Economic growth and convergence in Germany

KORZERE AUFS•TZE & KOMMENTARE SHORTER PAPERS & COMMENTS

Economic Growth and Convergence in Germany By

Bernhard Herz and Werucr RSger

I. Introduction

T he convergence of poor and rich regions is a key economic issue. Only if poor regions grow faster than richer ones, they can catch up and close existing income gaps. Standard neoclas-

sical growth models predict such an economic convergence: because of diminishing returns to capital, poor economies have high rates of return and therefore grow faster than richer regions. Contrarily, models of endogenous growth typically do not imply convergence of economies with different per capita income.

Empirical research on convergence has reported mixed results, depending on the data and sample of countries and regions considered (see, e.g., Baumol 1986; De Long 1988; Dowrick and Nguyen 1989; Barro and Sala-i-Martin 1990, 1991; Mankiw et al. 1990; Durlaufand Johnson 1991). Authors who examine a wide variety of countries/ regions typically find little or no indications for convergence, while studies focusing on relatively homogeneous samples of countries, e.g., the OECD countries, are more likely to find evidence for convergence. However, these studies report rather slow speeds of adjustment, with halving the difference between actual and steady-state income every 30 or so years (Barro and Sala-i-Martin 1990, 1991; Mankiw r al. 1990).

Remark: The first author thanks the Fritz Thyssen Foundation for financial support. The paper represents the views of the authors and not necessarily those of the EC. The authors thank an anonymous referee for helpful ~mments.

Herz/R6ger: Convergence 133

The empirical work on convergence has been confronted with two main problems. Firstly, steady-state per capita income may vary between regions because of regional differences, e.g., in tax or education policies. Thus, even neoclassical growth models predict only condi- tional convergence (Mankiw et al. 1990), i.e., regions are expected to converge only if differences in the determinants of steady-state income are accounted for. Studies of relatively heterogeneous regions without a reliable control for cross-country differences in steady-state income are thus likely to be biased against convergence. Secondly, the income data might be subject to considerable measurement errors, e.g., the data on US income from the last century or from LDC countries with inadequate statistics. Thus, the results are biased in favor of convergence (De Long 1988).

In this situation, a study of convergence in West Germany might have a number of merits. The country has been partitioned on the basis of labor market relations into 75 economically coherent subre- gions, the so-called "Raumordnungsregionen". Regional policies are also partly based on these areas. For these regions, there exists a relatively good data base so that measurement errors should be com- paratively minor. As the "Raumordnungsregionen" are determined on the basis of regional labor markets, they also provide a better basis for the analysis of convergence than possible alternative classifica- tions, especially the "Landkreise", which represent administrative units. Compared to the samples used in other studies, i.e., LDCs or the US states, the "Raumordnungsregionen" are also relatively homogeneous, thus lessening the problem of differences in steady-state incomes. On the policy side, the empirical analysis of convergence in West Germany is particularly interesting, as it could give some evidence for future growth in eastern Germany and possible convergence between the eastern and western part of the country.

The model which provides the theoretical basis for the empirical analysis deviates from previous studies of convergence by adopting a model of a small open economy. Given that we are analyzing relatively small regions which intensively trade goods with each other and with economic agents not facing any exchange rate risk or legal con- straints on the mobility of factors of production, this seems to be the more appropriate hypothesis. In order to explain that adjustment does not occur instantaneously, we additionally assume that firms face costs of adjustment for their capital stock, an hypothesis not inconsis- tent with time series evidence on investment behavior in West Ger- many (see Funke et al. 1989).

134 Weltwirtschaftl iches Archiv

H. A Model of a Small Open Economy

Regional convergence in Germany is analyzed in a model of a small open economy. Firms in each region are assumed to produce a commodity which is a perfect substitute for commodities from the neighboring regions. Capital is freely mobile so that firms can finance their investment at a given interest rate, i.e., regional saving does not constitute a constraint for investment in that region. Goods markets are perfectly competitive and region-specific growth rates of the population are allowed for. Under these conditions, regional demand does not influence prices and real interest rates (see Matsuyama 1987), and the model can be analyzed by entirely concentrating on the supply side. Deviating slightly from the conventional neoclassical growth model, firms are assumed to face convex costs of adjusting their capital stock. With this assumption small open regions do not converge instantaneously when there is free capital mobility. This framework will also be general enough to model certain barriers to investment more carefully.

