Can a Marginally Distorted Labor Market Improve Capital Accumulation, Output and Welfare?

Transcription

1 Can a Marginally Distorted Labor Market Improve Capital Accumulation, Output and Welfare? Tomas Sjögren Department of Economics Umeå School of Business and Economics Umeå University, SE Umeå, Sweden Abstract This paper sets up an intertemporal two-sector general equilibrium model where capital and labor are complements in production and where the labor market initially functions as a competitive spot market in both sectors of the economy. The purpose is to analyze how the introduction of a marginal distortion on the labor market in one sector of the economy (here represented by the formation of a trade with a bargaining power marginally above zero) affects factor prices, the allocation of perworker capital between the unionized and non-unionized sectors of the economy, aggregate saving, capital accumulation, output and welfare. It is shown that if the output elasticity w.r.t. capital is larger (smaller) in the unionized sector than in the non-unionized sector of the economy, then the formation of a weak trade union leads to a reallocation of capital and labor in such proportions that the per-worker capital stock increases (decreases) in the non-unionized part of the economy. This leads to a higher (lower) non-union wage, a reduced (increased) interest rate and an increase (decrease) in aggregate saving. The savings effect implies that the steady-state capital stock and the steady-state aggregate output both increase (decrease). It is also shown that if the output elasticity w.r.t. capital is larger (smaller) in the unionized sector than in the non-unionized sector of the economy, and if the steady-state capital stock initially exceeds the Golden Rule capital stock, then the formation of a weak trade union has a negative (positive) effect on the steady-state welfare. Keywords: Capital accumulation, trade unions, wage determination, employment. JEL Classifications: E 22, E 24, J 51. 1

2 1. Introduction Many studies emphasize the importance of trade unions for macroeconomic performance. A key question in this literature has been how the trade unions bargaining power affects capital accumulation. Although earlier theoretical studies have found mixed results, these studies nevertheless have one thing in common; the link between trade union bargaining power and capital accumulation is almost exclusively based on the assumption that trade unions dominate the entire labor market. This assumption is problematic because it disregards the empirical observation that trade unions usually only influence parts of a country s labor market. 1 By not taking into account the interaction between unionized and non-unionized segments of the labor market, a complete picture of the link between union bargaining power and capital accumulation is lacking. A key purpose of this paper is therefore to study how an increase in the bargaining power of trade unions affects the accumulation of capital in an economy which features a dual labor market where one sector is dominated by trade unions while the other sector functions as a competitive spot market. This paper also aims to study how an increase in union bargaining power affects other key macroeconomic variables such as savings, factor incomes, the non-union wage, the interest rates and the allocation of capital between the unionized and the non-unionized sectors of the economy. A brief welfare analysis is also conducted. In a partial equilibrium framework where the focus is on how the demand for capital is affected by trade unionism, several studies have demonstrated a negative relationship between trade unions and the capital stock. In an efficient bargaining model 2 with non-binding wage contracts 3, Grout (1984) shows that trade unions will have a negative effect on investment and capital accumulation, 1 For example, in 1990 the trade union density was 39% in Great Britain but only 16% in the United States (Booth 1995:4). 2 Under efficient bargaining, the firms and the unions bargain over both the wage and employment. 3 In a non-binding wage contract, the firm commits to a stock of capital before the wage negotiation takes begins. 2

3 and that an increase in union bargaining power will reduce the capital stock. A similar result is demonstrated by van der Ploeg (1987) within a right-to-manage 4 framework. 5 Devereux and Lockwood (1991) instead argue that by leaving out an analysis of what the effects are for the supply of capital, a potentially crucial aspect of trade unionism is omitted. Devereux and Lockwood analyze the effects of efficient bargaining in an overlapping generations (OLG) model with one production sector. At each point in time there are two generations alive; the young savers and the old dis-savers. Devereux and Lockwood show that if an increase in trade union power increases the young generation s share of total output at the expense of the old generation, then the aggregate saving may increase for a given interest rate. If this effect dominates the (plausible) inward shift in the demand for capital, an increase in trade union power may have a positive effect on the steady state capital stock. 6 A similar conclusion is reached by Koskela and Puhakka (2006) who use an OLG model to analyze the stability and dynamic properties in an economy where the labor market is characterized by right-to-manage wage bargaining. 7 4 In a right-to-manage bargain, the union and the firm bargain over the wage while the firm in the second stage chooses how many persons to employ. 5 The basic mechanism behind these results is that if unions manage to push up wages, then this tends to raise costs which reduces output and capital accumulation. See also Anderson and Devereux (1988) and Hoel (1990). Another mechanism is that if a firm invests in new capital, unionized workers may capture some of the returns on the project in the form of higher wages, thereby reducing investment incentives. 6 See also Aronsson et al (2001, 2002) and Lloyd-Braga and Modesto (2007). Aronsson et al analyze union wage setting in a Ramsey model with continuous time where labor and capital are complements in production. Their results imply that the steady state relationship between the wage rate and the capital stock is negative. The explanation is that an increase in trade union power pushes up the wage and reduces employment which, for a given stock of capital, reduces the marginal product of capital. Since the marginal product of capital in steady state equals the fixed rate of time preferences, the steady state capital stock is reduced. Lloyd-Braga and Modesto consider labor unions in a finance constrained economy with heterogenous agents and productive labor externalities. They show that an increase in union bargaining power has a positive effect on capital accumulation. 7 A related study is Palma and Seegmuller (2004) who analyze how the labor market structure and wage differentials affect endogenous fluctuations in an intertemporal OLG model with a dual labor market. In one sector of the economy, the wage is determined in a bargain between firms and unions while the wage is competitive, and equal to the workers reservation wage, in the other sector. They show that union bargaining power influences local indeterminacy (i.e. whether the steady-state is locally saddle-point stable or not) and they also show that this link is related to technological parameters in the production function. 3

