THE STUD OF GERMAN ECONOM WITHIN THE FRAME OF SOOW GROWTH MODE German precision in every little detail is reflected in the national economy V. V. Putin, ex-president of the Russian Federation. INTRODUCTION Since unification Germany has been performing sustainable economic growth. Analysis of latest macroeconomic trends in Germany may help us to see economic theories in practise. This paper aims to assess German economy within the frame of Solow growth model. It starts with identifying the state of economy, and then proceeds to the analysis of short-run and long-run effects of unemployment change on the performance of the economy. The measures for optimal saving rate at steady-state and the discussion of the outcomes of increased saving rate conclude the essay. The following abbreviations and notations are used in the text and data tables: OS ordinary least squares GDP gross domestic product MP marginal product of capital MP marginal product of labor CRS constant returns to scale rate of saving π inflation (GDP deflator δ rate of depreciation η rate of population growth u rate of unemployment All data for German economy for the period of study is retrieved from World Databan (www.databan.worldban.org. Data for capital stoc in Germany is received from Federal Statistical Office, Wiesbaden, via e-mail (vgr-vermoegen@destatis.de.
. STATE OF ECONOM Given the Cobb-Douglas production function, where the powers and show that the economy has constant returns to scale, OS estimations can be obtained for the unnown parameters. First, the function is to be transformed in order to mae it linear in parameters. Taing natural logs of terms will give me the following: ln ln ( ln To perform accurate and precise estimation I used data starting from 99 (instead of 003. All variables are in real term (price adjusted. As Solow growth model assumes that economy is closed and government expenditure is zero hence, I obtained figures for net real GDP subtracting from total GDP net exports (NX and government expenditure (G (see Appendix. OS regression of ln on ln and ln run in E-views performed the following results: Dependent Variable: OG( Method: east Squares Date: /3/0 Time: 7:56 Sample: 99 008 Included observations: 8 Variable Coefficient Std. Error t-statistic Prob. OG( 0.84309 0.0900 9.68578 0.0000 OG( 0.57673 0.5634 0.648 0.5409 R-squared 0.879434 Mean dependent var 8.08050 Adjusted R-squared 0.87899 S.D. dependent var 0.05693 S.E. of regression 0.00377 Aaie info criterion -4.84446 Sum squared resid 0.006643 Schwarz criterion -4.745486 og lielihood 45.59974 Durbin-Watson stat 0.733395 0.84 0.6 So, Germany has a production function. Analysis of the data for the years 99-008 showed the following results for saving rate (, depreciation rate (δ: 0.84 96 δ 0.090 i 0,093 d 7,834 69,644 * 533,70 Calculated in MS Excel The investment per worer (i exceeds depreciation per worer (d. Data for the period shows zero population growth in Germany, so (η is excluded from the function. It is clear that the German economy is currently below the steady state. I calculated the steady-state capital per
worer (* it shows that Germany is far below the steady-state (<*. Figure for * is calculated basing on the property of steady-state: δ δ δ in our case 0.84: * δ 0.6 Using the above formula I obtained the capital per worer units at different saving rates. Graph. represents the relationship between saving rate and capital per worer: Graph. Capital per Worer at Different Saving Rates (Germany,500,000 Capital per worer,,000,000,500,000,000,000 *f( 500,000 0 0.0 0. 0. 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0. 0. 0.4 0.5 0.6 0.7 0.8 0.9 0 Rate of saving Built in MS Excel As output y is the function of capital and positively related to it, I can say that increase in saving rate will increase the output substantially. 3. THE EFFECT OF CHANGE IN UNEMPOMENT TO THE OUTPUT Consider the production function ( for the economy. The effective government policy reduces the natural rate of unemployment from u to u. In section I considered employed labor force as labor input in production function (. Consequently, the decrease in the rate of unemployment will increase employed labor force (labor input. For study purposes and measurement precision I will tae a hypothetical economy where capital stoc ( is 8000, 3
