LABOR MARKET 127 general not relevant for the probability of x t+i = x g from the viewpoint of t +1.From the viewpoint of period t, the probabilities of the same event can be written as P t,t+i =(P t+1,t+i |x t+1 = x b ) · P(x t+1 = x b |I t ) +(P t+1,t+i |x t+1 = x g ) · P(x t+1 = x g |I t ) (3.A7) (where P(x t+1 = x g |I t )=1− p if x t = x g ,andsoforth).Thisallowsustoverifythe validity of the law of iterative ex pectations in this context. For i ≥ 2, we write E t+1 [x t+i ]=x b +(x g − x b )P t+1,t+i . (3.A8) At date t + 1, the probability on the right-hand side of (3.A8) is given, while at time t it is not possible to evaluate this probability with certainty: it could be (P t+1,t+i |x t+1 = x b ), or (P t+1,t+i |x t+1 = x g ), depending on the realization of x t+1 . Given the uncertainty associated with this realization, from the point of view of time t the conditional expectation E t+1 [x t+1+i ]isitselfarandom variable, and we can therefore calculate its expected value: E t [E t+1 [x t+i ]] = P(x t+1 = x b |I t )E t+1 [x t+i |x t+1 = x b ] +P(x t+1 = x g |I t )E t+1 [x t+i |x t+1 = x g ]. Inserting (3.A8), using (3.A7), and recalling (3.A6), it follows that E t [x t+i ]=x b +(x g − x b )P t,t+1 = E t [E t+1 [x t+i ]]. EXERCISES Exercise 30 Consider the production function F (k, l; ·)=(k + l)· − ‚ 2 l 2 − „ 2 k 2 . (a) Suppose a firm with that production function has given capital k =1, can hire l costlessly, pays g iven wage w =1,andmustpayF =1for each unit of l fired. If · t takes the values 4 or 2 with equal probability p =0.5, and future cash flows are discounted at rate r =1, what is the optimal dynamic employment policy? (b) Suppose capital depreciates at rate ‰ =1and can be costlessly adjusted to ensure that its marg inal product is equal to the cost of funds r + ‰. Does capital adjust- ment change the optimal employment pattern? What are the optimal levels of capital whe n · t =4and when · t =2? Exercise 31 Consider a labor market in which firms have a linear demand curve for labor subject to parallel oscillations, Ï(N, Z)=Z − ‚N. As in the main text, Z can take two values, Z b and Z g > Z b , and oscillates between these values with transition probability p. Also, the wage oscillates between two values, w b and w g >w b , and the oscillations of the wage are synchronized with those of Z. 128 LABOR MARKET (a) Calculate the levels of employment N b and N g that maximize the expected dis- counted value of the revenues of the firm if the discount rate is equal to r and if the unit hiring and firing costs are given by H and F respectively. (b) Compute the mobility cost k at which the optimal mobility decisions are consistent with a wage differential w = w g − w b when workers discount their future expected income at rate r. (c) Assume that the labor market is populated by 1,000 workers and 100 firms of which exactly half are in a good state in each period. What levels of the wage w b are compatible with full employment (with w g = w b + w as above), under the hypothesis that labor mobility is instantaneous? Exercise 32 Suppose that the marginal productivity of labor is given by Ï(Z, N)=Z − ‚N, and that the indicator Z t can assume three rather than two values {Z b , Z M , Z g }, with Z b < Z M < Z g , where the realizations of Z t are independent, while the wage rate is constant and equal to ¯ w in each period. Finally, hiring and firing costs are give n by H and F respectively. What form does the recursive relationship Î(Z t , N t )=Ï(Z t , N t ) − ¯ w + E t [Î(Z t+1 , N t+1 )] take if the parameters are such that only fluctuations from Z b to Z g or v ice versa induce the firm to adjust its labor force, while the employment level is unaffected for fluctuations from and to the average level of labor demand (from Z b to Z M or vice versa, or from Z M to Z g or vice versa)? Which are the two employment levels chosen by the firm?  