1Copyright © 2015 Kit Pasula. All rights reserved.7. A Note on Logarithms (with applications to growth accounting and elasticity)(a) Rates of ChangeConsider the functiony = θ ln xwhere ln denotes the natural logarithm and θ is a constant. The partial derivative and the totaldifferential are ,dy.xθdxx dxdy=θ = If both variables are in logarithmsln y = θ ln x ,then the total differential puts both variables in terms of percentage rates of change:.xdxydy= θThis transformation is used in so-called “growth accounting”, where the growth rate ofreal GDP is determined by (apart from some constant parameters) the rate of growth of thecapital stock, the rate of growth of hours worked, and the rate of growth of total factorproductivity. Consider the Cobb-Douglas production function,and take logs of both sides:1ln y = ln A + α ln K + 1( -α ln) L .Taking the total differential, one obtains the “growth accounting equation”:1( ) ,LdLKdKAdAydy= + α + – αwhere the rate of growth (or the percentage change) of production is determined by the rate ofgrowth of total factor productivity, the rate of growth of the capital stock, and the rate of growthof hours worked. The variable dA/A is often called the “Solow residual”. The Solow residualcaptures those factors that cause growth in real GDP apart from growth in K and L. Increases in1Two rules of logarithms are used: ln (EFG) = ln E + ln F + ln G, and ln (Eα) = α ln E.AK =y α L1-α2A are associated with factors such as increasing skills of workers, technological advances such asimprovements in the quality of machines, efficiency gains within firms associated withreorganization of the business, decreasing regulation that allows more workers to be involved inthe production of goods (and fewer workers dealing with the regulation), and so on.The growth accounting equation is typically used in the analysis of long-term economicgrowth. In the Canadian context, for example, the growth rate of real GDP over the period 1960to 2008 averaged about 3% per year. Over the same period, the average growth rate of hoursworked was about 1% per year and the average growth rate of the capital stock was about 3%. 2With α equal to 0.30, this implies that the Solow residual was, on average, about 1.4% per year.3(b) ElasticityConsider a demand function for goods with the following specification:Qd = P-θ x ,where Qd denotes quantity demanded, P denotes the price of the good, x denotes other factors,and θ is a constant parameter. If one takes logarithms,ln Qd = – θ ln P + ln x ,and then takes a partial derivative with respect to ln P, one obtains.//lnln= = -θdP PdQ Qd Pd Qd dThe (constant) price elasticity of demand is then equal to –θ. For example, if –θ equals -2 (that is,if the price elasticity of demand equals -2), then a 10% fall in price will, other factors fixed,result in a 20% increase in quantity demanded.Similarly, if one considers the labour demand function derived in this chapter,] 1[( ) ] [ ] ,/1( )[ /1 α α /1 α /1 αααα–= ––=w PAKw PA KLdit is clear that the elasticity of labour demand with respect to the real wage rate (w/P) will equal-1/α. That is, if one takes logarithms and then calculates the partial derivative, one obtains2Note that with both y and K growing about 3% per year, the capital-output ratio K/y was roughly constant overthis long period of time. One of the stylized facts of growth for most industrialized economies is that K/y is in factfairly close to constant over the long term.3For an application of this approach to East Asian economies over their period of very high economic growth(1966-1991), see Alwyn Young (1995), “The Tyranny of Numbers: Confronting the Statistical Realities of the EastAsian Growth Experience”, Quarterly Journal of Economics.3.1( / /() / )/ln( / )lnα–= =d w P w PdL Ld w Pd Ld d dIf α equals 0.3, the elasticity of labour demand with respect to the real wage rate equals -3.33. Inthis case, labour demand is very responsive to real wage changes. If the real wage rate falls by1%, the quantity of labour demanded will increase by 3.33% (holding other factors fixed).8. Calibration(a) The Basic ApproachSince some path-breaking research by Finn Kydland and Ed Prescott (1982) in the early 1980s4,it has become increasingly common for macroeconomists to utilize a technique known ascalibration. The technique involves the following steps:• The analysis begins with a key equation5 from the model, such as a first-order conditon, arearranged first-order condition with certain variables substituted out (using other parts ofthe model), or a (possibly rearranged) demand-suppy equilibrium relationship.• Calibrate the key equation, by assigning numbers to all of the parameters in the equation.The magnitude of the numbers utilized can be derived from the theory itself, from simplestatistical analysis, or from sophisticated empirical studies in the economics literature.• Collect data on the variables in the key equation.