4 This paper can also be related to the literature on endogenous growth and trade unions. In the majority of the theoretical models that use the basic one-sector OLG framework, there is either a negative 8 or no link 9 between unemployment and growth. An exception is Irmen and Wigger (2002) who show that if workers have a larger propensity to save than the capital owners, then the presence of trade unions may have a positive effect on growth if the trade unions are successful in shifting income away from the owners of capital to the workers. In models with a different set-up than the basic Cobb-Douglas one-sector OLG setting, some studies have pointed out that trade unions need not be harmful for growth. In a multi-sector economy where labor mobility is less than perfect, Agell and Lommerud (1993) show that a trade union which pursues an egalitarian wage policy can increase productivity growth by implementing a reduction in the wage differentials between low- and high-productivity sectors. De la Croix and Licandro (1995) analyze the effects of irreversibility and uncertainty in an OLG framework of a unionized economy. They find that an increase in union bargaining power has an ambiguous effect on capital. Palokangas (2004) studies how union power influences economic growth in a two-sector model where the firms in the hightech sector accumulate a stock of knowledge and bargain with unions over the wage, while there are no trade unions in the second sector where the wage is fixed. A key finding in that study is that an increase in union power boosts R&D and growth because firms in the high-tech sector try to escape wage increases through productivity improvements. 10 In this paper, we set up an OLG model with two-period lived agents where each generation of agents is made up of one firm-owner and a fixed number of workers. The labor market is made up of two sectors; one where the wage is determined in a bargain between the firm and a trade union, 8 See e.g. Daveri and Tabellini (2000). 9 See, for example, Corneo and Marquardt (2000). 10 Other studies which are concerned with trade unions and growth are Palokangas (1996), Dos Santos Ferreira and Lloyd-Braga (2002), Chang et al (2007) and Gori and Fanti (2009). 4

5 and another sector where the labor market functions as a competitive spot market. Workers who do not become employed in the unionized sector therefore find an alternative employment in the non-unionized sector. The firms in both sectors of the economy use labor and capital as inputs, and the aggregate capital stock available in a given time period is pre-determined by the savings made earlier. The approach used in this paper has two novelties compared with the earlier literature. First, previous studies assume that workers who do not become employed in the unionized part of the labor market either receive an unemployment benefit or are employed in an alternative sector where they are paid their reservation wage. In this paper, we instead assume that the market-clearing wage in the alternative sector exceeds the workers fixed reservation wage. This assumption has implications for how an increase in union bargaining power affects the workers alternative income. The reason is that an increase in the trade union s bargaining power (which leads to a higher wage in the unionized sector) will reduce employment in the unionized sector and increase the number of persons employed in the non-unionized sector. As long as labor and capital are complements in production in the two sectors, this reallocation of labor implies that also capital will be reallocated from the unionized to the non-unionized sector. Since more capital is now employed in the nonunionized sector, the capital stock per worker may increase in that sector. If this happens, an increase in trade union power has a positive effect also on the non-union wage. This possibility has not been addressed in the earlier literature and in this paper, we work out under what conditions this happens and what the implications will be for aggregate saving, capital accumulation, longrun output and welfare. The second novelty is to study the link between trade union bargaining power and capital accumulation on a marginally distorted labor market. More specifically, we analyze how the formation of a weak trade union (or equivalently, how the introduction of a marginal distortion) in 5

6 one sector of the labor market affects the whole economy. The following thought experiment is conducted. Assume that the economy is initially in a competitive equilibrium where there is no trade union in the economy (or equivalently, that the bargaining power of the trade union is initially zero). Assume now that the workers in one sector of the economy form a weak trade union, where the latter is defined as having a bargaining power which is positive but (infinitesimally) close to zero. If we let ρ [0,1] denote the bargaining power of the trade union, then the macroeconomic consequences of moving from the initial (undistorted) competitive equilibrium (where ρ = 0) to an equilibrium where one sector of the labor market is marginally distorted, can be analyzed by differentiating the equations which define the macroeconomic equilibrium w.r.t. ρ and then evaluating the resulting equations at the point where ρ = 0. This approach makes it possible to sign the effects on key macroeconomic variables by using envelope properties associated with the competitive equilibrium. The main results can be summarized as follows. If the output elasticity w.r.t. capital is larger (smaller) in the unionized sector than in the non-unionized sector, then a marginal increase in union bargaining power along the lines described above has a positive (negative) effect on the short-run 11 non-union wage and a negative (positive) effect on the short-run interest rate. This, in turn, has a positive (negative) effect on aggregate saving, the steady-state capital stock and on steady-state output. Another result is that if the output elasticity w.r.t. capital is much smaller in the unionized sector than in the non-unionized sector, then an increase in union bargaining power may cause a reallocation of labor and capital between the two sectors in such proportions that the per worker capital stock in the unionized sector is reduced. This latter outcome may have a negative effect on 11 At a given point in time, the short-run (temporary) equilibrium is defined conditional on the stock of capital accumulated up until that point in time whereas the long-run equilibrium prevails when capital accumulation has reached a steady-state. 6