FURTHER READING Theoretical implications of employment protection legislation and firing costs are potentially much wider than those illustrated in this chapter. For example, Bertola (1994) discusses the implications of increased rigidity (and less efficiency) in models of growth like the ones that will be discussed in the next chapter, using a two- state Markov process similar to the one introduced in this chapter but specified in a continuous-time setting where state transitions are described as Poisson events of the type to be introduced in Chapter 5. Economic theory can also explain why employment protection legislation is imposed despite its apparently detrimental effects. Using models similar to those discussed here, Saint-Paul (2000) considers how politico-economic interactions can rationalize labor market regulation and resistance to reforms, and Bertola (2004) shows that, if workers are risk-averse, then firing costs may have beneficial effects: redundancy payments not only can remedy a lack of insurance but also can foster efficiency if they allow forward-looking mobility decisions to be taken on a more appropriate basis. Of course, job security provisions are only one of the many institutional features that help explain why European labor markets generate lower employment than Amer- ican ones. Union behavior and taxation play important roles in determining high- wage, low-employment outcomes. And macroeconomic shocks interact in interesting LABOR MARKET 129 ways with wage and employment rigidities in determining the dynamics of employ- ment and unemployment across the Atlantic and within Europe. For economic and empirical analyses of the European unemployment problem from an international comparative perspective, see Bean (1994), Alogoskoufis et al. (1995), Nickell (1997), Nickell and Layard (1999), Blanchard and Wolfers (2000), and Bertola, Blau, and Kahn (2002), which all include extensive references.  REFERENCES Alogoskoufis, G., C. Bean, G. Bertola, D. Cohen, J. Dolado, G. Saint-Paul (1995) Unemployment: Choices for Europe, London: CEPR. Bean, C. (1994) “European Unemployment: A Survey,” Journal of Economic Literature, 32, 573–619. Bentolila, S., and G. Bertola (1990) “Firing Costs and Labor Demand: How Bad is Eurosclerosis?” Revie w of Economic Studies, 57, 381–402. Bertola, G. (1990) “Job Security, Employment and Wages,” European Economic Review, 34, 851–886. (1992) “Labor Turnover Costs and Average Labor Demand,” Journal of Labor Economics, 10, 389–411. (1994) “Flexibility, Investment, and Growth,” Journal of Monetary Economics, 34, 215–238. (1999) “Microeconomic Perspectives on Aggregate Labor Markets,” in O. Ashenfelter and D. Card (eds.), Handbook of Labor Economics , vol. 3B, 2985–3028, Amsterdam: North- Holland. Bertola, G. (2003) “A Pure Theory of Job Security and Labor Income Risk,” Revie w of Economic Studies, 71(1): 43–61. F. D. Blau, and L. M. Kahn (2002) “Comparative Analysis of Labor Market Outcomes: Lessons for the US from International Long-Run Evidence,” in A. Krueger and R. Solow (eds.), The Roaring Nineties: Can Full Employment Be Sustained? New York: Russell Sage, pp. 159–218. and A. Ichino (1995) “Wage Inequality and Unemployment: US vs Europe,” in B. Bernanke andJ.Rotemberg(eds.),NBER Macroeconomics Annual 1995, 13–54, Cambridge, Mass.: MIT Press. and R. Rogerson (1997) “Institutions and Labor Reallocation,” European Economic Review, 41, 1147–1171. Blanchard, O. J., and J. Wolfers (2000) “The Role of Shocks and Institutions in the Rise of European Unemployment: The Aggregate Evidence,” Economic Journal, 110: C1–C33. Nickell, S. (1997) “Unemployment and Labor Market Rigidities: Europe versus North America,” Journal of Economic Perspectives, 11(3): 55–74. and R. Layard (1999) “Labor Market Institutions and Economic Performance,” in O. Ashenfelter and D. Card (eds.), Handbook of Labor Economics, vol. 3C, 3029–3084, Amster- dam: North-Holland. Saint-Paul, G. (2000) The Political Economy of Labour Market Institutions,Oxford:Oxford University Press. 4 Growth in Dynamic General Equilibrium The previous chapters analyzed the optimal dynamic behavior of single consumers, firms, and workers. The interactions between the decisions of these agents were studied using a simple partial equilibrium model (for the labor market). In this chapter, we consider general equilibrium in a dynamic environment. Specifically, we discuss how savings and investment decisions by individual agents, mediated by more or less perfect markets as well as by institutions and collective policies, determine the aggregate growth rate of an economy from a long-run perspective. As in the previous chapters, we cannot review all aspects of a very extensive theoretical and empirical literature. Rather, we aim at