• Simulate the model: Use the model to generate time-series predictions of the endogenousvariable that is the focus of the analysis.• Compare the predictions of the model with the actual real world data on this variable.• Additional assessments can be conducted (sensitivity analysis, comparing simplecorrelations in the model with the actual correlations in the data, and so on).Let’s consider an example, based on the model in this chapter. The first-order conditionof profit maximization implies that the real wage rate equals the marginal product of labour(where K =1 has been assumed):= 1( – α)A L-α .w PThis equation, a first-order condition from profit maximization, forms the basis for the analysisof a number of different variables. The only parameter in the equation is α. The theoreticalanalysis suggests that (1-α) equals labour’s share of real income. In Canada, labour’s share hasbeen fairly constant over time, and a rough estimate would have α equal to 0.3 (so that labour’sshare is 0.70). The key equation is therefore written as4The paper is Finn Kydland and Ed Prescott (1982), “Time to Build and Aggregate Fluctuations”, Econometrica.5In more complicated calibration exercises, the model analyzed consists of multiple equations and uses advancedstatistical techniques to model expectations of future variables.4= 7.0 A L – 3.0 .w POne then collects time-series data on w, P, L, and y. The variable A is calculated as follows (usethe Cobb-Douglas production function with K=1.0):L 7.0yA =Simulations can then be conducted.(b) Some Calibration ResultsIf the main focus is on the real wage rate (or the marginal product of labour), one cancompare the actual real wage rate (w/P) with the predicted marginal product of labour (or thepredicted real wage rate) on the right-hand side of the equation. Using annual data over theperiod 1997-2008, one obtains the following result (the plot is from the file mpl.wmf).While there are advanced techniques for assessing the model’s predictions, theassessment here will be purely visual (even when advanced methods are used, it is still useful toconduct a visual analysis). The actual real wage rate and the predicted marginal product of labourtended to move together, with both variables increasing over this time period. Beginning in 2005,however, the actual real wage continued rising whereas the predicted marginal product tended tolevel off.REALWAGE MPLReal Wage and 0.7A/L**0.3 or Predicted Marginal Product of Labour1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 20080.8750.9000.9250.9500.9751.0001.0251.0501.0755If the focus is on labour demand, the equation can be rearranged so as to derive the labourdemand function. Solving for L, one obtains:] ./7.0[ 33.3w PAL =Now one can compare the actual L (the actual hours worked in the Canadian economy) with themodel’s predicted hours on the right-hand side of the equation. The plot in this case is shown inthe following Figure (the file is hours.wmf).While the actual hours rise throughout the time period, the predicted hours rise up to 2005 butthen fall sharply in 2006 and again in 2008. Indeed, predicted hours fall by about ten percentbetween 2005 and 2008, quite a large percentage drop. Why is the model predicting a fall inhours? From the labour demand function, it is clear that a rise in A increases labour demandwhile a rise in w/P decreases labour demand (with elasticities of 3.33 and -3.33, respectively).While A is generally rising, it was fairly constant between 2005 and 2008; on the other hand, thereal wage rate rises throughout the time period, with a very large increase in 2006. A rise in thereal cost of labour to firms (that is, a rise in the real wage rate), with almost no change in theexogenous factors that affect the marginal productivity of labour, generates a fall in labourdemand in the model.Similarly, the model predicted a recession near the end of the time period. Substitutingthe model’s predictions for L into the Cobb-Douglas production function,y = AL 7.0HOURS HRSPREDActual Hours and [0.7A/w/P]**3.33 = Predicted Hours1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 20084500000004750000005000000005250000005500000005750000006yields the model’s predictions for real GDP (or y). The figure for the actual y and the predicted yis shown below (the file is realgdp.wmf).Thus, the model was predicting a recession in Canada in 2006, with a slight recovery in 2007,followed by another recession in 2008.