7 the union wage and if this happens, the incentive to form a trade union is eliminated. It is also shown that if the output elasticity w.r.t. capital is larger (smaller) in the unionized sector than in the non-unionized sector of the economy, and if the steady-state capital stock initially exceeds the Golden Rule capital stock, then the formation of a weak trade union has a negative (positive) effect on the steady-state welfare. The outline of the paper is as follows. The basic model is presented in section 2. In Section 3 we analyze the short-run macroeconomic implications of introducing a marginal distortion in one sector of the economy. In Section 4, we characterize the long-run macroeconomic consequences (i.e. steady-state effects) and in Section 5 we look at how the steady-state level of welfare is affected. The paper is concluded in Section The Basic Model Consider a closed economy populated by two-period lived agents who are separated into workers and firm-owners, respectively. At any time t, there are two generations of workers and firm-owners alive; the young born in period t and the old born in period t 1. Population growth is zero and each cohort of workers is made up of N individuals while each cohort of firm-owners is normalized to one. Production in the economy takes place in two sectors. In sector 1 (called the unionized sector), the labor market is influenced by a trade union while in sector 2 (called the non-unionized sector), the labor market functions competitively. Following the convention in the literature on unionized labor markets, it is assumed that the production function in the unionized sector is characterized by decreasing returns to scale. This assumption is needed to obtain a well-defined demand function for labor. Decreasing returns to scale reflects the use of a fixed factor in the production and this fixed factor is provided by the firm-owner in return for the profit in the unionized sector. Since the fixed factor is assumed to be in limited supply (it is only held by the 7

8 firm-owner), it is not possible for other firms to enter the unionized sector and drive down the profit to zero. As for the non-unionized sector, we follow the convention in the literature by assuming that the production in the competitive sector is characterized by constant returns to scale. This assumption implies that the profit in sector 2 is zero. In the first period of life, a worker born at time t supplies one unit of labor inelastically (either in the unionized or in the non-unionized sector) in return for the wage w it in sector i = 1, 2. In the second period of life, the supply of labor is zero. As for the firm-owners, they can be viewed as a dynastic family which passes the ownership of the firm on between generations. A firm-owner born at time t manages the firm in the unionized sector when young in return for the profit income in that period, π t, but when old at time t + 1, the firm-owner passes the management of the firm on to the next generation. All agents have identical preferences and derive utility from the consumption of the goods produced in sectors 1 and 2. The two goods are assumed to be perfect substitutes in consumption which means that in equilibrium the prices of the two goods will be equalized and this price is normalized to one. By defining c t = x + x to be the sum in period t of an individual agent s consumption of good 1 (x ) and good 2 (x ), and by following Devereux and Lockwood (1991) in assuming that each generation has Cobb-Douglas preferences over consumption in the two time periods, the life-time utility function of an agent born in period t can be written as u(c t, c t+1 ) = c η 1 η t c t+1 (1) where c t, c t+1 are consumption levels when young and old, respectively, and where η (0,1) is a constant. An agent s first and second period budget constraints can be written as c t = m t s t and c t+1 = (1 + r t+1 )s t, respectively, where m t is first period income (hence m t = π t for the firmowner while m t = w it for a worker employed in sector i), s t is the amount saved when young and 8

9 r t+1 is the interest rate. The solution to an agent's intertemporal maximization problem defines the following saving and life-time indirect utility functions for a worker employed in sector i = 1, 2, and for the firm-owner, respectively s it = (1 η)w it, v it = v(w it, r t+1 ) = θw it (1 + r t+1 ) 1 η (2) s ft = (1 η)π t, v ft = v(π t, r t+1 ) = θπ t (1 + r t+1 ) 1 η (3) where θ = η η (1 η) 1 η and where the sub-index "f" denotes the firm-owner. Let us now turn to the production side of the economy. It is assumed that each firm is a pricetaker on the goods market and for notational convenience, the number of firms in each sector is normalized to one. The production functions are of Cobb-Douglas type; Y = F(N, K ) = N α K β and Y = G(N, K ) = N 1 γ K γ, where Y it is the output produced, N it is the number of workers employed and K it is the amount of physical capital used in sector i in period t. The parameters α and 1 γ are output elasticities w.r.t. labor while β and γ are output elasticities w.r.t. capital. Since the production technology in sector 1 is characterized by decreasing returns to scale, it follows that α + β < 1. The output elasticities w.r.t. to capital will play an important role in the results to be derived below and the sector with the largest output elasticity w.r.t. capital will be referred to as the more capital efficient sector. Using this terminology, the unionized (nonunionized) sector is more capital efficient if > γ (β < γ). The two sectors are said to be equally capital efficient if β = γ. Finally, we assume (without loss of generality) that there is no depreciation of the capital stocks in the two sectors The timing of Events The analysis in this paper is based on the assumption that the factor markets in both sectors of the economy initially (i.e. up until a specific point in time at which a trade union is formed in sector 9

10 1) function competitively. Since the competitive equilibrium can be viewed as the special case of the unionized economy where the trade union has zero bargaining power, we will here characterize the equilibrium in the unionized economy. Once a trade union has been formed in sector 1, the union wage is (in each time period t) determined in a bargain between the trade union 12 and the firm in sector 1. A key question in this context regards the degree of commitment allowed for in the wage contract in the unionized sector. In line with earlier studies, we focus on a non-binding wage contract which means that the firm commits to a level of capital before the wage is negotiated. 13 In the non-unionized sector 2, the wage is determined on a competitive spot market where the firm and the workers act myopically and treat w as exogenously given. The timing of events within each time period t can now be specified as follows. First, the agents observe that the wage in the unionized sector will exceed the wage in the non-unionized sector as long the trade union has any bargaining power. As this is common knowledge among the N workers in the economy, all N workers want to become employed in the unionized sector. They therefore join the trade union in the beginning of period t. Simultaneously, the firm in the unionized sector rents its capital, K, on the capital market. After the firm in sector 1 has installed its capital, the firm and the union bargain over w. We assume a decentralized bargain which means that the bargaining parties treat all macroeconomic variables (such as w, r t and r t+1 ) as exogenous. After w has been determined, the firm unilaterally determines how many workers, N, to employ. Who among the N union members that will become employed in the unionized sector is determined by a random draw and the workers who are not employed in sector 1 move to the non-unionized sector where they all will be employed, i.e. the number of employed persons in sector 2 is given by N = 12 Since the number of firms in the unionized sector is normalized to one, so is also the number of trade unions. 13 It turns out that whether the wage contract is binding or not does not affect the qualitative results derived below. 10