familiarizing readers with technical approaches and economic insights about the interplay of technology, preferences, market structure, and insti- tutional features in determining dynamic equilibrium outcomes. We review the relevant aspects in the context of long-run growth models, and a brief concluding section discusses how the mechanisms we focus on are relevant in the context of recent theoretical and empirical contributions in the field of economic growth. Section 4.1 introduces the basic structure of the model, and Section 4.2 applies the techniques of dynamic optimization to this base model. The next two sections discuss how decentralized decisions may result in an optimal growth path, and how one may assess the relevance of exogenous technological progress in this case. Finally, in Section 4.5 we consider recent models of endogenous growth. In these models the growth rate is determined endoge- nously and need not coincide with the optimal growth rate. The problem at hand is more interesting, but also more complex, than those we have considered so far. To facilitate analysis we will therefore emphasize the economic intuition that underlies the formal mathematical expressions, and aim to keep the structure of the model as simple as possible. In what follows we consider a closed economy. The national accounting relationship Y (t)=C (t)+I (t) (4.1) between the flows of production (Y ), consumption (C), and investment therefore holds at the aggregate level. Furthermore, for simplicity, we do not distinguish between flows that originate in the private and the public sectors. EQUILIBRIUM GROWTH 131 The distinction between consumption and investment is based on the concept of capital. Broadly speaking, this concept encompasses all durable factors of production that can be reproduced. The supply of capital grows in proportion with investments. At the same time, however, existing capital stock is subject to depreciation, which tends to lower the supply of capital. As in Chapter 2, we formalize the problem in continuous time. We can therefore define the stock of capital, K (t)attimet, without having to specify whether it is measured at the beginning or the end of a period. In addition, we assume that capital depreciates at a constant rate ‰. The evolution of the supply of capital is therefore given by lim t→0 K (t + t) − K (t) t ≡ dK(t) dt ≡ ˙ K (t)=I (t) − ‰K (t). The demand for capital stems from its role as an input in the productive process, which we represent by an aggregate production function, Y (t)=F (K (t), ). This expression relates the flow of aggregate output between t and t + t to the stocks of production factors that are available during this period. In principle, these stocks can be measured for any infinitesimally small time period t. However, a formal representation of the aggregate production process in a single equation is normally not feasible. In reality, the capital stock consists of many different durable goods, both public and private. At the end of this chapter we will briefly discuss some simple models that make this disaggregate structure explicit, but for the moment we shall assume that investment and consumption can be expressed in terms of a single good as in (4.1). Furthermore, for simplicity we assume that “capital” is combined with only one non-accumulated factor of production, denoted L (t). In what follows, we will characterize the long-run behavior of the economy. More precisely, we will consider the time period in which per capita income grows at a non-decreasing rate and in which the ratio between aggregate capital K and the flow of output Y tends to stabilize. The amount of capital per worker therefore tends to increase steadily. The case in which the growth rate of output and capital exceeds the growth rate of the population represents an extremely important phenomenon: the steady increase in living standards. Butinthischapterourinterestinthistypeofgrowthpatternstemsmorefrom its simplicity than from reality. Even though simple models cannot capture all features of world history, analyzing the economic mechanisms of a growing economy may help us understand the role of capital accumulation in the real world and, more generally, characterize the economic structure of growth processes. 