derivative of a product of two functions is the “ﬁrst times the derivative of the second, plus the second times the derivative of the ﬁrst.” The third row gives the “quotient rule.” In words, the derivative of a quotient is the “bottom times derivative of the top minus top times derivative of the bottom, divided by the bottom squared.” The ﬁnal row in Table A.4 gives what is called the “chain rule.” For a function of a function, the derivative is the “derivative of the outside times the derivative of the inside.” Example The chain rule is an important rule that will come in handy, particularly when we are doing multivariate optimization problems. Consider an example. Suppose a function is given by: Yt = ln [3 + 4X 2 t ] (A.28) Here, the “outside function” is ln(⋅), while the “inside function” is 3 + 4X 2 derivative of Yt with respect to Xt is: t. The dYt dXt = 8Xt 3 + 4X
2 t (A.29) Here, the part) and 8Xt is the derivative of the inside. 1 3+4X 2 t is the “derivative of the outside” part (evaluated at the inside In the analysis above, we considered derivatives of univariate functions – i.e. f (⋅) was a function of one variable, Xt. It is straightforward to apply the same rules outlined above to 973 multivariate functions. In particular, suppose that Yt = f (Xt, Zt). The partial derivative is a measure of how Yt changes as Xt changes, holding Zt ﬁxed. There is a similarly deﬁned partial derivative for how Yt changes as Zt changes, holding Xt ﬁxed. We will use the following notation: ∂Yt ∂Xt ∂Yt ∂Zt = fX(Xt, Zt) = fZ(Xt, Zt) (A.30) The partial derivative sign, ∂, is diﬀerent than d and denotes that all other variables are held ﬁxed. The subscripts X and Z under the f operator refer to the variable with respect to which one is diﬀerentiating. When calculating a partial derivative, you use the same rules as above, just treating the other variable as ﬁxed. Below are a couple of examples. Example Suppose that the function of interest is: Yt = ln Xt + Z α t (A.31) The partial derivatives are: ∂Yt ∂Xt ∂Yt ∂Zt = 1 Xt = αZ α−1 t Example Suppose that the function of interest is: Yt = X α t Z β t The partial derivatives are: 974 (A.32) (A.33) ∂Yt ∂Xt ∂Yt ∂Zt = αX α−1 t Z β t (A.34) = βX α t Z β−1 t Example Suppose that the function of interest is: Yt = ln [X α t + βZt] (A.35) In calculating the partial derivatives, we have to use the chain rule here. The partial derivatives are: ∂Yt ∂Xt ∂Yt ∂Zt = αX α−1
t + βZt X α t β + βZt X α t = (A.36) In these expressions, the and β are the “derivative of the inside” parts for both Xt and Zt. is the “derivative of the outside” part; αX α−1 1 t +βZt X α t For multivariate functions, a useful concept that will come in handy is the “total diﬀerential.” Whereas a partial derivative tells you how Yt changes as one variable changes, holding other variables ﬁxed, the total diﬀerential tells you how Yt changes as both variables change. Furthermore, whereas a partial derivative only tells you how Yt changes for a small change in Xt, the total derivative can be used to approximate the eﬀects on Yt of a large change in Xt. Formally, the total diﬀerential can be derived using a ﬁrst order Taylor series approximation. It says: dYt ≈ fX(X, Z)dXt + fZ(X, Z)dZt (A.37) Here, dYt = Yt −Y, dXt = Xt −X, and dZt = Zt −Z, where Y, X, and Z are particular values of these variables. The partial derivatives are evaluated at this point – i.e. here fX(X, Z) is a number, equal to the partial derivative fX(Xt, Zt) evaluated at the point (X, Z). In words, the total diﬀerential says that the change in Yt (relative to Y ) is approximately equal to the 975 sum of the partial derivatives times the change in each variable, where the partial derivatives are evaluated at (X, Y ). Example Suppose that the function is: Yt = ln [Xt + 3Z 2 t ] (A.38) The partial derivatives of this function are: ∂Yt ∂Xt ∂Yt ∂Zt = = 1 Xt + 3Z 2 t 6Zt Xt + 3Z 2 t (A.39) Suppose that we initially have Xt = 1 and Zt = 2. Then we have Yt = 2.5649. Suppose that both Xt and Zt change, to
1.1 and 2.1, respectively. The new value of the function is 2.6624. This means that dYt = 0.0975, and dXt = dZt = 0.1. Let’s see how well the total diﬀerential approximates this change. The partial derivatives evaluated at the initial values of Xt and Zt are 0.0769 and 0.9231, respectively. The total diﬀerential approximation would give us: dYt ≈ 0.0769 × 0.1 + 0.9231 × 0.1 = 0.1 (A.40) We can see that the total diﬀerential gives a good approximation (dYt = 0.1) to the actual change in output (dYt = 0.0975). The quality of the approximation will be worse (i) the bigger are the changes in the variables under consideration and (ii) the more non-linear the function is. If the function is linear, the total diﬀerential holds exactly – it is not an approximation. The concept of the total diﬀerential can be used to think about the growth rate of a sum. Suppose that we have: The total diﬀerential gives us: Yt = Xt + Zt dYt = dXt + dZt 976 (A.41) (A.42) Note that this holds exactly (not approximately), since Yt is a linear function of Xt and Zt. Suppose that the point to which you are comparing is last period’s value – i.e. dYt = Yt − Yt−1, dXt = Xt − Xt−1, and dZt = Zt − Zt−1. Then we can write this: Multiply and divide each term by its own lagged value, i.e.: ∆Yt = ∆Xt + ∆Zt Yt−1 Yt−1 ∆Yt = Xt−1 Xt−1 ∆Xt + Zt−1 Zt−1 ∆Zt (A.43) (A.44) = gY Note that ∆Yt Yt−1 sides by Yt−1, one gets: t – i.e. this is the growth rate. Taking note of this, and dividing both
gY t = Xt−1 Yt−1 gX t + Zt−1 Yt−1 gZ t (A.45) In words, what (A.45) says is that the growth rate of a sum equals the share-weighted sum of growth rates ( Xt−1 are the shares of X and Z in Y, respectively). An expression Yt−1 like this is useful for thinking about the contributions of diﬀerent expenditure categories to total GDP. and Zt−1 Yt−1 A.7 Optimization In economics we are often interested in ﬁnding optimums of functions. The optimum of a function, f (X), is the value of X, X ∗, at which f (X ∗) is either as large (the maximum) or as small (the minimum) as possible on the feasible set of values of X. Provided certain regularity conditions are satisﬁed, we can characterize optima using calculus. A necessary condition for X ∗ to be an interior optimum of f (X) is that f ′(X ∗) = 0. By “interior” we mean that we are not considering values of X that are on the “endpoints” of the feasible set of X values. This condition is what is called a ﬁrst order condition. The intuition for this is straightforward – for the case of a maximum, if a function were either increasing or decreasing at X ∗, then X ∗ could not possibly be an maximum. If f ′(X ∗) > 0, you could increase f (X) by increasing X ∗. If f ′(X ∗) < 0, you could increase f (X) by decreasing X ∗. We refer to points at which the ﬁrst order condition is satisﬁed as “critical points” – these are values of X at which the derivative of f (⋅) is equal to zero. Not all critical points are “global” optima – you could have multiple points where the ﬁrst order condition is satisﬁed, but only one represents the “global” optimum. We would refer to the other critical points as “local” maxima and minima. For most optimization problems encountered in this book, there 977 will only be one
optimum. The ﬁrst derivative being zero is necessary for either a maximum or a minimum. So how do we tell whether the critical point is a max or a min? The answer lies in looking at the second derivative. If the second derivative (evaluated at the critical point) is negative, then the critical point is a maximum. For a critical point to be a minimum, the second derivative (evaluated at that critical point) would be positive. We can think about maxima and minima intuitively by graphing a couple of functions. First, consider the function Y = X 2, where X can take on any real value (positive or negative). The plot of this function is shown in Figure A.2. One can clearly see that X = 0 is the minimum value of the function. Figure A.2: Y = X 2 Next, consider a more interesting function. Suppose that Y = ln X − 2X. The function is only deﬁned for positive values of X. The plot is shown below. One can observe from the ﬁgure that the optimum occurs somewhere around X = 1/2. 978 -2-1.5-1-0.500.511.52X00.511.522.533.54YY=X2 Figure A.3: Y = ln X − 2X Let’s work through the ﬁrst and second derivatives of each function and verify that calculus gives us the right answers that we can see graphically. Example The function is Y = X 2. The ﬁrst derivative is 2X. The critical value at which this equals zero is X ∗ = 0. Is this a minimum or a maximum? The second derivative is 2, which is positive. This tells us that this critical point is a minimum. This is consistent with what we can see in Figure A.2. Example The function is Y = ln X − 2X. The ﬁrst derivative is 1 − 2. For this to equal X zero, we must have X ∗ = 1/2. Is this a minimum or a maximum? The second derivative of this function is − 1 X 2. This is negative. Hence, this critical point is a maximum, which is consistent with what we observe in Figure A.3. One can usually write minimization problems as maximization problems and vice-versa. One does this by simply multiplying the function to be optimized by −