11 N N. Simultaneously with hiring labor, the firm in the non-unionized sector rents its capital, K. The total amount of capital available in the economy in period t, K t, is pre-determined by the saving made in the previous time period. Therefore, K t is treated as exogenous at time t and the capital market is in equilibrium when K t = K + K Determination of Factor Inputs and Factor Prices In the unionized sector, the firm s profit is given by π t = F(N, K ) w N r t K where the firm treats the rental price of capital, r t, as exogenous. To obtain time consistent solutions for N, K and w, we solve this part of the model using backward induction. Stage 3: Determination of N. Conditional on w, K and r t, the firm in the unionized sector chooses how many persons to employ. Maximizing π t w.r.t. N produces the first-order condition F N = w, where F N = F/ N, and this first-order condition can be used to derive the conditional labor demand function, N c 1 (w, K ), and the conditional profit function, π c (w, r t, K ), as N c = N c 1 (w, K ) = α 1/(1 α) 1/(1 α) β/(1 α) w K (4) π c t = π c (w, r t, K ) = (1 α)α α/(1 α) α/(1 α) β/(1 α) w K rt K (5) where the super-index c denotes conditional. Stage 2: Determination of w. Conditional on K, the wage is determined in a decentralized bargain between the union and the firm. The assumption of a decentralized bargaining structure implies that both the firm and the trade union act myopically and treat macroeconomic variables (such as w, r t and r t+1 ) as exogenous. In the bargain, the firm aims to maximize the conditional c profit π t while the trade union aims to maximize the expected utility of the representative union member. Since the union recognizes that the workers who do not become employed in sector 1 can obtain a job in sector 2, the union s objective function, V, can be written as 11

12 V = N c N v(w, r t+1 ) + (1 N c ) v(w N, r t+1 ) (6) where the (life-time) indirect utility function v(w it, r t+1 ) is defined in (2) for i = 1, 2. The wage bargain is determined within the context of the right-to-manage framework. This approach requires that the fall back payoffs are specified in the event that the bargaining parties do not reach an agreement. If no agreement is reached, the firm is closed during period t while the workers seek employment in the non-unionized sector. Since the firm already has installed its capital, the firm needs to pay the rental cost. This means that the firm s fall back profit is given by r t K. The fall back utility for the representative union member is v(w, r t+1 ). This implies that the firm s rent from the bargain is given by π t ( r t K ) while the union s rent is given by (v v )N c /N. The joint maximand is the Nash product, Ω t, of these rents Ω t = [ N 1 c ρ (w,k ) [v(w N, r t+1 ) v(w, r t+1 )]] [π c (w, r t, K ) + r t K ] 1 ρ (7) where the parameter ρ [0,1] measures the bargaining power of the union relative to the firm. Substituting v(w, r t+1 ) = θw (1 + r t+1 ) 1 η and v(w, r t+1 ) = θw (1 + r t+1 ) 1 η, as well as equations (4) and (5), into equation (7) and maximizing the resulting expression w.r.t. w produces w = Φ(ρ)w, where Φ(ρ) = 1 + ρ(1 α)/α. The function w = Φ(ρ)w will be referred to as the wage setting equation and it shows that the bargained wage will be a mark-up over the nonunion wage as long as the union has any bargaining power (ρ > 0). Note in particular that the wage setting equation does not contain K. This means that the firm in stage 1 cannot use K to influence the outcome of the wage bargain. Observe also that when ρ = 0, then the wage setting equation reproduces the outcome in the competitive equilibrium where labor mobility ensures that the wages are equalized between the two sectors, i.e. w = w. 12

13 Stage 1: Determination of K. In the first stage, the firm chooses K to maximize the conditional profit function π t c = F(N 1 c (w, K ), K ) w N 1 c (w, K ) r t K subject to w = Φ(ρ)w. Since both w and Φ(ρ) are treated as exogenous, the firm will treat w as given in Stage 1. By using that the optimal choice of labor in Stage 3 satisfies F N = w, it follows that the first-order 2 condition for K becomes F K = r t and the second-order condition implies F NN F KK F NK > Turning to sector 2, we use the normalizations k = K /N and g(k ) = G(N, K )/N to write the necessary conditions for the firm in sector 2 as w = g(k ) k g k (k ), r t = g k (k ) (8) Since G(N, K ) = N 1 γ K γ, the equations in (8) reduce to w = (1 γ)k γ and r t = γk γ The Formation of a Weak Trade Union: Short-Run Equilibrium Effects In this Section, we address the question of how the formation of a weak trade union in sector 1 affects the key macroeconomic variables in this model economy. A weak trade union is defined as having a bargaining power, ρ, which is positive but (infinitesimally) close to zero and we therefore conduct the following thought experiment. Assume that the labor market in both sector 1 and in sector 2 has functioned as a competitive spot market up until period t 1 and that the economy in period t 1 is in the competitive long-run equilibrium (steady-state). This means that up until 14 To show the latter result, observe that the first-order condition w.r.t. K in Stage 1 is given by π t = F K K r t + (F N w ) N t = 0 K where N t c / K = F NK /F NN and where the first-order condition in Stage 3 implies that F N = w. The secondorder condition is obtained by differentiating this equation w.r.t. K c 2 π t K c N 2 = F KK + 2F t NK + F K NN ( N c t ) 2 = F K KK 2 F 2 2 NK F + F NK F NN NN F 2 = 1 NN (F F NN F KK F 2 NK ) < 0 NN where we have used that N c t / K = F NK /F NN. Since F NN < 0, the last inequality above implies F NN F KK F 2 NK. 13