(c) A Note on FilteringThe analysis above was conducted using the actual variables. Sometimes, when analystsare focusing on the business cycle, researchers “filter” the data so as to separate the long-termchanges in the variables with the short-run fluctuations associated with the business cycle. Thatis, the actual variable y (for example) would be split into two components, a trend componentytrend and a cyclical component yc: .ctytrendytyt=+ One of the most common filters used is the Hodrick-Prescott (or HP) filter. The HP filterallows the long-run or trend component to be time-varying yet fairly smooth. With real GDP, forexample, the long-run component would not simply increase by 3% every year, but may increaseby 2.7% in one year, 3.5% in another, and so on. The cyclical component, when plotted, wouldindeed look very much like a business cycle (and would have a zero mean).REALGDP YPREDActual Real GDP and [A*L**0.7 using predicted L = Predicted Real GDP1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 20089e+119.5e+111e+121.05e+121.1e+121.15e+121.2e+121.25e+121.3e+121.35e+127CANSIM DataVariable L:wL:Hours, V2529224 (annual)Wages, Salaries, Supplementary Income, V1996534 (monthly); this variable iswage times hours, and is used in calculating w, the wage rate.Consumer Price Index, V41693271 (annual)6Real GDP, V1992067 (quarterly).P:y: Recall that the variable A was calculated as y/L0.7.In the calibration analysis, the data must all be in the same frequency. In this case, all data mustbe annual (you can easily convert monthly data or quarterly data to annual data, but doing theopposite is not quite as straightforward). The software that I used (called RATS, RegressionAnalysis for Time Series) automatically converted the monthly and quarterly data to annual data.Not all software would do this conversion automatically. When one is collecting data for anempirical project, one must determine what data are available and determine what frequency willactually be used for the empirical analysis (and ensure that one frequency can easily beconverted to another. In this case, data on hours were not readily available on a quarterly basistherefore, one had to work with annual data.There are data on hours on a quarterly basis, but only for the “business sector” (the hoursused in this chapter are aggregate hours for the entire Canadian economy). For the “businesssector”, a unified set of data on these variables is available in the Cansim Table 3830008(Cansim provides a lot of footnotes, explaining what the terms actually mean). The following listprovides the Cansim numbers for the relevant variables:Business Sector Variables Real GDP:Hours Worked:Total compensationper hour workedImplicit price deflator(the price level)V1409154V1409155V1409158V20805660 6The CPI data are also available on the monthly basis. The series used here is the annual CPI.8Appendix: Tom Sargent and Labour Adjustment CostsThis setup is taken from Sargent (1978), “ Estimation of Dynamic Labor Demand Schedulesunder Rational Expectations”, published in the Journal of Political Economy. Sargent was aproponent of using quadratic models (quadratic objective functions, with linear constraints), asthis setup generated linear first-order conditions that can then be utilized in econometricanalyses.The production function is quadratic: 2The variable A is total factor productivity (Sargent treats A as unobservable; after Kydland andPrescott, economists regard A as observable (use a Cobb Douglas production function tocalculate A). The symbols f0 and f1 are constant parameters (just numbers); these parameterswould have to be set in the calibration exercise.Note that the marginal product of labour, f’(L) is 0 .Note that the parameters would have to be set so this condition holds (given the real-worldnumbers for A and L). The marginal product of labour must be positive, so the calibration of thesymbols f0 and f1 must be done such that marginal product is positive. In addition, f’’(L) must benegative- that is, there are diminishing returns: 0 ,so that the parameter f1 must be positive.Sargent adds labour adjustment costs to a standard intertemporal problem of a firmchoosing L so as to maximize liftetime profit (in real terms). The adjustment costs in period t aresimply specified as follows (again, a quadratic functional form is used): !” # $” %!$ 2 & ,where d is a constant parameter (that would need to be calibrated). Changing the quantity oflabour employed, either up or down, is costly. It is costly to add workers and it is costly to reducethe number of workers.The Lagrangian, denoted Z, for maximizing lifetime profit is as follows: (only the firsttwo terms are shown; W is the nominal wage rate, P is the price level, and r is the real interestrate).9‘ ()*,+ -+.+ / & (012+ *+3)&4)5+3) ,,&6+3)7+3) *+3)& 8,*+3)&*+ ,19+ …The first-order condition is as follows:‘ :; & 1 1 0.With labour adjustment costs, the condition of profit maximization is not the marginal product oflabour equals the real wage rate. To maximize lifetime profit, the firm will choose the level of Lwhere the marginal product of labour equals the real wage rate with adjustments for changes inlabour adjustment costs in both the current and the following period.Solving the first-order condition for Lt yields the following equation that can becalibrated: . ; & 1 : 1 1 Once one sets the parameters (f0, d, f1), and collects data on L, A, W, P, and r), one can then do acalibration analysis.