Firms in region i try to maximize the present value of their cash flow by appropriately choosing investment and employment strategies. Formally, they solve the following decision problem:

max V o = S [ F ( K i t , A i t N # s # ) - w # N i t - l # ]e -~8"d~d t . (1) {i, N} 0

F() is a linear homogeneous production function in capital Kit and labor Nit, with Air indicating the level of technology. In addition, the skill level of the labor force s# may differ because of regional differences in the educational infrastructure and the demand for educational services, e.g., between metropolitan and rural areas. For sim- plicity, we assume F() to be a Cobb-Douglas function

Yi, = KTt(Ai , N~, s/,) x - ' . (2)

Both technology and the labor force growth with constant rates. Their evolution is governed by the following equations.

Air = Aio e "t (3)

Nit = Nio e n''. (4)

The term Aio reflects different levels of technology in the initial period which may persist over time. The growth rate of the population nl can vary from region to region, w is the real wage rate and I denotes investment expenditures. Output and investment prices are assumed


to be identical. Firms face costs of adjusting their capital stock, therefore investment expenditures consist of two components, the cost of new investment goods J and adjustment costs, which are a positive function of the rate of change of the capital stock

lit = .]it (1 + 0.5q~ K/t)lit , (5)

where ~b is the adjustment cost parameter. The capital stock evolves according to

I ( = lit -- (~ K i t , (6)

with O the rate of depreciation of fixed capital. The Hamiltonian at t is

H u = F ( ' ) - w i t N i t - I i t ( 1 + 0 . 5 d ? ~ ) + q i t ( l u - 6 K i , ) , (7)

where q is the shadow price of capital.l The first-order conditions are

\K,]

- + q i = O

Ftc - w = O

and the transversality condition is

lim qt K t e - rt = O . t "* oO

(8)

(9)

(10)

(11)

Let us define a variable X in efficiency units per capita as x = X~ (A. N), then we can formulate the dynamic system governing the evolution in each region using (6), (8), and (9) as

f,= [q,~l (~+~,,+~)]k~ ~ =(r+8)q~, ~,z.~,-~,~-,~_ 0.5(

- ~ " . t , ~i q i , - 1) 2.

(12) (13)

t For a more detailed derivation and interpretation of q see for example Hayashi (1982).

136 Weltwirtschaftliches Archiv

The steady-state values for q and k are given by the following equations:

q* = 1 + ~b(6+n,+Tt) (14)

k* = s, (:[(r +6)q* - O'-~ (q*-l)2]} 'It'-'). (15)

Since y~' = k*~s I -~ steady-state output is

y* = s, {: [(r +~5)q* - O'-~ (q*- l )2]} "/'-1' (16)

and measured in per capita terms, steady-state output Y/N is defined as A i si Ci with

C,= {! [(r + tS) q* - ~-~ (q*- l )2 ]} "/''-~'. (17)

From (17) the effect of the population growth rate on per capita output can be determined. First note, that q approaches a constant in steady state (see 14) and K* grows with the rate ~ + n (see 15). For the transversality condition to be satisfied, the condition r > (~t + n) must hold. Inserting (14) into (17) and using this condition shows that n exerts a negative effect on per capita income in the steady state. The requirement that the real rate of interest exceeds the rate of technical progress and the growth rate of the population is also a necessary condition for the steady-state growth path of the economy not to be dynamically inefficient. The economic interpretation of a negative effect of n on per capita output is of course different in this model when compared to a model of a closed economy. In a closed economy and for a given rate of savings out of current income, a higher rate of the population growth will necessarily lead to a lower capital intensity in the long run. 2 In the open economy, investment is not restricted by saving within the same region. Since capital is freely mobile, it can be imported to satisfy the investment needs of the particular region. The only costs associated with higher population growth are the adjustment costs for capital, 4-

z The same result will hold when we allow the rate of savings to depend on the rate of interest. In this case, an increase in current savings will be associated with a higher rate of interest which in turn lowers capital intensity in the long run.