14 period t 1, the union s bargaining power has been zero (ρ = 0) and the wages have been equalized between the two sectors. Assume that the workers who enter sector 1 in the beginning of period t now form a weak trade union. This will have a direct and positive effect on the wage in sector 1 which, in turn, influences the other macroeconomic variables in this model economy. To study how the transition from a competitive labor market to a partially 15 and marginally 16 distorted labor market affects this model economy, we can differentiate the equations which define the equilibrium w.r.t. ρ and evaluate the resulting expressions at ρ = 0. A distinction will be made between the short-run (temporary) equilibrium and the long-run equilibrium. The former is the equilibrium that holds in each time period t and this equilibrium is conditioned on the amount of aggregate capital, K t, that has been accumulated up until time t. The long-run equilibrium holds when the accumulation of the aggregate stock of capital has reached a steady-state. To evaluate how the formation of a weak trade union in sector 1 affects the economy, let us begin by characterizing the capital market equilibrium (CME) and the labor market equilibrium (LME) in more detail. Beginning with the CME, we note that the returns on capital will be equalized between the two sectors when the capital market is in equilibrium; F K = g k. Substituting the adding-up conditions K t = K + K and N = N + N into F K = g k produces Ψ(N, K ) = F K (N N, K t K ) g k ( K N ) = 0 (9) The function Ψ(N, K ) = 0 implicitly determines all combinations of K and N (and also all combinations of K = K t K and N = N N ) which are associated with equilibrium on the 15 Recall that the trade union is only present in sector Recall that the trade union s bargaining power is (infinitesimally) close to zero. 14

15 capital market. Using equation (9) to solve for K as a function of N and K t produces a relationship which will be referred to as the CME-locus; K CME (N, K t ). Analogously, we can combine the wage setting equation w = Φ(ρ)w with the equations w = F N, w = g k g k, K t = K + K and N = N + N to obtain Δ(N, K, ρ) = F N (N N, K t K ) (1 + ρ 1 α ) [g α (K ) K g N N k ( K )] = 0 (10) N The function Δ(N, K, ρ) = 0 determines all combinations of K and N (and also all combinations of K = K t K and N = N N ) which are associated with equilibrium on the labor market. Using equation (10) to solve for K as a function of N, K t and ρ produces a relationship which will be referred to as the LME-locus; K LME (N, K t, ρ). The slopes of the CME-locus and the LME-locus in (N, K ) space are given by CME K N = Ψ N2 = k g kk /N F NK > 0 (11) Ψ K 2 [ F KK g kk /N ] LME K N = Δ N2 Δ K 2 = [ F NN Φ(ρ)k 2 g kk /N ] > 0 (12) Φ(ρ)k g kk /N F NK where Ψ N2 < 0, Ψ K2 > 0, Δ N2 > 0 and Δ K2 < 0 denote partial derivatives w.r.t. N and K. The fact that the CME-locus and the LME-locus both have a positive slope is a consequence of that capital and labor are complements in production in both sectors of the economy. The short-run equilibrium is defined by the intersection of the CME-locus with the LME-locus and the super-index "*" will be used to denote an equilibrium value in the short-run equilibrium. We will apply the correspondence principle 17 and only consider equilibria which are stable in the 17 Samuelson (1947). 15

16 sense that if a shock hits the system, then the equilibrium will be restored. It can be shown that the stability condition implies that the LME-locus is steeper than the CME-locus. 18 Equations (9) and (10) together define the equilibrium levels of capital and labor in sector 2, K and N, as functions of the trade union s bargaining power (ρ) and as functions of the size of the accumulated aggregate stock of capital (K t ); K = K 2 (ρ, K t ) and N = N 2 (ρ, K t ). To characterize how an increase in trade union power affects the capital - labor allocation in sector 2, we differentiate equations (9) and (10) w.r.t. K, N and ρ. This produces [ Ψ K 2 Ψ N2 Δ K2 Δ N2 ] H [ K / / ] = [ 0 N ( 1 α α ) w ] (13) where H = Ψ K2 Δ N2 Δ K2 Ψ N2 = Ψ K2 Δ K2 ( K LME N K CME ) > 0 (14) N and where the sign H > 0 is based on the premise that the LME-locus is steeper than the CMElocus. By using Cramer s rule in conjunction with K t = K + K and N = N + N, we obtain K K = (1 α) w Ψ N 2 > 0, α H = (1 α) w Ψ N 2 < 0, α H N N = (1 α) w Ψ K 2 > 0 α H = (1 α) w Ψ K 2 < 0 (15) α H The explanation for the comparative static results in (15) is straightforward. When the workers in sector 1 in the beginning of period t form a weak trade union (i.e. ρ increases marginally from 18 If the opposite would occur and the CME-locus would be steeper than the LME-locus, then the rightward shift of the LME-locus which accompanies an increase in trade union power would have a positive effect on both the number of persons employed, and on the amount of capital hired, in the unionized sector. This is not a logically consistent response to a higher union wage and gives the economic intuition behind the assumption that the LME-locus is steeper than the CME-locus. 16