1. Suppose that you want to minimize the function Y = X 2. You could alternatively maximize the function Y = −X 2. The ﬁrst derivative is −2X and the critical value is X ∗ = 0 (i.e. multiplying the function by −1 does not aﬀect the ﬁrst order condition). The second derivative is now −2, which is negative. This says that X ∗ = 0 is the maximum of the function Y = −X 2. Equivalently, X ∗ = 0 is the minimum of Y = X 2. The basic rules of optimization that we have encountered apply equally well to multivariate problems. Suppose you have a function of two variables, f (X, Z). The ﬁrst order conditions 979 00.20.40.60.811.21.41.61.82X-3.4-3.2-3-2.8-2.6-2.4-2.2-2-1.8-1.6YY=lnX!2X are to set the partial derivatives with respect to both arguments equal to zero: fX(X, Z) = 0 and fZ(X, Z) = 0. The second order conditions are a little more complicated, but basically get at the same point. Technically the second order conditions place restrictions on the Hessian, which is a matrix of second derivatives. We won’t concern ourselves with any of that in this textbook. It’s a little more diﬃcult to graphically see the optima for a multivariate function, so we’ll work through a simple example: Example Suppose that the function we want to optimize is: Y = X αZ 1−α − aX − bZ (A.46) Here, α, a, and b are parameters. Find the ﬁrst partial derivatives: ∂Y ∂X ∂Y ∂Z = αX α−1Z 1−α − a = (1 − α)X αZ −α − b Setting these derivatives equal to zero implies: αX α−1Z 1−α = a (1 − α)X αZ −α = b The ﬁrst condition implies that: α−1 ( X Z ) = a α The second condition implies that: Divide (A.50) by (A.49)
to get: α ) = (X Z b 1 − α 980 (A.47) (A.48) (A.49) (A.50A.51) This optimality condition gives us the ratio of X Z that is consistent with the function being maximized. However, it is not possible to determine the levels of X or Z consistent with the function being maximized – you can see this by solving (A.51) for either X or Z and plugging it into one of the ﬁrst order conditions, where the X or Z will drop out. Often times in economics we will be interested in constrained optimization problems. Constrained optimization is at the heart of economics. Economics is about how agents maximize some objective (e.g. well-being, proﬁt) subject to the scarcity they face (e.g. limited income, limited time). Generally, we would like to maximize some multivariate function where the values of the variables we can choose are constrained in some way. Below is a simple example of a constrained optimization problem: max X,Z ln X + ln Z s.t. X + Z ≤ 1 Here, the “max” operator means that we want to maximize the function; the subscript X and Z refer to the fact that these are the variables we get to choose. The “s.t.” means “subject to.” The constraint is that the sum of X and Z must be weakly less than 1. One can see why the constraint matters here – if there were no constraint, the maximizing values of X and Z would be ∞ (inﬁnity) – i.e. you’d just want these variables to be as big as possible. The constraint puts a bound on how big these can be. For the optimization problems considered in this book, we will handle constrained optimization problems in the following way. We will assume that the constraint “binds,” which means holds with equality. Then solve for one variable in terms of other variables, and substitute back into the objective function (the function we want to optimize). This renders the constrained problem unconstrained. Then we ﬁnd the ﬁrst order conditions as usual. In this particular example, we can see that if the constraint binds, Z = 1 − X. Plug this into the objective, which renders the problem an unconstrained one in just choosing X
: max X ln X + ln(1 − X) 981 The ﬁrst order condition is: Now solve for XA.52) (A.53) We can then solve for the optimal value of Z by plugging this back into the constraintA.54) An alternative way to solve a constrained optimization problem is to use the method of Lagrange multipliers. Let λ be a number which references the value you would place (in terms of the objective function) on being able to “relax” the constraint (i.e. making the right hand side of the inequality bigger than 1). The Lagrangian is: L = ln X + ln Z + λ(1 − X − Z) (A.55) The Lagrangian is the objective function (ln X + ln Z) plus λ times the “big” side of the weak inequality minus the “small” side (where “big” refers to the “greater than or equal to” side and “small” refers to the “less than or equal to” side). Take the derivatives with respect to X, Z, and λ: ∂L ∂X ∂L ∂Z ∂L ∂A.56) The derivative with respect to the λ just gives you back the constraint. At an optimum, 982 all of these conditions must be equal to zero. This gives us two equations in two unknownsA.57) The ﬁrst condition tells us that X = Z. But if X = Z, the second condition tells us that X = Z = 1 2. This is exactly the same solution we got using the method of substituting the constraint into the objective function. The method of Lagrange multipliers is most useful in situations where the constraint may not bind (which would mean λ = 0). We will not be dealing with such cases, but the two methodologies will yield the same answers, as we will see in the example below. Example Consider a simple consumer optimization problem. A household can consume two goods, X and Z. She gets utility from those two goods, but faces a constraint that her expenditure on those two goods cannot exceed her income. The problem is: max X,Z U = ln X + Z s.t. PXX + PZZ ≤ Y Here PX and PZ are the prices of each good
, and Y is income available (which is taken as exogenous). ln X + Z is the utility function. Solve for Z in terms of X: Z = Y − PXX PZ (A.58) Plug this into the objective function, rendering this an unconstrained problem: max X U = ln X + Y − PXX PZ The ﬁrst order condition is: 983 1 X = PX PZ (A.59) To see how one can characterize this optimum using a Lagrangian, set up the Lagrangian: L = ln X + Z + λ (Y − PXX − PZZ) (A.60) The ﬁrst order conditions are: ∂L ∂X ∂L ∂Z = 1 X − λPX = 0 = 1 − λPZ = 0 Solving the second ﬁrst order condition for λ yields: λ = 1 PZ Plugging this in to the ﬁrst order condition for X yields: 1 X = PX PZ (A.61) (A.62) (A.63) (A.64) This is the same as (A.59), which was obtained simply assuming that the constraint holds with equality. This condition has a popular name in economics. It is a “MRS = price ratio” condition, where MRS stands for the marginal rate of substitution. The marginal rate of substitution is equal to the ratio of marginal utilities of two goods. In this case, the marginal utility of X is ∂U X and the marginal utility ∂X of Z is ∂U X. The price ratio is simply the ratio ∂Z of prices of the two goods. We can use (A.59) to solve for X: = 1. Then the MRS is ∂U ∂X / ∂U ∂Z = 1 = 1 X = PZ PX (A.65) Now plug this into the budget constraint to solve for Z: 984 PZ + PZZ = Y Z = Y PZ − 1 (A.66) (A.65) and (A.66) give us the demand functions for X and Z. The demand for X is decreasing in its own price and increasing in the price of Z. It does not depend on how much income the household has. The demand for Z is decreasing in its
own price and increasing in income. That X does not depend on income is not a general result but rather results because we have assumed a special kind of utility function here called quasilinear utility. Exercises 1. Express the following equations as log-linear functions, i.e. take logs and simplify. (a) Y = zK αN 1−α. (b) Z = certβK. 2. Calculate the ﬁrst and second derivative of the following functions: (a) f (c) = ln c. (b) u(c) = c1−σ 1−σ. (c) h(w) = (−6w3 + 17w − 4)β − ln(θwβ). 3. Calculate all the ﬁrst, second, and cross derivatives of the following functions: (a) F (K, N ) = θK αN 1−α. (b) F (K, N ) = ln θ + α ln K + (1 − α) ln N. (c) F (Z, X) = θZ βX γ. 4. Solve the following constrained maximization problem. Hint: Argue the constraint binds and then substitute the constraint into the objective function. First ﬁnd the optimality conditions. Then plug those optimality conditions back into the constraint, expressing x, w, and z as functions of parameters. max x,w,z U = α ln(x) + β ln(w) + (1 − α − β) ln(z) subject to pxx + pww + pzz ≤ y. 985 5. Consider an individual who receives utility from consumption, c, and leisure, l. The individual has ¯L time to allocate to work, n, and leisure. The individual’s consumption is a function of how much he works. In particular, c = n. The individual’s maximization problem is √ U = ln(c) + θl max c,l,n subject to √ n c = n + l = ¯L where θ > 0. Solve the maximization problem. Hint: Substitute both constraints into the objective function. 6. Evaluate: j=0 2j. j=0 j2. (a) ∑3 (b) ∑3