132 EQUILIBRIUM GROWTH 4.1. Production, Savings, and Growth The dynamic models that we consider here aim to explain, in the simplest pos- sible way, on the one hand the relationship between investments and growth, and on the other hand the determinants of investments. The production process is defined by Y (t)=F (K (t), L (t)) = F (K (t), A(t)N(t)), (4.2) where N(t) is the number of workers that participate in production in period t and A(t) denotes labor productivity; at time t each of the N(t)workers supplies A(t) units of labor. Clearly, there are various ways to specify the concept of productive efficiency in more detail. The amount of work of an individual may depend on her physical strength, on the time and energy invested in production, on the climate, and on a range of other factors. How- ever, modeling these aspects not only complicates the analysis, but also forces us to consider economic phenomena other than the ones that most interest us. To distinguish the role of capital accumulation (which by definition depends endogenously on savings and investment decisions) from these other factors, it is useful to assume that the latter are exogenous. The starting point of our analysis is the Solow (1956) growth model. This model is familiar from basic macroeconomics textbooks, but the analysis of this section is relatively formal. We assume that L(t) grows at a constant rate g , ˙ L(t)=gL(t), L(t)=L (0)e gt , and for the moment we abstract from any economic determinant for the level or the growth rate of this factor of production. Furthermore, we assume that the production function exhibits constant returns to scale, so that F (ÎK , ÎL )=ÎF (K , L ) for any Î. The validity of this assumption will be discussed below in the light of its economic implications. Formally, the assumption of constant returns to scale implies a direct relationship between the level of output and capital per unit of the non-accumulated factor, y(t) ≡ Y (t)/L (t)andk(t) ≡ K (t)/L(t). Omitting the time index t,wecanwrite y = F (K , L ) L = LF(K /L , 1) L = f (k), EQUILIBRIUM GROWTH 133 which shows that the per capita production depends only on the capital/labor ratio. The accumulation of the stock of capital per worker is given by ˙ k(t)= d dt  K (t) L(t)  = ˙ K (t)L (t) − ˙ L(t)K (t) L(t) 2 = ˙ K (t) L(t) − ˙ L(t) L(t) K (t) L(t) . Since ˙ K (t)=I (t) − ‰K (t)and ˙ L(t)=gL(t), we thus get ˙ k(t)= I (t) L(t) − ( g + ‰)k(t). Assuming that the economy as a whole devotes a constant proportion s of output to the accumulation of capital, C(t)=(1− s )Y (t), I (t)=sY(t), then I(t)/L(t)=sY(t)/L(t)=sy(t)=sf(k(t)), and thus ˙ k(t)=sf(k(t)) − ( g + ‰)k(t). The main advantage of this expression, which is valid only under the simpli- fying assumptions above, is that it refers to a single variable. For any value of k(t), the model predicts whether the capital stock per worker tends to increase or decrease, and using the intermediate steps described above one can fully characterize the ensuing dynamics of the aggregate and per capita income. The amount of capital per worker tends to increase when sf (k(t)) > ( g + ‰)k(t), (4.3) and to decrease when sf(k(t)) < ( g + ‰)k(t). (4.4) Having reduced the dynamics of the entire economy to the dynamics of a single variable, we can illustrate the evolution of the economy in a simple graph as shown in Figure 4.1. Clearly, the function sf(k) plays a crucial role in these relationships. Since f (k)=F (k, 1) and F (·) has constant returns to scale, we have f (Îk)=F (Îk, 1) ≤ F (Îk, Î)=ÎF (k, 1) = Îf (k) for Î > 1, (4.5) where the inequality is valid under the hypothesis that increasing L,the second argument of F (·, ·), cannot decrease production. Note, however, that the inequality is weak, allowing for the possibility that using more L may leave production unchanged for some values of Î and k. If the inequality in (4.5) is strict, then income per capita tends to increase with k, but at a decreasing rate, and f (k) takes the form illustrated in the figure. If a steady state k ss exists, it must satisfy sf(k ss )=(g + ‰)k ss . (4.6) 134 EQUILIBRIUM GROWTH Figure 4.1. Decreasing marginal returns to capital 4.1.1. BALANCED GROWTH The expression on the right in (4.3) defines a straight line with slope ( g + ‰). In Figure 4.2, this straight line meets the function sf(k)atk ss : for k < k ss , ˙ k = sf(k) − ( g + ‰)k > 0, and the stock