17 zero), this has a positive effect on the wage in sector 1. This induces the firm in sector 1 to hire fewer workers which means that more workers will be employed in sector 2. Since labor and capital are complements in production in both sectors of the economy, this reallocation (compared with the allocation observed in the previous time period t 1) of labor between the two sectors will be accompanied by a reduction of capital in sector 1 and an increase of capital in sector 2. Since the increase in trade union power reduces the amounts of labor and capital employed in sector 1 but increases the amounts of labor and capital employed in sector 2, the net effects on the capital stocks per worker in the two sectors, k = K /N and k = K /N, are ambiguous. To evaluate the effects on these variables, it is instructive to first look at what happens with the size of the capital stock per worker in the unionized sector compared with that in the non-unionized sector. Therefore, we define the per worker capital ratio as k /k and it is given by 19 k k = Φ(ρ) β(1 γ) αγ (16) where we recall that Φ(ρ) = 1 + ρ(1 α)/α. Since the expression on the RHS of equation (16) is monotonously increasing in ρ, the following result is readily available; Proposition 1: An increase in trade union bargaining power leads to an increase in the per worker capital ratio k /k. This result reflects that if labor becomes more expensive in sector 1 than in sector 2, then the firm in sector 1 will use relatively more capital than labor in the production process. Another 19 To derive equation (16), first use that the following equality holds; w /r t = Φ(ρ)w /r t. Then, substitute w = F N = αn α 1 K β and r t = F K = βn α K β 1 into the left hand side of the equality, and substitute w = (1 γ)k γ and r t = γk γ 1 into the right hand side of the equality. Using the definitions k = K /N and k = K /N, and solving for the ratio k /k, produces equation (16). 17

18 observation from equation (16) is that the more capital efficient the technology in sector 1 is relative to the technology in sector 2 (i.e. the larger β is relative to γ), the larger will be k /k. Let us now take a closer look at how an increase in trade union power affects the capital stock per worker in the non-unionized sector of the economy. The effect on k is of fundamental importance for the remaining part of the analysis because if there exists a positive relationship between ρ and k, then an increase in trade union power also has a positive effect on the non- union wage. Evaluating the effect of ρ on k is equivalent with determining how the move in Figure 1 from point A to point B along the CME-locus affects k = K /N. Note, therefore, that the levels of k which can be observed along the CME-locus are defined by k CME = K CME (N, K t )/N. Substituting N = N 2 (ρ, K t ) into this definition implies that we can write the short-run equilibrium level of the capital stock per worker in the non-unionized sector as k = k 2 (ρ, K t ). Differentiating the resulting equation for k w.r.t. ρ, and using equation (11) to rewrite the resulting expression, produces k N N F KK +g kk = k F KK +F NK (17) Since N / > 0 and N F KK + g kk < 0, equation (17) implies that whether an increase in trade union power has a positive or negative effect on the capital stock per worker in the non-unionized sector depends on the sign of the expression k F KK + F NK. The sign of this expression is ambiguous. This ambiguity reflects that a movement to the right along the CME-locus (which is equivalent with a reduction in K and N ) affects the marginal product of capital in sector 1, F K, in two ways. First, the reduction in K has a positive effect on F K, and since the equality F K (K /N ) = g k (k ) holds at each point along the CME-locus, this effect works in the direction of pushing down k. This is captured by the term k F KK in equation (17). Second, the reduction 18

19 in N which is observed along the move to the right on the CME-locus has a negative effect on F K (K /N ). Since F K (K /N ) = g k (k ) holds at each point along the CME-locus, this effect works in the direction of pushing up k. This is captured by the term F NK in equation (17). Which of these opposing effects that will dominate in equilibrium depends both on the characteristics of the production functions, and on the initial level of ρ. In the Appendix, it is shown that the sign of k F KK + F NK is determined by sign [k F KK + F NK ] = sign [(β γ) + ρβ(1 γ) ( 1 α )] (18) α We can now combine equations (17) and (18) to make the following claim regarding how k is affected by the formation of a weak trade union; Proposition 2: Assume that ρ = 0 holds initially. In this situation, the formation of a weak trade union in sector 1 has the following effect on the capital stock per worker in the non-unionized sector of the economy: (i) If the unionized sector is more (less) capital efficient than the non-unionized sector in the sense that β > γ (β < γ), then k / > 0 ( k / < 0). (ii) If the unionized and the non-unionized sectors are equally capital efficient in the sense that β = γ, then k / = 0. Note that if the trade union initially already has some bargaining power (ρ > 0 holds initially), then equations (17) and (18) together imply that k / is positive also when β = γ. Observe also that since we can rewrite equation (16) to read k = k Φ(ρ)β(1 γ)/(αγ), it follows that k = β(1 γ) [(1 α)k α 2 γ + αφ(ρ) k ] (19) 19