(c) ∑5 j=1 (d) ∑1000 j=1 5. 7. Show that: (2j − 3). ∑i ∑i (a) (b) (Xi + Yi) + ∑i Xi − ∑i Yi ∑i Xi + 2XiYi + Y 2 ) − ∑i i ∑i 8XiYi (X 2 i = 2. (X 2 i − 2XiYi + Y 2 i ) = 1 2. 986 Appendix B Probability and Statistics Appendix An important feature of modern economics is the comparison of models to data. To make these comparisons it is important to know some basic statistics. It is also useful to know some rules of probability when dealing with decision-making under uncertainty. This appendix reviews some basic statistical and probabilistic concepts and deﬁnitions. B.1 Measures of Central Tendency: Mean, Median, Mode The mean, median, and mode are diﬀerent ways of describing what is usually referred as a measure of central tendency of a distribution of a variable. That is, they reﬂect the typical values a variable takes. The mean (arithmetic mean to be more accurate) is usually calculated as the sum of the values divided by the total number of values. In terms of notation, the population mean is usually denoted by µ while the sample mean is denoted by ¯x (the sample is just a subset of the total population). Suppose we have a variable x for which we have N observations, which corresponds to the entire population. Therefore, xi represents observation i for i = 1, 2,..., N. The average for x is calculated as, µ = ∑N i=1 xi N. (B.1) If we have a population of 7 observations (N =7) given by: 8, 6, 15, 14, 13, 48, and 8, the mean can be calculated as: µ = 8 + 6 + 15 + 14 + 13 + 48 + 8 7 = 16. As you may realize, a limitation of the mean as a measure of of central tendency is that it is sensitive to outliers, i.e. a value that diﬀers greatly from the others. Suppose for instance we have a sample with the annual income of 100 individuals, 99 of whom have an income that varies between $40,000 and $80,000. The 100th
, however, has an income of $500,000. Clearly, if we use the mean, the income of the typical household would be signiﬁcantly over-estimated. 987 In our previous example, we can see that most values are very close to each other, with the exception of 48. If we are only looking at one speciﬁc measure of the distribution, we need to make sure the value obtained (16, in the case of the mean) is not reﬂecting the one that is very distinct and higher than the rest (48, in the example). The median, the value such that half of the observations are above and half of the observations are below, is not aﬀected by outliers. Obtaining the median is simple. We ﬁrst order observations from the smallest to the largest value. Again, with our previous example the ordering would be: 6, 8, 8, 13, 14, 15, and 48. Since we have an odd number of observations, the median is just the middle value: 13. Note that half of the values are above 13 and half of the values are below it. Note that the value obtained here is below the value obtained for the mean. As we mentioned, the median is not aﬀected by outliers. Since 48 is a value signiﬁcantly above the other ones, it was expected that the value of the median is lower. Obviously, when we have millions of observations, what is ‘expected’ is not so clear. Now, if we had an odd number of observations, the median is calculated by taken the average of the two middle observations. For instance, if our set of data was composed by 6, 8, 8, 13, 14, 15, 23, and 48. The median would be calculated as (13+14)/2 = 13.5. Finally, the mode is the most commonly observed value within our set. In the previous case, that would be 8. As you may be wondering, nothing prevents us from having a distribution that has more than one mode, i.e. a distribution in which there is two or more most commonly observed values. For instance, if we had 6, 8, 8, 13, 14, 15, 15, and 48, the mode would be 8 and 15. We refer this as a multimodal distribution and, more speciﬁcally,
bimodal. B.2 Expected Value Expected value is closely related to the mean. Formally, the expected value of a random variable is the probability-weighted arithmetic mean. Let us be concrete by oﬀering an example. Suppose that a random variable X can take on three values – X = 1, X = 3, and X = 11. We sometimes refer to these three diﬀerent possible realizations as “states of nature,” or just “states” for short. Suppose that the probability of these three states occurring are p1 = 1 6. Note that probabilities must sum to one – some state of nature must occur. The expected value of X, which we shall denote with E[⋅], is: 2, and p3 = 1 − p1 − p2 = 1 3, p2 = 1 E[X] = p1 × 1 + p2 × 3 + p3 × 11 = 11 3 = 3.6667 (B.2) More generally, suppose that X can take on N diﬀerent discrete values. Index these realizations by i = 1,..., N, and suppose that the probability of each realization is given by 988 pi, where ∑N i=1 pi = 1. Then the expected value is: E[X] = N ∑ i=1 Note that the expected value of X is in general not the simple arithmetic mean of the possible realizations of X. In the example used above, the arithmetic mean of all possible realizations is 5. It is important to weight potential realizations by their probability of occurring. (B.3) piXi However, the arithmetic mean of a sample of realizations of a random variable ought to correspond to its expected value in a large enough sample. Suppose that you are a statistician and observe a time series of X, denote the realizations by Xt, for t = 0,... T. If T is suﬃciently T t=0 xt big, the arithmetic average of Xt, i.e. ∑ T +1, should correspond to the expected value of X. This is because one should observe fewer realizations of 11, in the example considered above, than the other two realizations given the assumed probability structure. In essence, the frequency of observations in a random sample does the probability weighting for you, and the arithmetic mean of a
random sample (provided that sample is suﬃciently large) corresponds to the expected value of the random variable. There are a couple of useful properties of expected value. We will illustrate these in the context of the example considered above. First, expected value is a linear operator, which means that the expected value of a linear transformation of the series equals the transformation of the expected value of the series. Suppose that we consider multiplying all possible realizations of X by a constant parameter, call it a. Suppose that a = 3. Then the expected value is: E[aX] = p1 × a + p2 × 3a + p3 × 11a = a (p1 + p2 × 3 + p3 × 11) = a E[X] = 11 (B.4) Alternatively, suppose that you consider subtracting (or adding) a constant to each possible realization of X. In particular, we are interested in E[X − a]. In this particular example, we would have: E[X − a] = p1 × (1 − a) + p2 × (3 − a) + p3 × (11 − a) = p1 + p2 × 3 + p3 × 11 − a(p1 + p2 + p3) = E[X] − a = 2 3 (B.5) Since expected value is a linear operator, the expected value of a non-linear transformation 989 of a random variable is in general not equal to the non-linear transformation of the expected value; i.e. E[f (X)] ≠ f (E[X]). Suppose that we are interested in computing the expected value of 1 X. We would have: E [ 1 X ] = p1 × 1 1 + p2 × 1 3 + p3 × 1 11 = 0.5152 (B.6) In contrast, note that the inverse of the expected value of X is = 0.2727. This fact of expectations operators comes in handy when thinking about precautionary saving, for example. 1 E[X] Lastly, suppose that we have two random variables. Let these variables be Z and Y. To make things a little cleaner, suppose that each series can only take on two discreet values. Because it is a linear transformation, for example the expected value of the sum (or diﬀerence) of Z and Y would equal the sum (or diﬀerence) of the expected
values, i.e. E[Z + Y ] = E[Z] + E[Y ]. But suppose, instead, that we are interested in a non-linear transformation, such as the expected value of the product, E[ZY ]. In general, E[ZY ] ≠ E[X] E[Y ]. This would be true if the series were independent, but not in general. 2 and p2 = 1 − p1 = 1 Suppose, as an example, that Z and Y are independent. Z can take on values of 2 or 4, with probabilities p1 = 1 2 being the probabilities of these two states being realized. Y can take on two values, 0 or 3, with probabilities q1 = 2 3 and q2 = 1 3, respectively. The expected value of Z is just E[Z] = 3, while the expected value of Y is E[Y ] = 1. The product of expectations, E[X] E[Y ], is 3. But what about the expected value of the production, i.e. E[XY ]? With two possible states of nature for two diﬀerent random variables, there are in essence four joint possible states of nature – Z could be 2 and Y could be 0, Z could be 2 and Y could be 3, Z could be 4 and Y could be 0, or Z could be 4 and Y could be 3. If the realizations of Z and Y are independent, then the probabilities of these states occurring is simply the product of the expectations. Hence, we would have: E[ZY ] = p1q1 × (2 × 0) + p1q2 × (2 × 3) + p2q1 × (4 × 0) + p2q2 × (4 × 3) = 3 (B.7) For this particular example, the expected value of the product is equal to the product of the expected values. But what if, in contrast, the series are not independent? In particular, suppose that Z is more likely to be high when Z is high and vice-versa. Here, we need to discuss conditional probabilities. Let P r(Y ∣ Z) denote the probability of a particular realization of Y given a particular realization of Z. Suppose that Z can again take on two values of 2 or 4, each with probability of 1 2. Y can again take on two values of 0 or 3. But suppose that Y is more