of capital tends to increase towards k ss ; for k > k ss ,onthecontrary, ˙ k < 0, and in this case k tends to decrease towards its steady state value k ss . Figure 4.2. Steady state of the Solow model EQUILIBRIUM GROWTH 135 The speed of convergence is proportional to the vertical distance between the two functions, and thus decreases in absolute value while k approaches its steady-state value. In the long-run the economy will be very close to the steady state. If k ≈ k ss =0,thenk = K /L is approximately constant; given that d dt K (t) L(t) =  ˙ K (t) K (t) − ˙ L(t) L(t)  K (t) L(t) ≈ 0 ⇒ ˙ K (t) K (t) ≈ ˙ L(t) L(t) , the long-run growth rate of K is close to the growth rate of L. Moreover, since F (K , L ) has constant returns to scale, Y (t) will grow in the same proportion. Hence, in steady state the model follows a “balanced growth” path, in which the ratio between production and capital is constant. For the per capita capital stock and output, we can use the definition that L(t)= A(t)N(t). This yields Y (t) N(t) = Y (t) L(t) L(t) N(t) = f (k t )A(t), K (t) N(t) = k t A(t). In terms of growth rates, therefore, we get the expression (d/dt)[Y(t)/N(t)] Y (t)/N(t) = (d/dt) f (k t ) f (k t ) + ˙ A(t) A(t) . When k t tends to a constant k ss , as in the above figure, then df(k t )/dt = f  (k t ) ˙ k tends to zero; only a positive growth rate ˙ A(t)/A(t) can allow a long- run growth in the levels of per capita income and capital. In other words, the model predicts a long-run growth of per capita income only when L grows over time and whenever this growth is at least partly due to an increase in A rather than an increase in the number of workers N. If we assume that the effective productivity of labor A(t) grows at a positive rate g A , and that g ≡ ˙ L L = ˙ A A + ˙ N N = g A + g N , then the economy tends to settle in a balanced growth path with exogenous growth rate g A : the only endogenous mechanism of the model, the accumula- tion of capital, tends to accompany rather than determine the growth rate of the economy. A once and for all increase in the savings ratio shifts the curve sf(k) upwards, as in Figure 4.3. As a result, the economy will converge to a steady state with a higher capital intensity, but the higher saving rate will have no effect on the long-run growth rate. In particular, the accumulation of capital cannot sustain a constant growth of income (whatever the value of s )ifg =0and f  (k) < 0. For simplicity, consider the case in which L is constant and ‰ =0.Inthatcase, ˙ Y Y = f  (k) ˙ k f (k) = sf  (k), (4.7) 136 EQUILIBRIUM GROWTH Figure 4.3. Effects of an increase in the savings rate and an increase in k clearly reduces the growth rate of per capita income. Asymptotically, the growth rate of the economy is zero if lim k→∞ f  (k)=0, or it reaches a positive limit if for k →∞the limit of f  (k)=∂ F (·)/∂ K is strictly positive. Exercise 33 Retaining the assumption that s is constant, let ‰ > 0.Howdoesthe asymptotic behavior of ˙ Y /Y depend on the value of lim k→∞ f  (k)? 4.1.2. UNLIMITED ACCUMULATION Even if f  (k)isdecreasingink, nothing prevents the expression on the left of (4.3) from remaining above the line ( g + ‰)k for all values of k, implying that no finite steady state exists (k ss →∞). For this to occur the following condition needs to be satisfied: lim k→∞ f  (k) ≡ f  (∞) ≥ g + ‰ s , (4.8) so that the distance between the functions does not diminish any further when k increases from a value that is already close to infinity. Consider, for example, the case in which g = ‰ = 0: in this case the steady- state capital stock k is infinite even if lim k→∞ f  (k) = 0. This does not imply that the growth rate remains high, but only that the growth rate slows down so much that it takes an infinite time period before the economy approaches something like a steady state in which the ratio between capital and output remains constant. In fact, given that the speed of convergence is determined [...]