20 Combining the results in Proposition 2 with equation (19), it follows that the per worker capital stock in sector 1 will be affected as follows by the formation of a weak trade union; Proposition 3: Assume that ρ = 0 holds initially. In this situation, the formation of a weak trade union in sector 1 has the following effect on the capital stock per worker in sector 1: (i) If the unionized sector is more capital efficient than the non-unionized sector (β > γ), or if the unionized and the non-unionized sectors are equally capital efficient (β = γ), then k / > 0. (ii) If the unionized sector is less capital efficient than the non-unionized sector (β < γ), then k / > 0 (< 0) if (1 α)k > (<) αφ(ρ) k /. The intuition for the result in part (i) is that firm 1 s cost of labor in this case increases relative to that of capital 20 which induces firm 1 to employ relatively more capital per worker than before. In the scenario outlined in part (ii), both the cost of capital and the cost of labor increase. 21 Then, depending on whether it is the cost of capital or labor that increases the most, the per worker capital stock in sector 1 may either increase or decrease. The results in Proposition 2 have implications for the factor price responses. By substituting k = k 2 (ρ, K t ) into the equations in (8), we can write w and r t as functions of ρ and K t ; w = w 2 (ρ, K t ) and r t = r(ρ, K t ). Furthermore, since the equations in (8) imply that w is an increasing function of k and that r t is a decreasing function of k, we can combine these observations, together with the results in Proposition 2, to establish the following results; 20 Below it will be shown that an increase in trade union bargaining power reduces the interest rate when β > γ, and either reduces or has no effect on the interest rate when β = γ. 21 In Proposition 4 below, it is stated that an increase in trade union bargaining power has a positive effect on the interest rate when β < γ and ρ = 0 hold initially. 20

21 Proposition 4: Assume that ρ = 0 holds initially. In this situation, the formation of a weak trade union in sector 1 has the following effects on the non-union wage and on the interest rate: (i) (ii) (iii) If β > γ, then w If β < γ, then w If β = γ, then w > 0 and r t < 0. < 0 and r t > 0. = r t = 0. Observe in particular that if the unionized sector is more capital efficient than the non-unionized sector (β > γ), then the increase in trade union power has a positive effect on the non-union wage. This result implies that the unionization of one part of the labor market has a positive effect on the non-union wage if the resulting reallocation of capital and labor causes the capital stock per worker (and hence the marginal product of labor) to increase in the non-unionized sector of the economy. This result is worth emphasizing because in models where capital is absent, or in models without capital mobility between the unionized and non-unionized sector, an increase in trade union power cannot have a positive effect on the non-union wage. Note also that the comparative static responses in parts (ii) and (iii) correspond to more traditional results where an increase in trade union power has a negative (or no) effect on the non-union wage. The results in Proposition 4 also have implications for the union wage. To illustrate this, let us substitute w = w 2 (ρ, K t ) into the wage setting equation for the union wage. This gives us the equilibrium union wage as w ρ produces = w 1 (ρ, K t ) = Φ(ρ)w 2 (ρ, K t ). Differentiating this equation w.r.t. w = 1 α w α + (1 + ρ 1 α α ) w (20) 21

22 The first term on the RHS is the direct positive effect on the union wage of increased bargaining power while the second term on the RHS is a feedback effect which arises because the union wage is a mark-up over the non-union wage. From Proposition 3 we know that if the unionized sector is more capital efficient than the non-unionized sector in the sense that β > γ, then w / > 0. In this situation the feedback effect induces the trade union to opt for a wage increase which exceeds the direct effect. If the two sectors instead are equally capital efficient (β = γ), then the feedback effect is zero but if the unionized sector is less capital efficient than the non-unionized sector (β < γ) then the feedback effect is negative. In the latter case, the feedback effect induces the trade union to opt for a wage increase which is smaller in size than the direct effect. The fact that the feedback effect is negative when β < γ begs the question if there can be a situation when the negative feedback effect dominates over the positive direct effect, thereby implying that an increase in trade union power causes w to decrease? In the Appendix we show that when ρ = 0 holds initially, then sign w = sign[(1 α γ)k + (1 α β)k ] (21) Since the assumption of decreasing returns to scale in sector 1 implies 1 α β > 0, it follows from equation (21) that if α + γ > 1 (which can only happen if β < γ) 22, then the RHS of (21) is ambiguous in sign. In the latter situation, one cannot rule out that the negative feedback dominates and w / < 0. If this would be the case we conclude that the workers in sector 1 have no incentive to form a trade union. 22 To see this, observe first that if β > γ, then α < 1 γ which, in turn implies α + γ < 1. In this situation, both terms inside square brackets in (21) are positive which means that w / cannot be negative when β > γ. This implies that a negative w / may only occur if β < γ because this opens up for the possibility that α + γ > 1. 22

23 Let us finally look at how the formation of a weak trade union affects the aggregate saving in the economy. Define FI t to be the aggregate factor income that accrues to the agents who are young in period t. Since the young agents save a fixed proportion 1 η of their first period income (see the saving functions defined in 2 and 3), the aggregate saving is given by S t = (1 η)fi t. Observe also that FI t is equal to the aggregate output minus the factor income paid out to the old generation, i.e. FI t = Y + Y r t K t. By using this definition, the aggregate saving in period t is written S t = S(ρ, K t ) = (1 η)[f(n N, K t K ) + G(N, K ) r t K t ] (22) where N it = N i (ρ, K t ) and K it = N i (ρ, K t ). Differentiating equation (22) w.r.t. ρ and evaluating the resulting expression at ρ = 0 produces S t r t = (1 η)k t (23) As can be seen, the effect on aggregate saving only depends on whether the formation of a weak trade union has a positive or a negative effect on the interest rate. This simple result follows because in the initial competitive equilibrium, the marginal products of labor and capital are both equalized between the two sectors. As a consequence, the first-order effect on aggregate output of a reallocation of capital and labor from sector 1 to sector 2 is zero (i.e. aggregate output is unchanged). This means that the only thing which determines the sign of S t / is how the aggregate income that accrues to the old generation, r t K t, changes. Since K t is predetermined by the savings made in period t 1, the effect on r t K t is solely determined by the effect on the interest rate. If r t / > 0 ( r t / < 0), then a smaller (larger) part of FI t accrues to the young, which means that aggregate saving decreases (increases). Combining equation (23) with the results in Proposition 4 produces; 23