likely to be high when Z is high and more likely to be comparatively 990 3 when Z = 4 and 0 with low when Z is low. Suppose that Y equals 3 with probability 2 3 when Z = 2. Assume that Y = 0 with probability 1 when Z = 2. Formally, this probability 1 means we are assuming P r(Y = 3 ∣ X = 4) = 2 3 and P r(Y = 3 ∣ X = 2) = 0. This means that the probabilities of the four states occurring are as follows. Z = 2 and Y = 0 with probability 3, and Z = 4 1 2 and Y = 0 with probability 1 3. Note that the probabilities of the four states occurring sum 2 to 1. The expected value of Z is again just 2. The expected value of Y is somewhat more complicated, because we need to condition on Z. In particular, we have: × 1, Z = 2 and Y = 3 with probability 1 2 × 0, Z = 4 and Y = 3 with probability 1 2 × 1 × 2 E[Y ] = 1 2 (1 × 0 + 0 × 3) + 1 2 (2 3 × 3 + 1 3 × 0) = 1 (B.8) In other words, the expected value of Y is probability Z = 2 times the sum of probabilityweighted realizations of Y conditional on this plus the probability Z = 4 times the sum of probability-weighted realizations of Y conditional on this. The expected value of Y for this particular example again works out to 1, just as in the case where the realizations of Z and Y were assumed to be independent. But what about the expected value of the product of X and Y? In this example, this is given by: E[ZY ] = 1 2 × 1 × (2 × 0) + 1 2 × 0 × (2 × 3) + 1 2 × 1 3 × (4 × 0) + 1 2 × 2 3 × (4 × 3) = 4 (B.9) In this example, the expected value of the product (4) is larger than the product of expected values (3). As we shall see below, if two random variables covary positively with one another, the expected value of a product will be greater than the product of expectations, while the if they covary negatively with one another, the reverse will be true. To understand what we mean by covary, it is useful to extend the concept of
conditional probabilities to conditional expectations. Formally, let E[Y ∣ Z] denote the expected value of Y conditional on the realization of Z. In the particular example we have been considering, we have E[Y ∣ Z = 2] = 0, while E[Y ∣ Z = 4] = 2. If the conditional expectations of Y are diﬀerent than the unconditional expectation (which we calculated above and which in this example is just 1), then Y and Z covary with one another. Evidently they covary positively with one another since the expectation of Y conditional on Z being high is larger than the unconditional expectation of Y (and vice-versa conditioning on Z being low). 991 B.3 Measures of Dispersion: Variance and Standard Deviation As useful as the measures of central tendency are, they provide an incomplete picture of the distribution. For instance, knowing that the GDP per capita in the U.S. is $51,000 just tells you the average income. Some individuals have income well above the average while others have income that is signiﬁcantly below the average. If we are interested in what the distribution of income looks like, then the mean and the median provide little information. We need a measure of dispersion that tells us how dispersed, or spread out, the observations are. A simple way of capturing dispersion would be to calculate the average “distance” each realization of a random variable is from its mean. One could measure “distance” by the absolute value of the diﬀerence between the realization of a random variable and its mean. A downside of this approach is that it places equal weight on realizations near the mean as those far from the mean. An alternative to using absolute value to measure distance is to measure distance with squared deviations from the mean. This is what economists do when we calculate variance. Formally, the variance of a random variable is the expected value of squared deviations of a random variable from its mean. Formally: var(X) = E[X − E[X]]2 (B.10) Note that we can equivalently write (B.10) as: var(X) = E [X 2 + E[X]2 − 2X E[X]] (B.11) We can distribute the outer expectation operator as: var(X) = E[X 2] + E[E[X]2] − E[2X E[
X]] (B.12) In simplifying (B.12), note that the expected value of an expected value is just the expected value. For example, there is no uncertainty over what E[X]2 is; hence, E[E[X]2] = E[X 2]. Furthermore, since there is no uncertainty over E[X], we can write the last term in (B.12) as E[X] E[2X] = 2 E[X]2 – in other words, we can take the E[X] inside the outer expectations operator outside of that expectations operator. Making use of this, we can write (B.12) as: var(X) = E[X 2] − E[X]2 (B.13) Let us work with one of the examples used above. Suppose that X can take on three values 3, or approximately 3.67. 6. The expected value is 11 – 1, 3, and 11 – with probability 1 2, and 1 3, 1 992 The variance can be calculate as the probability-weighted sum of squared deviations from the mean, or: 2 2 (1 − 11 3 var(X) = 1 3 (3 − 11 3 A diﬃculty in interpreting the variance is that it is expressed in squared units of the mean of a series. An easier metric to interpret is called the standard deviation, and is simply the square root of the variance. In particular: (11 − 11 3 = 11.56 + 1 2 + 1 6 (B.14) ) ) ) 2 sd(X) = √ var(X) (B.15) In the example above, the standard deviation of the random variable X would be 3.39. To interpret this statistic, it means that, on average, X is 3.39 units away from its mean. It is common to use the standard deviation as a measure of volatility (in a time series context, i.e. how much does a series tend to move around over time) and a measure of dispersion (in a cross-sectional context, i.e. how diﬀerent do individuals on average look from one another at a point in time). There are a couple of useful properties of variance. First, the variance of a constant times a random variable is the constant squared times the variance of the random variable. In particular: var(aX) = a2var(X
) (B.16) The standard deviation of a constant times a random variable is just the constant times the standard deviation of the variable: sd(a) = asd(X) (B.17) The variance of a constant plus a random variable is just the variance of the random variable: var(X + a) = var(X) (B.18) (B.18) turns out to be a special case of a formula relating variance and covariance of sums of random variables, which we discuss further in the section below. We can see clearly from (B.17) something that was mentioned above – the units of the variance are squared units of the mean, while units of the standard deviation are simply units of the mean. These units are controlled by the constant a. For series with diﬀerent means, it is diﬃcult to compare volatilities of series by comparing standard deviations. For example, 993 the standard deviation of X in the example above is 3.39. But the standard deviation of 2X would be 6.78. 2X would appear more volatile than X, but this appearance is illusory because the series have diﬀerent means. One way to deal with this issue is to compute what is called the coeﬃcient of variation, or cv. The coeﬃcient of variation is deﬁned as the ratio of the standard deviation of a random variable to its mean. The coeﬃcient of variation of X above is 0.92545 (i.e. 3.39/3.67). The coeﬃcient of variation of 2X is also 0.92545 (i.e. 7.68/(22/3)). Computing coeﬃcients of variation allows one to compare volatilities of series with diﬀerent means. Another way to compare volatilities of series with potentially diﬀerent means is to instead compute the variance / standard deviation of the natural log of a series. In particular: var(ln X) = E [ln X − E[ln X]]2 (B.19) In looking at (B.19), note that E[ln X] ≠ ln E[X] (see the discussion on expected value above). Why is it that computing standard deviations of logs can deal with the problem of series have diﬀerent means? As noted