... consider models for which the macroeconomic dynamics are well-defined (but not necessarily optimal from the aggregate point of view) in the absence of perfectly competitive markets and in the presence of increasing returns to scale 4.5 Endogenous Growth and Market Imperfections To obtain an income share for the non-accumulated factor that is not reduced to zero in the long-run and at the same time allow for. .. > 0 The steady-state growth path describes the optimal dynamics of the economy without any transitional dynamics if f (k) = bk for each 0 ≤ k ≤ ∞ We should note, however, that the constant b is not allowed to be a function of ˜ L if F (ÎK , ÎL ) = ÎF (K , L ) Hence, F (K , L ) = b K = F (K ), and the nonaccumulated factor L cannot be productive for the economy considered, that grows at a constant rate... the firm are therefore given by f f K j (t) L j (t) − K j (t) L j (t) = r (t), K j (t) f (K j (t)/L j (t)) = w(t), L j (t) which are valid for each t and each j Since all firms face the same unit costs of capital and labor, every firm will choose the same capital/labor ratio, K j /L j ≡ k In equilibrium firms therefore can differ only as regards the scale of their operation: if L is the 1 46 EQUILIBRIUM GROWTH... constant returns to scale, even if it does not have the Cobb–Douglas form Unfortunately, the functional form (4.28) implies that ˜ lim f (k) = lim A·k ·−1 = 0 k→∞ k→∞ if · < 1, that is if „ > 0 and labor realistically receives a positive share of national income Given that the labor share is approximately constant (around 150 EQUILIBRIUM GROWTH 60 % in the long-run), the empirical evidence does not seem supportive... ) (4.17) Since the law of motion for capital is given by ˙ k = f (k) − c , (4.18) we can therefore study the dynamics of the system in c , k-space 4.2.2 STEADY STATE AND CONVERGENCE The steady state of the system of equations (4.17) and (4.18) satisfies f (ks s ) = Ò, c s s = f (ks s ), if it exists For the dynamics we make use of a phase diagram as in Chapter 2 On the horizontal axis we measure the... crucial condition for the decentralization of the socially optimal savings and investment decisions Allowing for increasing returns to scale means (in general) that we lose this result It becomes important therefore to confront the optimal growth path of the economy with the growth path that results from decentralized investment decisions In addition, we need to pay attention to the criteria for the distribution... be evident by simply observing the final product, and it is not easy to prevent or punish reproduction of these aspects by third parties Many recent growth models allow for increasing rather than constant returns to scale, and are therefore naturally forced to study markets and productive structures characterized by non-rivalry and non-excludability of certain factors 4.5.2 INVOLUNTARY TECHNOLOGICAL PROGRESS... invest rather than to consume now has a precise economic interpretation For simplicity, we assume that g = 0, so that normalizing by population as in (4.10) is equivalent to normalizing by the labor force Assuming that ‰ = 0 too, the accumulation constraint, ˙ f (k(t)) − c (t) − k(t) = 0, (4.11) implies that higher consumption (for a given k(t)) slows down the accumulation of capital and reduces future... detail Let us assume for now that K is a private factor of production The property rights of EQUILIBRIUM GROWTH 145 this factor are owned by individual agents who in the past saved part of their disposable income The economy is populated by infinitely lived agents, or “households,” which for the moment we assume to be identical The typical household, indexed by i , owns one unit of labor For simplicity,... = 0 t→∞ (4.14) 4.2.1 ECONOMIC INTERPRETATION AND OPTIMAL GROWTH Equations (4.12) and (4.13) are the first-order conditions for the optimal path of growth and accumulation In this section we provide the economic intuition for these conditions, which we shall use to characterize the dynamics of the economy The advantage of using the present-value shadow price Î(t) is that we can draw a phase diagram in . Economy of Labour Market Institutions,Oxford:Oxford University Press. 4 Growth in Dynamic General Equilibrium The previous chapters analyzed the optimal dynamic behavior of single consumers, firms,. point of our analysis is the Solow (19 56) growth model. This model is familiar from basic macroeconomics textbooks, but the analysis of this section is relatively formal. We assume that L(t) grows. also well defined for Î < 0. In this case, the term in parentheses tends to infinity and, since its exponent (1 − Î)/Î is negative, lim k→∞ f  (k)=0 .For = 0 the functional form (4.9) raises