employed labor force 80, unemployment reduces from u 0 to u 0, and the economy.8 experiences a Solow growth production function 0 (immediate impact of the change in unemployment to the level of output. (a Short-term effects of reduced unemployment 0.. et us, first, see the short-term In the short-run an increase in will cause a fall in capital stoc per worer measure the change in output per worer by defining it as a function of unemployment rate.. I can If I denote total labor force as F, then the employed labor at any given unemployment rate can be expressed as a function of F and u t : f F, u F( u (3. t ( t t So, the percentage change in labor input due to change in unemployment will be: F( u F( u F( u Rearranging the terms will give a percentage change in as a function of unemployment rate: u u (3. u In our hypothetical economy this will be: 0. 0. 0. 0.5 0. 0.8 i.e. due to reduction in unemployment the labor input increased by.5 Next, I express the percentage change in capital stoc per worer as function of : Replacing the s with its expressions in 3. will result in: F( u F( u F( u (3.3 u u u Thus, for our example the percentage change in capital stoc per worer is: 0.0. 0. 0. i.e.. 0. 0.9 And, similarly, I can write output per worer as a function of unemployment. The Solow growth model states that output per worer y is a function of capital per worer : y f( 4
Hence, the percentage change in y will be: ( ( ( y ( ( Employing 3. again I will get: F( u u y ( (3.4 F u u In our hypothetical example the percentage change in output per worer due to change in unemployment will be: 0.8 0.8 ( 0. 0.8 0.8 y (0.89 0.9-0.09 i.e. 9 ( 0. 0.9 So, the 0 fall in unemployment entails.5 increase in employed labor,. decrease in capital stoc per worer and final 9 decrease in output per worer. The short-run effects of a decrease in unemployment on the economy in steady-state are represented in Graph 3a.: Graph 3a. Output per worer, y Investment, i Depreciation, d yf( d(δn y if( i Capital per worer, One should be careful in interpretation of this decrease. The percentage shows not the fall in output (although it virtually seems to be a decrease in output, but indicates fall in productivity i.e. amount of output produced by a worer. Thus, increased employment reduces labor productivity. 5
The graph shows how decrease in capital stoc per worer reduces output per worer y and investment per worer i. Economy moves to under-steady-state, where investment exceeds depreciation. In contrast to the decrease in output per worer, total output will rise. The change in total output is the marginal changes in both factors of production. Algebraically: MP MP (3.5 In short-run the change in total output is due to change in labor only: MP MP if divide by : y MP (3.5- MP is the differentiation of production function with respect to : MP ( ( ( d d (3.6 For I use equation 3.3. And substituting all figures will end up with: u u y ( as y I divide both sides by y to get: u u u y ( (3.7 3 u In our example the percentage change in total output will be: 0. 0. 0.0 0. 0.05 i.e..5 0. 0.8 y y y y y y Change in investment can be found using iy: i y 3 For another formula for the short-run percentage change in total output due to change in unemployment, see Appendix (B, section (a. 6
Thus, the short-run effect of reduced unemployment on the total output is favorable. In our hypothetical economy 8000, 80. Output level at this state is: 0.8 0. 0.8 0. (8000 (80 3,85 Decrease in unemployment by 0 will cause the employed labor ( to rise by.5. Thus, new level of labor input is 90. The capital stoc is fixed at 8000. And new short-term output level will be: (8000 (90 0.8 0. 0.8 0. 3,6 Percentage change in output is: 3,63,85 0.049 3,85 i.e..49 As the Graph 3a. showed, due to increase in employment the economy moved to under-steadystate. In response to this capital stoc starts increasing and in the long-run change in unemployment will have even greater impact on total output, what is discussed next. (b ong-term effects of reduced unemployment Capital stoc ( remained stable in the short run, but it will increase in order to get the initial steady-state level of investment per worer. Following the basic logic, if / and denominator ( changes by some percentage, then numerator ( is to change the same percentage in order to get initial value of. Hence, in the long-run percentage change in capital will be equal to percentage change in labor: An algebraic proof of the above equation is given in Appendix (B, section (b. ong-run change in total output ( will be due to changes in both inputs labor ( and capital (. Rewriting the equation 3.5: MP MP (3.5 7