in Appendix A, the diﬀerence in logs is approximately the percentage diﬀerence between the values of the variable. In terms of (B.19), ln X − E[ln X] is approximately the percentage diﬀerence of X about its mean. Percentages are, by construction, unitless. We can see why this works using the example considered above. In particular, the var(ln 2X) = var(ln X + ln 2) = var(ln X) (making use of (B.18)). In other words, the variance of the log of a series is independent of how that series is scaled (i.e. what its mean is). As long as a variable cannot go negative (meaning that one can in fact take the natural log of the series), macroeconomists almost exclusively focus on measures of volatility/dispersion based on natural logs of a series rather than using the coeﬃcient of variation. As deﬁned, the variance and standard deviation are properties of distributions of a random variable. It is also possible to compute sample variances and standard deviations. As above, let µ be the sample mean. Suppose that one observes T diﬀerent observations. Then the sample variance is: ∑T i=1 (Xi − µ)2 σ2 = T The sample variance is just the arithmetic average of squared deviations about the sample mean. In a suﬃciently large sample, this ought to correspond to the population variance, similar to how the sample mean ought to correspond to the expected value of a series in a suﬃciently big sample of data. The sample standard deviation is typically denoted σ and is simply the square root of the sample variance. (B.20) 994 B.4 Measures of Association: Covariance and Correlation In discussing expectations above, we referred to how two series covary with one another. In this section we formalize this concept and introduce the concepts of covariance and correlations as measures of association between two diﬀerent random variables. Formally, suppose that one has two random variables, X and Y. The covariance is deﬁned as: cov(X, Y ) = E [ (X − E[X]) (Y − E[Y ]) ] (B.21) In other words, the covariance between two series is the expected value of the product of deviations
about the mean. Note that if X = Y, then (B.21) collapses to the general formula for variance, (B.10). It is a measure of association between two series and conveys information about how series are associated with one another. If it is positive, it means that one series being above its average value means that, on average, the other series will also be above its mean. Note that one can write (B.21) as: cov(X, Y ) = E [XY − E[X]Y + E[X] E[Y ] − X E[Y ]] (B.22) The outer expectation operator in (B.22) can be distributed as follows: cov(X, Y ) = E[XY ] − E[X] E[Y ] + E[X] E[Y ] − E[X] E[Y ] (B.23) To do this, we are making use of two rules. One was documented above, and this is that the expected value of a sum is the sum of expected values. The other rule hinges on the fact that E[X] and E[Y ] are known – there is no uncertainty over these expectations, and thus they are in essence constants. Thus, E[E[X]Y ] = E[X] E[Y ] – i.e. the E[X] can be taken “outside” the outer expectations operator since it is a constant. Simplifying (B.23), we get: cov(X, Y ) = E[XY ] − E[X] E[Y ] (B.24) can be re-written as follows: E[XY ] = E[X] E[Y ] + cov(X, Y ) (B.24) (B.25) In other words, (B.25) tells us that the expected value of a product is equal to the product of the expected values plus a covariance term. If this covariance term is 0, then the expected value of a product equals the product of the expectations. If two series co-vary positively, the expected value of the product will be greater than product of expectations. To see 995 this concretely, return to the example considered about with Z and Y. Z can take on two values, each with probability 1 2. Let these two values by 2 and 4; hence, the expected value is E[Z] =
3. Suppose that Y can also take on two values. Y equals 3 with probability 2 3 3 when Z = 4. Y = 0 with probability 1 when Z = 2. As when Z = 4 and 0 with probability 1 documented above, the unconditional expectation of Y is E[Y ] = 1. What is the covariance 2, Z = 4 and Y = 3 between these two series? We can have Z = 2 and Y = 0 with probability 1 with probability 1 6. Hence, the covariance 2 is: 3, and Z = 4 with Y = 0 with probability cov(Z, Y ) = 1 2 (2 − 3)(0 − 1) + 1 3 (4 − 3)(3 − 1) + 1 6 (4 − 3)(0 − 1) = 1 (B.26) For this particular example, the covariance between Z and Y works out to 1. Note this covariance is consistent with (B.25) given that E[XY ] = 4 while E[X] E[Y ] = 3. We can use the formula relating covariance, the expectation of a product, and the product of an expectation, i.e. (B.24), to derive an expression for the variance of a sum of two random variables. In particular, suppose that we are interested in the variance of X + Y. This can be written: var(X + Y ) = E [X + Y − E[X] − E[Y ]]2 (B.27) Working this out in long hand, we get: var(X + Y ) = E [X 2 + XY − X E[X] − X E[Y ] + XY + Y 2 − Y E[X] − Y E[Y ] − E[X]X − E[X]Y + E[X]2 + E[X] E[Y ] − E[Y ]X − Y E[Y ] + E[Y ] E[X] + E[Y ]2] (B.28) (B.28) may be simpliﬁed: var(X + Y ) = E [X 2 + Y 2 + E[X]2 + E[Y ]2 + 2XY − 2X E[X] − 2Y E[Y ] − 2X E[Y ] − 2Y E[X]] (B.29) We can now distribute the outer expectation operator, again
making use of the fact repeatedly that E[E[X]Y ] = E[X] E[Y ] (i.e. the “inner” expectation operator can be moved outside the outer expectation operator). Doing so, we get: 996 var(X + Y ) = E[X 2] − E[X]2 + E[Y 2] − E[Y ]2 + 2 E[XY ] − 2 E[X] E[Y ] (B.30) Using (B.13), we can write (B.30) as: var(X + Y ) = var(X) + var(Y ) + 2 (E[XY ] − E[X] E[Y ]) (B.31) Now, using (B.25), we may write (B.31) as: var(X + Y ) = var(X) + var(Y ) + 2cov(X, Y ) (B.32) In words, (B.32) says that the variance of a sum equals the sum of variances plus two times the covariance between two variables. The result given above that var(X + a) = var(X) follows from this. If a is a constant, its variance is zero and its covariance with X is also zero. Hence, (B.18) is just a special case of (B.32). The sign of a covariance conveys information about whether two series tend to move together (positive covariance), opposite one another (negative covariance), or are unrelated (zero covariance). But it is diﬃcult to interpret magnitudes of a covariance. Similar to issues with variances and standard deviations, the covariance depends upon the means of the series under consideration. For this reason, it is common to instead measure the association between two series using the correlation coeﬃcient. The correlation coeﬃcient is deﬁned as the ratio of the covariance between X and Y divided by the product of the standard deviations of X and Y. In particular: corr(X, Y ) = cov(X, Y ) sd(X)sd(Y ) (B.33) The correlation coeﬃcient is constructed to lie between -1 and 1. A correlation of 0 means the series exhibit no (linear) relationship to one another and the covariance is zero. If two series are
identical, then cov(X, Y ) = var(X) and sd(X)sd(Y ) = var(X), so the correlation coeﬃcient is 1. Series that co-move negatively have a negative correlation. The strength of co-movement between two series is measured by how close the correlation is to 1 (or -1, in the case of negatively correlated variables). The correlation coeﬃcient is scale invariant. One can show that, for two constants a and b, cov(aX, bY ) = ab × cov(X, Y ). Similarly, the standard deviations of these constant times the random variables are sd(aX) = asd(X) and sd(bY ) = bsd(Y ). Hence, the correlation between aX and bY is invariant to the values of a and b. As with variances and means, it is possible to construct sample equivalents of covariances 997 and correlation coeﬃcients given a sample of observed data. Let µX and µY denote the sample arithmetic means of X and Y, and σX and σY denote the sample standard deviations (all deﬁned above). Then the sample covariance is: ̂cov(X, Y ) = ∑T i=1 (Xi − µX)(Yi − µY ) T (B.34) The “hat” appears atop the covariance operator in (B.34) to refer to the fact that it is an estimated measure of covariance based on an observed sample of data. The sample correlation coeﬃcient is typically denoted via ρ(X, Y ) and is deﬁned similarly to (B.33): ρ(X, Y ) = ̂cov(X, Y ) σXσY (B.35) 998 Appendix C The Neoclassical Model with an Upward-Sloping Y s Curve In the main text, we make an assumption on preferences that allows us to write the labor supply curve as a function only of the real wage, wt, and an exogenous variable which may be interpreted as an exogenous shock to preferences. This variable is called θt. Formally, the kind of preference speciﬁcation needed to motivate such a speciﬁcation in a micro-founded model is