The dynamic behavior of the economy can be analyzed by lineariz- ing the model around its steady state, which yields the linear dynamic system [o .8.

= --fkk r * + 6 J L ( q - q * ) J "

Since the determinant of the coefficient matrix is negative (fa ~ < 0), the system has the saddle path property. There exists a unique conver- gent path to the steady state

(k , - k*) = (k o - k*)e x : (19)

or kt = (1 - e~ ' t ) k* + e~'t k o , (20)

where 21 is the negative eigenvalue from the coefficient matrix in (18). We can reformulate this equation in logarithmic form by approximat- ing

kt/k* - 1 ~. lnk t - Ink*. (21)

Thus

and

Inki, = ( 1 - e ~'t) Ink* + eX: lnklo (22)

Inyit - Inyio = (I - e ~'t) In y* - (I - e x't) Inyi0. (23)

The negative parameter 21, which governs the speed of adjustment to the steady state, depends on the underlying parameters of the model:

221 = (r + 6) - : ](r + 6) 2 4fkk (24) ~OCm- 1 I( 71/e . - 1 ) ) 1/2 �9 s 1) +

Note, that the formulation of the adjustment process in a small open economy in (23) is observationally equivalent to the equation derived by Mankiw et al. (1990, eq. 14) for a closed economy. As is the case for the closed economy model, the rate of convergence 21 as expressed in (24) declines with a rise in the returns to scale of reproducible factors of production (a), since fkk declines and converges to zero for a = 1. In this model, however, there is also another determinant of convergence, namely adjustment costs for fixed factors of production. As can be inferred from (24) an increase in these adjustment costs, ~b, unambiguously lowers the speed of convergence. The


adjustment cost model therefore provides an alternative interpretation for low speeds of convergence. Low estimates for 2t should not necessarily be interpreted as evidence in favor of increasing returns, they could also result from high adjustment costs.

In estimating the convergence regression, we assume that the growth rate of technology, ~, the rate of capital depreciation, 6, and the adjustment costs, ~b, are constant across the regions, it reflects primar- ily the advancement of knowledge, which is not region-specific and there are not any strong reasons to expect depreciation rates to differ greatly across regions. Equally the determinants of adjustment costs, e.g., tax policies and regulations, can be expected to be very similar in the different regions.

Contrarily, the level of technology Air does not only encompass technology but also resource endowments and institutional aspects which may differ across regions. We assume that these persistent regional differences can be expressed via the constant of (3) in the following way

lnAio = ao + at xi + el, (25)

where a o is a constant, a t measures the impact of observable components xi , and e~ is a purely regional shock. Similarily, the skill level of the labor force sit may vary between the regions because of differences in the education infrastructure and demand for education, i.e., between metropolitan and rural areas. We therefore express s~ as a function of school attainment z~

s i = b zi + # i , (26)

where b measures the efficiency of education, and #~ is a random shock. A higher skill level of the labor force and the initial resource endowment both increase steady-state per capita income and the convergence parameter 21 (see 24). An increase in the adjustment costs, 4, on the other hand lowers the speed of convergence.

Substituting for y* in (23) gives

lnya -- lnYio = flo + fll xi + f12 lnzi + fla lnYio + ~i (27)

with - - (1 -e l')(a0 = - (1 - e

/ h = (1 = (1

b 2 = (1 - # " ) b

Equation (27) forms the basis for our estimation.

Herz/Rrger: Convergence 139

1H. Data

The data are mainly from the "laufende Raumbeobachtung" of the Bundesforschungsanstalt for Landeskunde und Raumordnung. The data set includes nominal income, population, and schooling since 1970. Data before 1970 is from the Statistische Landes~imter. The income variable is the gross domestic product. This might give rise to measurement problems because of commuters. However, the "Raumordnungsregionen" are divided up on the basis of local labor markets which should lessen potential data problems. As we do not have price indices for the individual regions, we deflate the regional nominal incomes by the national consumer price index of the respec- tive year. As an indicator for school attainment, we use the proportion of persons with "Abitur." To account for regional differences in the initial level of technology, resource endowments, and institutions, Aio, the following dummies are constructed. The sample of the 75 "Raumordnungsregionen" is split into three groups according to their initial per capita real income. It is thus assumed that initial differences in per capita real income can be attributed to regional differences of the technology and resource endowments. Thus, the first dummy X 1 comprises the 25 regions with the lowest initial per capita real income and the second dummy X 2 the group of 25 regions with medium per capita incomes.