where MP is the change in total output due to change in capital stoc and MP is the change in total output due to change in labor. Theory states that when the production function has constant returns to scale (CRS a percentage change in inputs (, will result in the same percentage change in total output (. So: For precision I can provide the following proof of the above. A little manipulation can be applied to rearrange the function 3.5 into: MP MP y MP MP (3.5- which is the function of change in output per worer in the long-run. MP and MP can be obtained by differentiating production function to and respectively: ( with respect d MP (3.8 d MP ( ( ( d d (3.6 In part (a I obtained function for percentage change in labor ( (equation 3.3. And putting 3.3, 3.6, and 3.8 into 3.5- will result in: y y ( ( ( ( ( ( ( ( ( ( as y I can divide the above by to get The above function shows that long-run percentage change in total output due to change in unemployment is equal to the percentage change in labor 4. 4 See Appendix (B, section (c for alternative proof of this statement. 8
et me illustrate it in the example of our hypothetical economy. Percentage change in labor ( was.5. Hence, the percentage change in total output is also.5. Recall our assumption that in our hypothetical economy 8000, 80. Output level at this state is: 0.8 0. 0.8 0. (8000 (80 3,85 Reduction of unemployment by 0 will cause the employed labor ( to increase by.5 (see part (a. Thus, new level of labor input is 90. To get to its steady state level (00 units of capital stoc per worer the capital stoc also increases by.5 to 9000. And new output level will be: 0.8 0. 0.8 0. (9000 (90 3,583 Percentage change in output is: 3,583 3,85 0.5 3,85 i.e..5 The immediate and over-time impact of increase in labor on the total output is depicted in the following graph: Graph 3. Short-run and ong-run Effects of Increased abor to the Total Output (Change in Production Possibility Curve 3 a c b The production possibility curve shows maximum level of output the economy can produce with given amount of two factors capital (vertical axis and labor (horizontal axis. In the graph an increase in labor from to shifts the production curve from to (the level output moves from point a to a higher point b. Over time capital increases from to so that initial / ratio restores. This shifts production curve to 3 and the output level moves to point c. Thus, the short-run change in total output is the difference between a and b and the long-run change is difference between a and c. In short, the long-run effect of reduced unemployment on the total output of the economy is much greater then the short-run impact. 9
4. STEAD-STATE CAPITA, OUTPUT, CONSUMPTION AND SAVING RATES (a Steady-state capital stoc per worer, output per worer and consumption per worer functions I am given the following production function with assumption of zero population growth rate and no technological progress:.3 0.7 0 n 0, g 0 In order to get per worer variables I divide the function by : Replacing the decimals with lower case letters for per worer units I get: (4. f( (4. y From the macroeconomic theory I now that change in capital stoc equals the investment minus depreciated capital. In per worer representation that is: i δ (4.3 where δ is the rate of depreciation. In steady-state the change in capital per worer is zero ( 0. Hence: i δ (4.4 I also now that investment per worer is equal to savings per worer. Savings are the proportion of income per worer saved. I have income per worer as a function of capital per worer (equation 4.. Expressing all these mathematically: i s s y y i s where is the rate of saving (i.e. marginal propensity to save. Replacing investment in equation 4.4 I will get: δ Following rearrangement gives me the capital stoc per worer as a function of saving and depreciation rates: δ 0.7 δ 0 0.7 7 ( δ 0 7 0 7 * (4.5 δ 0