based on Greenwood, Hercowitz, and Huﬀman (1988). Other speciﬁcations of preferences, in contrast, make labor supply considerably more complicated. In particular, there may be an intertemporal dimension to labor supply. Labor supply may be a function of the real interest rate, rt. For example, suppose that rt increases. A higher rt, other things being equal, likely means that a household would like to increase its saving (we say “likely” because this conclusion rests on another assumption that the substitution eﬀect dominates the income eﬀect in terms of consumption). In a model where income is exogenous, increasing saving requires reducing consumption. But if the household can inﬂuence its income through an endogenous labor supply choice, if it wants to increase its saving in response to an increase in rt it stands to reason that it may wish to increase its labor supply. This appendix explores the ways in which allowing for this possibility impacts the graphical presentation of the neoclassical model. We also look at how exogenous shocks might have diﬀerent eﬀects on endogenous variables. In a nutshell, allowing for an intertemporal dimension of labor supply results in the Y s curve being upward-sloping rather than vertical. This allows IS shocks to have eﬀects on output even in the neoclassical model, and results in supply shocks having smaller eﬀects on output than they would in the world where the Y s curve is instead vertical. The graphical presentation here with an upward-sloping Y s curve is very similar to the real interetemporal model in Williamson (2014). We prefer the presentation in the text with a vertical Y s curve for a couple of reasons. First, it greatly simpliﬁes the analysis – one needn’t worry about secondary eﬀects in the labor market and it removes some ambiguities related to how diﬀerent exogenous shocks impact endogenous variables. Second, it seems plausible to us that labor supply is only 999 very weakly impacted by the real interest rate. As such, the Y s curve is likely quite steep. Therefore, the assumptions giving rise to a vertical Y s curve do not seem to be too unrealistic. Third, the assumption of a vertical Y s greatly simpliﬁes the analysis of the New Keynesian model. If shocks to
the IS curve may impact Y f t, then both AD and AS curves will shift in response to these shocks. This does not fundamentally change much but signiﬁcantly complicates the analysis. C.1 The Neoclassical Model with an Intertemporal Dimension to Labor Supply The equations characterizing the equilibrium of the neoclassical model are identical to what is presented in Chapter 18 with the exception of the labor supply function. These are presented below for completeness: Ct = C d(Yt − Gt, Yt+1 − Gt+1, rt) Nt = N s(wt, θt, rt) Nt = N d(wt, At, Kt) It = I d(rt, At+1, Kt) Yt = AtF (Kt, Nt) Yt = Ct + It + Gt Mt = PtM d(rt + πe t+1, Yt) rt = it − πe t+1 (C.1) (C.2) (C.3) (C.4) (C.5) (C.6) (C.7) (C.8) The only diﬀerence relative to our earlier presentation involves (C.2), which features an argument related to the real interest rate. For the purposes of this appendix we assume that the partial derivative with respect to the real interest rate here is positive, ∂N s > 0, whereas ∂rt in our standard treatment in the main body of the text we (implicitly) assume that this partial derivative is zero. Our alternative assumption in this appendix reﬂects the reasonable idea that a household wishing to save more due to a higher interest rate will both consume less and work more. (⋅) The demand side of the economy is identical to what is presented in the main text and is not repeated here. The IS curve can be used to graphically summarize (C.1), (C.4), and (C.6). Money will still be neutral and the classical dichotomy will still hold; hence, we can analyze (C.7)-(C.8) after determining the equilibrium values of real endogenous variables. What will 1000 be diﬀerent is the supply side of the economy. Intuitively, there will be a relationship between rt and Yt on the supply side – a higher r
t will stimulate labor supply, which results in more Nt and hence more Yt. Figure C.1 graphically derives the Y s curve under these assumptions. Start with a particular value of the real interest rate, call it r0,t. Given θt, this determines the position of the labor supply curve (upper left plot). Find the level of labor input consistent with being on both the labor demand and supply curves. Call this N0,t. Plug this into the production (lower left plot). This gives a value of output, call it Y0,t. Reﬂect this onto the horizontal axis (lower right plot), and this gives a you a pair, (r0,t, Y0,t), consistent with (C.2), (C.3), and (C.5) all holding. Figure C.1: The Y s Curve: Derivation with Intertemporal Labor Supply One can consider higher or lower values of the real interest rate. A higher value results in the labor supply curve shifting out, which results in more labor input and hence more output. A lower real interest rate causes the labor supply curve to shift left, which has the 1001 𝑤𝑤𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝑌𝑌𝑠𝑠 𝑟𝑟0,𝑡𝑡 𝑟𝑟2,𝑡𝑡 𝑟𝑟1,𝑡𝑡 𝑤𝑤0,𝑡𝑡 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝐴𝐴𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑁𝑁0,𝑡𝑡
𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡,𝑟𝑟0,𝑡𝑡) 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴𝑡𝑡,𝐾𝐾𝑡𝑡) 𝑌𝑌0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡,𝑟𝑟2,𝑡𝑡) 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡,𝑟𝑟1,𝑡𝑡) 𝑤𝑤1,𝑡𝑡 𝑤𝑤2,𝑡𝑡 𝑁𝑁2,𝑡𝑡 𝑁𝑁1,𝑡𝑡 𝑌𝑌2,𝑡𝑡 𝑌𝑌1,𝑡𝑡 𝑌𝑌2,𝑡𝑡 𝑌𝑌1,𝑡𝑡 opposite eﬀect on labor input and output. Connecting the dots in the upper right plot, we get an upward-sloping Y s curve. A higher value of rt is associated with a larger value of Yt because of the eﬀect of rt on labor supply. Note that the Y s curve is nevertheless fairly steep. One could of course draw things diﬀerently, but to get the Y s curve far from vertical one would need the labor supply curve to shift quite signiﬁcantly with the real interest rate. The full equilibrium of the real side of the model can be characterized graphically as in Figure C.2 using the
same ﬁve part graph as in the main text. The only diﬀerence is that the Y s curve is upward-sloping rather than vertical. 1002 Figure C.2: IS − Y s Equilibrium with Upward-Sloping Y s Curve As noted above, the classical dichotomy continues to hold and nominal endogenous variables may be determined after real endogenous variables. We can do so using the same money demand-supply graph as in the main text, shown below in Figure C.3. Given values of rt and Yt (determined at the intersection of the IS and Y s curves), the position of the (upward-sloping) money demand curve is determined. The Pt where this intersects the 1003 𝑤𝑤𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑠𝑠 𝐼𝐼𝐼𝐼 𝑤𝑤0,𝑡𝑡 𝑟𝑟0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑁𝑁0,𝑡𝑡 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴𝑡𝑡,𝐾𝐾𝑡𝑡) 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡,𝑟𝑟0,𝑡𝑡) 𝐴𝐴𝑡𝑡𝐹𝐹(𝐾𝐾
𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑(𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟𝑡𝑡)+𝐼𝐼𝑑𝑑(𝑟𝑟𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡)+𝐺𝐺𝑡𝑡 (vertical) money supply curve is the equilibrium price level. Figure C.3: Equilibrium in the Money Market C.2 Eﬀects of Shocks with Upward-Sloping Y s We are now in a position to analyze the eﬀects of changes in diﬀerent exogenous variables on the endogenous variables of the model. In the process, we can examine how the results compare to the model presented in the main text where the Y s curve is vertical. Consider ﬁrst an exogenous increase in At. These eﬀects are shown in Figure C.4. In terms of how the Y s curve shifts, things work out exactly as in the text. Holding the real interest rate ﬁxed, labor demand shifts right and the production function shifts up. With higher Nt and higher At, Yt is higher for a given rt. As a consequence, the Y s curve shifts horizontally to the right. Because the horizontal shift is derived holding rt ﬁxed, the horizontal shift of the Y s curve here is exactly the same as presented in the text when the Y s curve is vertical. The rightward shift of the Y s curve causes Yt to rise and rt to
fall. The lower rt stimulates autonomous expenditure (consumption and investment are both higher) so that the expenditure line shifts up in the upper right graph (shown in green). What is diﬀerent relative to the version of the model with a vertical Y s curve is that output increases by less than the horizontal shift of the Y s curve, and consequently rt falls by less than it would in that model. Furthermore, there is a secondary eﬀect in the labor market that must be taken into account. The lower rt causes the labor supply curve to shift in. This inward shift of labor supply is the reason why output increases by less than the horizontal shift of the Y s curve – in equilibrium, Nt goes up by less when rt changes than if rt is held ﬁxed. The 1004 𝑀𝑀𝑡𝑡 𝑃𝑃𝑡𝑡 𝑀𝑀0,𝑡𝑡 𝑃𝑃0,𝑡𝑡 𝑃𝑃𝑡𝑡𝑀𝑀𝑑𝑑(𝑟𝑟0,𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌0,𝑡𝑡) 𝑀𝑀𝑠𝑠 equilibrium quantity of labor input must be consistent with the equilibrium level of output. The real wage is higher in equilibrium. Figure C.4: Increase in At with Upward-Sloping Y s Curve Relative to the version of the model with a vertical Y s curve presented in the main text, in this version of the model output increases by less, the real interest rate falls by less, the real wage rises by more, and labor input rises by less. Indeed, it is conceivable that Nt could 1005 𝑤𝑤𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡�