IV. Results

Table 1 shows OLS regressions of (27) for the 75 "Raumord- nungsregionen" in West Germany for the period 1957-88. The coefficient of log per capita income, which is highly significant, implies a convergence coefficient of 2 = 0.044. This corresponds to halving the difference between actual and steady-state income every 16 years, which is smaller than the estimates for the US and Europe by Barro and Sala-i-Martin (1990, 1991) and the findings of Mankiw et al. (1990) obtained within the framework of an augmented Solow model for the US. The measures for the skill level of the labor force as well as the two dummies for the initial resource endowment and technology enter significantly into the regression equation. These four vari- ables explain more the 60 percent of the cross-region variation in per capita income.

Like Barro and Sala-i-Martin (1991) for the US and Europe, we also find declining rates of convergence in later periods in Germany. When splitting the initial estimation period into the two subperiods

140 Weltwirtschaftliches Archiv

Table 1 - Estimation of the Convergence Equation for 75 "Raumordnungsregionen"

constant y'57 /70 z I (schooling) X o (technology) X 1 (technology) R 2

s s r Implicit half-life (years)

1957-88 1957-70 1970-88

7.75 (0.82) 6.41 (0.72) 3.15 (1.10) -0.76 (0.09) -0.64 (0.08) -

- - -0.29 (0.11) 1.56 (0.49) 0.95 (0.32) 0.80 (0.41)

-0.10 (0.05) -0.17 (0.04) 0.03 (0.05) -0.09 (0.03) -0.11 (0.03) -0.01 (0.03)

0.61 0.51 0.27 0.86 0.60 0.55

16 9 37

1957-70 and 1970-88, the coefficient of log income is significant in both regressions and is, as expected, distinctly higher in the first sub- period. The estimation results imply an convergence coefficient of 4=0.078 for the period 1957-70 and 4=0.019 for the interval 1970- 88. This corresponds to a half-life of income difference of 9 and 37 years respectively, thus indicating a pronounced deceleration in the speed of convergence within Germany. The proxy for the skill level of the labor force is again significant in both subperiods, while the dummies for the initial resource and technology endowment are only significant in the first interval.

As the figures in Table 2 indicate, the estimated speed of convergence over the period 1957-70 coincides well with a theoretical pre- diction for standard parameter values of the production function and relatively low values for the adjustment cost parameter ) As outlined in Section II, there exist two possible explanations for this phe- nomenon, namely an increase in returns to scale for reproducible factors of production like physical and human capital and/or a significant increase in adjustment costs. While it is theoretically conceivable that the economy moves to a state with increased returns to scale to capital over time - e.g., if the technology is characterized by a "So- below" production function (see Sala-i-Martin 1990) - we are not aware of independent evidence, e.g., from production studies, which could provide empirical support for such a specification.

3 These time-series estimates imply unrealistically high values for 4. As pointed out by Chirinko (1993), this is a notorious problem in Q investment studies which can possibly be explained by excess volatility of financial asset prices. This creates a measurement problem for the Q model insofar as investment decisions are based on fundamentals.

Herz/R6ger: Convergence

Table 2 - Determinants of the Speed of Convergence ~

141

1957-70 1970-88

~ b ~ ,ll ~ b ~ ~

3.5 0.4 --0.080 3.5 0.4 --0.080 10.5 0.4 --0.041

3.5 0.7 --0.038 10.5 0.7 --0.025

�9 The following assumptions on macroeconomic conditions in both periods are made: population growth rate (p.a.) 0.8% [0.1% during the second period], rate of technical progress (p.a.) 3% [1%], rate of depreciation (p.a.) 2.5% [2.5%], real interest rate (p.a.) 3.5% [3.5%]. - b For an average investment to capital ratio of 8% in 1960-70 and 6% in 1970-88 a value of 3.5 for ~ implies that adjustment costs are roughly 14% in the first and 10% in the second period. This share increases linearly in 4).