Similarly, replacing in equation 4. I obtain output (income per worer as function of saving and depreciation rates: y y δ 3 7 0 7 y * (4.6 δ And equation 4.6 enables us to retrieve consumption per worer as a function of rates of saving and depreciation: c y y c y( 3 7 c * ( (4.7 δ The above equation again supports the economic theory that consumption of a worer is negatively related to the saving rate. (b Optimal saving rate for a given depreciation rate in steady-state The economy has a depreciation rate of 0 percentδ 0.. Using the functions in section 4 part (a (namely equations 4.5, 4.6, and 4.7 I can calculate capital stoc per worer, output per worer and consumption per worer for different saving rates. Table 4b. shows the results of calculations: Table 4b. * y* δ* c* MP 5 MP-δ 0.0 0.0000 0.0000 0.0000 0.0000 0..0000.0000 0.000 0.9000 000 0.000 0..698.3459 0.69.0767 0.500 0.0500 4.8040.603 0.4804.09 0.000 0.0000 0.4 7.458.84 0.746.0869 0.0750-0.050 0.5 9.966.993 0.9966 0.9966 0.0600-0.0400 0.6.934.55.93 0.86 0.0500-0.0500 0.7 6.70.304.67 0.6907 0.049-0.057 0.8 9.504.4380.9504 0.4876 0.0375-0.065 0.9 3.0783.5643.3078 0.564 0.0333-0.0667.0 6.870.687.687 0.0000 0.0300-0.0700 saving rate at which output per worer (y* will be maximized (at 00 saving rate saving rate at which consumption per worer (c* will be maximized (30 - Golden Rule saving rate Calculated in MS Excel 5 The function for marginal product of capital (MP in the above table was obtained by differentiating the production function.3 0.7 0 with respect to(see Section 3 part (b, page 7, eq. 3.8
In this case, the Golden Rule saving rate is 30 - at this rate the consumption per worer is maximized. Optimum saving rate can be directly obtained from consumption per worer function by differentiating it with respect to saving rate (δ. Output per worer is always maximized at 00 saving rate but this has not an economic meaning, as entire income cannot be saved without any consumption. Chart 4b. summarizes the figures for output per worer and consumption per worer from the above table: Graph 4b. Steady-state Output per Worer and Consumption per Worer at Different Saving Rates 3.0000 Output per worer, y, consumption per worer, c.5000.0000.5000.0000 y 0.5000 c 0.0000 0.00 0.0 0.0 0 0.40 0.50 0.60 0.70 0.80 0.90.00 Rate of saving Built in MS Excel So, at the steady-state the optimal (Golden Rule saving rate is (30 the consumption per worer is about.. An alternative method of obtaining optimum saving rate is given in Appendix (B, section (d.
5. INCREASED SAVING RATE AND THE OVERA ECONOMIC WEBEING One of the prior aims of every government is to ensure economic welfare of its citizens. One of the best ways to raise national wealth is to encourage savings, thus, investment. Policymaers may choose to promote increase in public savings by running budget deficit, encouragement private saving by reducing the tax on income from lending capital, or the combination of both. However, the statement that investing the larger proportion of the national income will entail higher productivity and living standards is not always true. et us consider the economy, which is at a steady-state below the Golden Rule level. At this point the consumption is not maximized. If the government chooses to rise saving rate, the effect will be upward shift of investment curve. Eventually, an increase in saving rate will decrease consumption, as larger proportion of income is being saved now. However, over time due to the excessive investment the economy will move towards the new steady-state. Moreover, extra investment will entail increase of production and following growth of employment. Thus, in the long-run the whole society will benefit from the extended saving. The immediate and long-term effects of the increase in saving rate are shown in the following graph: Graph 5. Output, y Consumption, c Investment, y t 0 t Time, t The saving rate increases Consumption restores initial level 3
Government policy increases saving rate at time t o, what cause immediate rise of investment and fall of consumption. The time period from t 0 to t is so-called sacrifice period where the consumption is below the initial level (which is not preferable for present generation of consumers. In contrast, in the long-run (after t consumption raises even higher then its initial level. It is clear that there is a trade-off between the short-run and long-run effects of the increased saving rate. It is the decision of policymaer whether to rise saving rate, sacrificing the welfare of present consumer in favor of higher steady-state in the perspective future or to stay inactive. One should tae into account the drawbacs of sacrifice period and equalize the interests of present and future generations. In contrast to the