�� 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑠𝑠 𝐼𝐼𝐼𝐼 𝑤𝑤0,𝑡𝑡 𝑟𝑟0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑁𝑁0,𝑡𝑡 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴0,𝑡𝑡,𝐾𝐾𝑡𝑡) 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡,𝑟𝑟0,𝑡𝑡) 𝐴𝐴0,𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟1,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+
𝐺𝐺𝑡𝑡 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴1,𝑡𝑡,𝐾𝐾𝑡𝑡) 𝐴𝐴1,𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑌𝑌𝑠𝑠′ 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡,𝑟𝑟1,𝑡𝑡) 𝑤𝑤1,𝑡𝑡 𝑁𝑁1,𝑡𝑡 𝑌𝑌1,𝑡𝑡 𝑟𝑟1,𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟1,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟1,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 actually fall when At increases. For this to happen, the Y s curve would have to be suﬃciently ﬂat (i.e. labor supply would have to be quite sensitive to the real interest rate). This is not how we have drawn the ﬁgure (which shows Nt rising); but even if it rises, labor input nevertheless rises by less than it would if the Y s curve were vertical. The e
ﬀect of higher At on the price level is qualitatively the same as it is the version of the model with a vertical Y s curve. A lower rt and higher Yt both work to stimulate money demand (i.e. the money demand curve pivots to the right). This means that Pt must fall (equivalently, the price of money in terms of goods, 1, must rise). This is shown in Figure Pt C.5. Figure C.5: Increase in At: the Money Market Consider next a shock which causes the IS to curve to shift to the right. In Figure C.6, we consider an increase in government spending, Gt, though qualitatively the diagram would be the same for an increase in At+1 or a decrease in Gt+1. The rightward shit of the IS curve along an upward-sloping Y s curve results in rt rising and Yt rising. The increase in Yt is diﬀerent relative to the version of the model considered in class. The mechanism through which output increases is that the higher rt causes the labor supply curve to shift to the right, which results in wt falling and Nt rising. The higher rt works in the opposite direction of the exogenous impetus to desired expenditure (in this example, an increase in Gt), but it does not completely oﬀset it. This is shown in green in the upper right plot. How Ct and It react depends on the exact exogenous shock causing the IS curve to shift. In the case of an increase in Gt, they both must decline in equilibrium. Yt increases by less than Gt (recall from the text that the horizontal shift of the IS curve is the change in Gt, and since 1006 𝑀𝑀𝑡𝑡 𝑃𝑃𝑡𝑡 𝑀𝑀0,𝑡𝑡 𝑃𝑃0,𝑡𝑡 𝑀𝑀𝑑𝑑(𝑟𝑟0,𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌0,𝑡𝑡) 𝑀𝑀𝑠𝑠 𝑀𝑀�
�𝑑(𝑟𝑟1,𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌1,𝑡𝑡) 𝑃𝑃1,𝑡𝑡 Yt increases by less than the horizontal shift of the IS curve in equilibrium Yt − Gt is lower) and rt increases, both of which work to reduce Ct. A higher rt works to reduce It. Figure C.6: Positive IS Shock with Upward-Sloping Y s Curve Without additional assumptions, it is not possible to say deﬁnitively what ought to happen to Pt in response to a positive IS shock. On the one hand, Yt is higher, which stimulates the demand for money and would put downward pressure on the price level. But on the other 1007 𝑤𝑤𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑠𝑠 𝐼𝐼𝐼𝐼 𝑤𝑤0,𝑡𝑡 𝑟𝑟0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝑁𝑁0,𝑡𝑡 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴𝑡𝑡,𝐾𝐾𝑡𝑡) 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡,
𝑟𝑟0,𝑡𝑡) 𝐴𝐴𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺0,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟1,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺0,𝑡𝑡 𝐼𝐼𝐼𝐼′ 𝑌𝑌1,𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺1,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺1,𝑡𝑡 𝑌𝑌
𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺1,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟1,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟1,𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺1,𝑡𝑡 𝑟𝑟1,𝑡𝑡 𝑁𝑁1,𝑡𝑡 𝑤𝑤1,𝑡𝑡 hand, rt is higher, which depresses the demand for money and puts upward pressure on the price level. Which eﬀect dominates is not clear, and so Figure C.7 is drawn with a “?” to denote this ambiguity. Figure C.7: Positive IS Shock: the Money Market C.3 Sources of Output Fluctuations with an Upward-Sloping Y s Curve In the text we argued that the neoclassical / real business cycle model relies on supply shocks to be the main drivers of output in order to be at all consistent with known facts from the data. In the version of the model considered in the text, this conclusion is not particularly deep – with a vertical Y s curve, only supply shocks can impact output. But what if the Y s curve is instead upward-sloping? Since IS shocks can impact equilibrium output in this speciﬁcation of the model, does this open the door to demand-driven theories of output ﬂuctuations even within the conﬁnes of the neoclassical model? The answer turns out to be no. It is true that an upward-sloping Y s curve permits demand shocks to have eﬀects on output. But there are several problems as pertain the data. First, it is unlikely that the Y s curve is very ﬂat (i.e. that labor supply is highly sensitive to
the real interest rate). This means it would require extremely large shocks to the IS curve to generate reasonably-sized output ﬂuctuations. Second, as shown above in Figure C.6, conditional on IS shocks the real wage is countercyclical in the neoclassical model. In contrast, in the data the real wage is moderately procyclical, probably even moreso than conventional measures of the real wage would indicate (in particular, recall the discussion on the “composition bias” 1008 𝑀𝑀𝑡𝑡 𝑃𝑃𝑡𝑡 𝑀𝑀0,𝑡𝑡 𝑃𝑃0,𝑡𝑡 𝑃𝑃𝑡𝑡𝑀𝑀𝑑𝑑(𝑟𝑟0,𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌0,𝑡𝑡) 𝑀𝑀𝑠𝑠 𝑃𝑃𝑡𝑡𝑀𝑀𝑑𝑑(𝑟𝑟1,𝑡𝑡+𝜋𝜋𝑡𝑡+1𝑒𝑒,𝑌𝑌1,𝑡𝑡)? IS shock has ambiguous effect on price level with upward-sloping 𝑌𝑌𝑠𝑠 curve in Chapter 20). Third, in the data consumption, investment, and output are all strongly positively correlated with one another. IS shocks will typically result in either Ct and It moving opposite Yt (e.g. in response to an increase in Gt) or Ct and It moving opposite one another (e.g. this is what would happen conditional on anticipated changes in future government spending). This co-movement problem means that even if one entertains an upward-sloping Y s curve, for the neoclassical model to produce quantitatively reasonable co-movements among endogenous variables it must be predominantly driven by productivity shocks. 1009 Appendix D The New Keynesian Model with Sticky Wages In the main text we generate a non-vertical AS curve in
the short run by assuming that the price level is sticky (either completely or partially). In this appendix we document how a sticky nominal wage results in a similar non-vertical AS curve. The real wage, wt, gives the units of goods that a ﬁrm must pay the household in exchange for one unit of labor. The nominal wage, Wt, gives the units of money (i.e. dollars) that the ﬁrm must pay the household in exchange for one unit of labor. The real and nominal wage are connected via. If Wt = 6 dollars, and the price of a good in terms of dollars is the identity that wt = Wt Pt Pt = 2, then one unit of labor costs the ﬁrm 6/2 = 3 goods. In the sticky wage model, we assume that the nominal wage is set in advance and therefore exogenous and ﬁxed within a period. We denote the exogenous nominal wage as ¯Wt. With a ﬁxed nominal wage, it is in general impossible to simultaneously be on the labor demand and supply curves. We assume that the “rules of the game” are as follows. Once ¯Wt is set, the household commits to supply as much labor as the ﬁrm demands at this nominal wage. This means that the household will not be on its labor supply curve. Relative to the neoclassical model, we replace (25.1) with the condition that wt = ¯Wt, where ¯Wt is exogenous. The Pt following equations therefore characterize the supply side of the sticky wage New Keynesian model: wt = ¯Wt Pt Nt = N d(wt, At, Kt) Yt = AtF (Kt, Nt) (D.1) (D.2) (D.3) The AS curve is deﬁned as the set of (Pt, Yt) pairs consistent with these three equations holding. We can derive the AS curve graphically using a four part graph. In the upper left quadrant we plot wt against Nt. Given ¯Wt and ¯Pt, the real wage is determined. Given this real wage, we determine labor input oﬀ of the labor demand curve. Given this level of labor input, we determine output from the production function. 1010 Figure D.1: The Sticky Wage AS Curve: Derivation Figure D.