24 Proposition 5: Assume that ρ = 0 holds initially. In this situation, the formation of a weak trade union in sector 1 affects aggregate saving as follows: (i) If the unionized sector is more (less) capital efficient than the non-unionized sector in the sense that β > γ (β < γ), then S t / > 0 ( S t / < 0). (ii) If the unionized and the non-unionized sectors are equally capital efficient in the sense that β = γ, then S t / = 0. These short-run saving effects will influence the accumulation of capital and let us therefore turn our attention to how the formation of a weak trade union affects the economy in the long-run. 4. The Formation of a Weak Trade Union: Steady-State Effects In a closed economy without depreciation, the aggregate investment is given by K t+1 K t. The aggregate saving made by the young is given by the saving function defined in equation (22), S(ρ, K t ), while the aggregate saving by the old (which per definition is the same as the increase in their financial wealth) is K t. Since the total aggregate saving (S t K t ) equals the aggregate investment (K t+1 K t ), it follows that the aggregate stock of capital is determined by K t+1 = S(ρ, K t ). In a steady state, the equality K t+1 = K t holds at each point in time and by using this definition in K t+1 = S(ρ, K t ), we obtain K = S(ρ, K ) where S = S(ρ, K ) = (1 η)[f(n N 2, K K 2) + G(N 2, K 2) r K ], N i = N i (ρ, K ), K i = K i (ρ, K ) and r = r(ρ, K ) are the equilibrium functions defined in Section 3 evaluated at K t = K and where "~" denotes a steady state level. The equation K S(ρ, K ) = 0 implicitly determines the aggregate steady state capital stock as a function of the trade union s bargaining power; K (ρ). We consider a stable steady-state where the assumption of stability implies that the saving function K t+1 = S(ρ, K t ) crosses the line K t+1 = K t from above in (K t, K t+1 ) space, or put differently, that the slope of the saving function 24

25 is less than one in the stable steady-state; 0 < S / K < 1. To determine how K is affected by the formation of a weak trade union in sector 1, we differentiate K S(ρ, K ) = 0 w.r.t. ρ. This produces K = S / 1 S / K (24) Since the denominator of equation (24) is positive in the stable steady-state, it follows that sign K S = sign (25) Combining equation (25) with the properties of S/ that are summarized in Proposition 5 allows us to characterize the steady-state effects on capital accumulation as follows; Proposition 6: Assume that ρ = 0 holds initially. In this situation, the formation of a weak trade union in sector 1 has the following effects on the aggregate capital stock in steady-state: (i) If the unionized sector is more (less) capital efficient than the non-unionized sector in the sense that β > γ (β < γ), then K / > 0 ( K / < 0). (ii) If the unionized and the non-unionized sectors are equally capital efficient in the sense that β = γ, then K / = 0. The explanation for these results is straightforward. From the analysis in Section 3 we know that an increase in trade union bargaining power has a positive (negative) effect on aggregate saving when β > γ (β < γ) whereas the aggregate saving is unchanged when β = γ. Since it is the aggregate saving which determines the size of the aggregate stock of capital, the results in Proposition 6 follow immediately. 25

26 Let us now look at the steady-state effects on the factor prices and on the capital stock per worker in the non-unionized sector. Substituting the function K (ρ) into the equilibrium functions w = w 1 (ρ, K t ), w = w 2 (ρ, K t ), r t = r(ρ, K t ) and k = k 2 (ρ, K t ) allows us to define the corresponding steady-state levels as w 1(ρ) = w 1 [ρ, K (ρ)], w 2(ρ) = w 2 [ρ, K (ρ)], r (ρ) = r[ρ, K (ρ)] and k 2(ρ) = k 2 [ρ, K (ρ)]. Differentiating these functions w.r.t. ρ gives us the steadystate effects as dw 1 dρ = w 1 [ρ,k (ρ)] = w 1 + w 1 [ρ,k (ρ)] K K, = w 1 K dw 2 dρ = w 2 [ρ,k (ρ)] = w 2 + w 2 [ρ,k (ρ)] K K = w w K (26) dr = r[ρ,k (ρ)] dρ = r + r[ρ,k (ρ)] K = r K K, dk 2 = k 2 [ρ,k (ρ)] dρ = k 2 + k 2 [ρ,k (ρ)] K K = k 2 K (27) The first term on the RHS in each derivative corresponds to the short-run effect discussed in Section 3 while the second term on the RHS in each derivative captures the additional effect that arises as a consequence of the change in the aggregate capital stock that takes place in the long-run. To sign these additional terms, we first observe that the following comparative static results hold 23 w 1 K, w 2 K, k 2 K > 0, r K < 0 (28) Let us provide some intuition for these properties. If the aggregate capital stock increases, some of this new capital will be invested in sector 1 and some will be invested in sector 2. As a result, the capital stock per worker increases in both sectors of the economy, i.e k 1 / K > 0 and k 2 / K > 0. All else equal, these changes have positive effects on the marginal products of labor and on the wages. Hence w 1 / K > 0 and w 2 / K > 0. Finally, the increase of the capital stock in each 23 These comparative static properties are formally derived in the Appendix. 26