Table 3 - Q-Investment Equation for Germany (Manufacturing sector) a

Dep. Variable constant qb d c

I/K 0.056 (0.004) 0.018 (0.003) --0.0136 (--0.004)

Statistics R 2 = 0.75, DW = 1.39

�9 Estimation period: 1965-84. - b Marginal tax-adjust~i Q as calculated in Funke et al. (1989). - c Dummy variable with values equal to zero for t<1970 and equal to Q for t_> 1970.

There exists, however, some evidence that adjustment costs have increased over time in Germany. Using the data set on tax-adjusted marginal q for German industry constructed by Funke et al. (1989) and also using their figures on gross investment ratios (I/K) it can clearly be seen that the responsiveness of investment to changes in q has declined significantly over time. This in turn implies that adjustment costs must have increased. According to the regression results in Table 3, ~b is more than three times higher in the period 1970-84 than in 1957-70. Given the lower investment rates in the later period, this would roughly correspond to a doubling of actual adjustment costs. Interestingly, both the cross-section convergence results and the time- series investment regressions imply changes in adjustment costs which are roughly in the same order of magnitude. An increase of ~b by about


200 percent could explain a 50 percent reduction in the speed of convergence (see Table 2). Alternatively, an increase in returns to capital could be an additional contributing factor to explain the decline in convergence. However, as indicated in Table 2 this increase would have been substantial.

V. Conclusions

The empirical analysis gives clear evidence of regional convergence in West Germany: poorer regions tend to grow faster than richer ones. In the period 1957-88, the speed of convergence was around 4 percent per year, implying a halving of the difference between actual and steady-state income every 16 years. While our empirical findings on convergence are of a similar magnitude as found by studies for the US and Europe by Barro and Sala-i-Martin (1991) and Mankiw et al. (1990), they indicate however a somewhat faster speed of adjustment for Germany. Also the pattern of a deceleration of the speed of convergence in recent years is similar to the developments found in these two regions (Barro and Sala-i-Martin 1991).

References

Barro, R. (1991). Economic Growth in a Cross Section of Countries. Quarterly Journal of Economics, Vol. 106, pp. 407-443.

Barro, R., and X. Sala-i-Martin (1990). Economic Growth and Convergence Across the United States. NBER Working Paper, No. 3419. Cambridge, Mass.

- - (1991). Convergence Across States and Regions. Brookings Papers on Economic Activity, Vol. 1, pp. 107-158.

Baumol, W J. (1986). Productivity Growth, Convergence, and Welfare: What the Long- Run Data Show. American Economic Review, Vol. 76, pp. 1072-1085.

Bundcsministerium f'tir Winschaft (1993). Jahreswirtschaftsbericht t993. Bonn.

Chirinko, R.S. (1993). Business Fixed Investment Spending: A Critical Survey of Modeling Strategies, Empirical Results, and Policy Implications. Federal Reserve Bank of Kansas City Working Paper, No. 93-301, Kansas City.

De Long, J.B. (1988). Productivity Growth, Convergence, and Welfare: Comment. American Economic Review, Vol. 78, pp. 1138-1154.

Dowrick, S., and D.-T. Nguyen (1989). OECD Comparative Economic Growth 150-85; Catch-Up and Convergence. American Economic Review, Vol. 79, pp. 1010-1030.

Durlauf, S.N., and P. Johnson (1991). Local versus Global Convergence across Na- tional Economies. Center for Economic Policy Research Working Paper, No. 269, London.


Funke, M., A. R/iU, and D. Willenbockel (1989). Taxation and Capital Formation in West German Industries: A Q-Theory Approach. In: M. Funke (ed.), Factors in Business Investment. Berlin: Springer, pp. 76-103.

Hayashi, E (1982). Tobin's Marginal q and Average q: A Neoclassical Interpretation. Econometrica, Vol. 50, pp. 213-224.

Matsuyama, K. (1987). Current Account Dynamics in a Finite Horizon Model. Journal oflnternational Economics, Vol. 23, pp. 299-313.

Mankiw, N.G., D. Romer, and D.N. Weil (1980). A Contribution to the Empirics of Economic Growth. NBER Working Paper, No. 3541, Cambridge, Mass.

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Economic growth and convergence in Germany