1 graphically derives the sticky wage AS curve. Start with a particular price level, P0,t. Given the exogenous nominal wage, ¯Wt, this determines a real wage,. Given this real wage, we determine labor input from the labor demand curve, N0,t. We then evaluate the production function at this level of labor input, giving Y0,t. Next, consider a lower price level, P1,t < P0,t. This results in a higher real wage. From the labor demand curve, this results ¯Wt P0,t 1011 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝑤𝑤𝑡𝑡 𝑃𝑃𝑡𝑡 𝑃𝑃0,𝑡𝑡 𝑃𝑃1,𝑡𝑡 𝑃𝑃2,𝑡𝑡 𝑊𝑊�𝑡𝑡𝑃𝑃0,𝑡𝑡� 𝑊𝑊�𝑡𝑡𝑃𝑃1,𝑡𝑡� 𝑊𝑊�𝑡𝑡𝑃𝑃2,𝑡𝑡� 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡=𝐴𝐴𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴𝑡𝑡,𝐾𝐾𝑡𝑡) 𝐴𝐴𝐴𝐴 in a
lower level of labor input, N1,t. The lower labor input results in a lower level of output, Y1,t. Next, consider a higher price level, P2,t > P0,t. This results in a lower real wage. The lower real wage induces the ﬁrm to hire more labor. More labor input results in a higher level of output, Y2,t. Connecting these diﬀerent (Pt, Yt) pairs yields an upward-sloping AS curve. The AS curve will shift if exogenous variables relevant for equations (D.1)-(D.3) change. Consider ﬁrst a change in At. This is shown graphically in Figure D.2. A higher At shifts the labor demand curve out. Holding the price level ﬁxed at P0,t, there is no change in the real wage. With the labor demand curve shifted out, the ﬁrm ﬁnds it desirable to hire more labor at the ﬁxed real wage. The production function also shifts up. Combining higher labor input with the shifted production function results in a higher level of output for a given price level. Put diﬀerently, the AS curve now crosses through a (Pt, Yt) pair to the right of the original point. In other words, the AS curve shifts out to the right. 1012 Figure D.2: The Sticky Wage AS Curve: Increase in At Note that changes in θt will not aﬀect the position of the AS curve in the sticky wage model. This is because θt is relevant only for the labor supply curve, and we are not on the labor supply curve in the sticky wage model. There is a new exogenous variable relevant for the position of the AS curve here, and that is ¯Wt. Suppose that ¯Wt were to increase. Holding the price level ﬁxed at P0,t, this would result in a higher real wage. Along the 1013 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡�
� 𝑤𝑤𝑡𝑡 𝑃𝑃𝑡𝑡 𝑃𝑃0,𝑡𝑡 𝑊𝑊�𝑡𝑡𝑃𝑃0,𝑡𝑡� 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡=𝐴𝐴0,𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴0,𝑡𝑡,𝐾𝐾𝑡𝑡) 𝐴𝐴𝐴𝐴 𝑌𝑌𝑡𝑡=𝐴𝐴1,𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴1,𝑡𝑡,𝐾𝐾𝑡𝑡) 𝐴𝐴𝐴𝐴′ downward-sloping labor demand curve, this would entail a reduction in labor input, from N0,t to N1,t. There is no shift of the production function. Lower labor input, however, means a reduction in output for a given price level. This means that, for a given price level P0,t, output will be lower at Y1,t. In other words, the AS curve will shift to the left when ¯Wt increases. Figure D.3: The Sticky Wage AS Curve: Increase in ¯Wt 1014 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡
𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝑤𝑤𝑡𝑡 𝑃𝑃𝑡𝑡 𝑃𝑃0,𝑡𝑡 𝑊𝑊�0,𝑡𝑡𝑃𝑃0,𝑡𝑡� 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡=𝐴𝐴𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴𝑡𝑡,𝐾𝐾𝑡𝑡) 𝐴𝐴𝐴𝐴 𝑊𝑊�1,𝑡𝑡𝑃𝑃0,𝑡𝑡� 𝐴𝐴𝐴𝐴′ Table D.1 summarizes how the changes in relevant exogenous variables qualitatively shift the sticky wage AS curve. Table D.1: Sticky Wage AS Curve Shifts Change in Variable Direction of Shift of AS ↑ At ↑ ¯Wt ↑ θt Right Left No Shift t At this point it is useful to compare and contrast the sticky price and sticky wage models. In the sticky price model of the text, we replace labor demand with the AS curve that Pt = ¯Pt + γ(Yt − Y f ). In the sticky wage model presented here, we replace labor supply with the condition that wt = ¯Wt, where ¯Wt is exogenous. Either setup generates a non-vertical AS Pt curve because higher Pt results in higher Yt, and thus both versions will allow demand shocks (both real demand shocks to the IS curve and monetary shocks) to aﬀect output. But the two models do have diﬀerent implications for the behavior of the
real wage in response to exogenous shocks. D.1 Equilibrium Eﬀects of Shocks in the Sticky Wage Model Eight equations characterize the equilibrium of the sticky wage economy. These are shown below: Ct = C d(Yt − Gt, Yt+1 − Gt+1, rt) wt = ¯Wt Pt Nt = N d(wt, At, Kt) It = I d(rt, At+1, Kt) Yt = AtF (Kt, Nt) Yt = Ct + It + Gt Mt = PtM d(rt + πe t+1, Yt) rt = it − πe t+1 (D.4) (D.5) (D.6) (D.7) (D.8) (D.9) (D.10) (D.11) The endogenous variables of the model are Yt, Nt, Ct, It, rt, it, Pt, and wt. This is eight 1015 endogenous variables (with eight equations). The exogenous variables of the model are At, t+1, and ¯Wt. We could also consider θt to be an exogenous variable, but At+1, Gt, Gt+1, Mt, πe it will have no eﬀects on the equilibrium of the economy since we are not on the labor supply curve in the sticky wage model. These expressions are identical to the neoclassical model, with the exception of (D.5), which replaces the labor supply curve. The full equilibrium is depicted graphically in Figure D.4: 1016 Figure D.4: Sticky Wage IS-LM-AD-AS Equilibrium There is a similarity to the sticky price model here in how the equilibrium quantity of labor input is determined, though there are some diﬀerences. Output is determined by the joint intersection of the AD and AS curves. The level of labor input must be consistent with this quantity of output. The real wage is read oﬀ of the labor demand curve at this level of labor input, instead of the labor supply curve as in the sticky price model. 1017 𝑤𝑤𝑡𝑡 𝑃𝑃𝑡𝑡 𝑌𝑌𝑡𝑡 �

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