above situation, if the economy is already at (or even above the Golden Rule steady-state, then the increase in savings will have the opposite effects. In the short-run the consumption will fall, but long-run restoration of consumption will be below the initial level. Productivity of capital (in terms of marginal product of capital MP will fall. The government is expected to maintain optimal (Golden Rule saving rate, where MPngδ i.e. slope of the output per worer curve will be equal to the slope of depreciation per worer line. Thus, the practical value of the idea of improving economic wellbeing of the society through investing larger share of national income depends on the state of economy and trends it is being ruled by. 4
APPENDIX (A Series Name (Economy of concern - Germany 003 004 005 006 007 008 Output, GDP deflator (π 04 05 06 06 08 0 Nominal GDP, current CU, (N,63,800,000,000,0,900,000,000,4,00,000,000,35,00,000,000,48,00,000,000,495,800,000,000 Exports of goods and services, current CU, (X 77,30,000,000 849,90,000,000 9,400,000,000,05,740,000,000,37,90,000,000,77,040,000,000 Imports of goods and services, current CU, (M 685,380,000,000 736,990,000,000 80,850,000,000 9,0,000,000 966,0,000,000,0,570,000,000 Net exports, (NXX-M 85,930,000,000,930,000,000 9,550,000,000 3,50,000,000 70,970,000,000 54,470,000,000 General government final consumption expenditure, current CU, (G 46,850,000,000 45,860,000,000 49,960,000,000 45,430,000,000 435,640,000,000 45,680,000,000 Net nominal GDP, current CU, (NNN-NX-G,66,00,000,000,68,0,000,000,70,690,000,000,768,50,000,000,8,590,000,000,889,650,000,000 GDP, price adjusted, (00*NN/π,597,34,65,385,60,009,53,80,606,3,30,755,668,066,037,736,686,657,407,407,77,863,636,364 Capital, Capital stoc, price adjusted, ( 0,48,93,69,308 0,485,95,38,095 0,535,48,3,076 0,70,339,6,64 0,697,009,59,59 0,73,490,909,09 Depreciation, D (δ Consumption of fixed capital, price adjusted, (D 309,904,36,93 97,535,696,98 78,08,373,600 90,670,4,579 30,446,869,565 3,984,53,63 Depreciation rate, (δd/ 0.097 0.084 0.064 0.07 0.0300 0.030 abor, (η abor force, total (T 40,766,06 40,60,890 4,67,80 4,894,75 4,,94 4,375,9 Unemployment, total, of total labor force, (u 0 0 9 8 Employed labor force, (T*(-u/00 36,689,456 36,35,68 37,039,860 37,704,848 38,4,967 38,985,09 Population growth rate, (η 0 0 0 0 0 0 Per worer units, (, y, MP Capital stoc per worer, (/ 84,49 90,65 84,436 83,89 78,409 74,80 Output per worer, (y/ 43,53 44,333 43,367 44,40 43,898 44,065 Marginal product of capital, (MP(y -y /( - 0.684 56 0.687 -.434 0.063-0.046 5
6 APPENDIX (B (a Alternative function for short-run change in total output due to change in unemployment rate: ( ( u u u F u F using figures for our hypothetical economy: 0.04.04 0. 0. 0. i.e..4 (b Proof of as (c Alternative proof of (in the long-run ( ( ( ( as (see previous section: ( ( (
7 (d Alternative derivation of Golden Rule saving rate The saving rate, at which consumption per worer is maximized, can be derived by differentiating the function for consumption per worer (c* with respect to saving rate. In section 4, part (a I obtained steady-state consumption per worer at given depreciation rate (δ as a function of saving rate: ( * 7 3 δ c If I generalize it: ( * δ c A little rearrangement will bring it to: ( ( δ δ * c So, in order to get the optimal (Golden Rule saving rate I differentiate the above with respect to and equalize it to zero: ( ( ( ( ( ( ( ( ( ( ( ( δ δ optimal d dc 0 0 * At steady-state with given depreciation rate (δ the Golden Rule saving rate (optimal saving rate is equal to the capital intensity level (the power of in production function.
REFERENCES Abel, A, Bernane, B & Croushore, D, 008, Macroeconomics, Pearson Education Inc., Boston, US. Federal Statistical Office, Wiesbaden, 00, National Accounts Annual Report, published on August 8 th 00, received on November 0 th 00, <http://www.destatis.de>. Gärtner, M, 009, EurMacro: Economics Site, last modified: August 3 st 009, viewed November 8 th 00, <http://www.fgn.unisg.ch/eurmacro/tutor/solow_index.html> Gordon, R, 006, Macroeconomics, Pearson Education Inc., Boston, US. Maniw, N, G, 003, Macroeconomics, 5 th edition, Worth Publishers, N, USA, chapters 8-7, pages 80-34. World Development Indicators & Global Development Finance Database, 00, WorldDataBan, last updated September 8 th 00, viewed November 4 th 00, <http://databan.worldban.org>. 8