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Monopsony in the labor market, minimum wages and the time horizon: some unresolved issues.

Introduction

Until now, most researchers in the minimum wage field, such as Card and Krueger (1997) or Manning (2006), Boal and Ransom (1997) and Neumark and Wascher (2008), have focused on a short-run perspective of the labour market and considered only one variable factor of production and neglected the effects of substitution when two or more variable factors of production are there. In some cases, authors, like Metcalf (2008, p. 497), briefly addressed the issue. An important exceptional contribution in this discourse is the study of Maurice (1974). (1) His analysis, however, is only convincing as long as the non-discriminating monopsony is treated with one variable factor of production. A second memorable exception is the contribution of Barr and Roy (2008). They tackled the question of whether the relatively low wages paid by monopsonists can lead to sub-optimal levels of investment in physical or human capital.

The monopsonist will choose, in the absence of minimum wages, extremely high capital intensity. As the monopsonist offers low wages, labour supply will be low too. (2) Once confined by a minimum wage, labour becomes more expensive. Now, he will choose a lower capital intensity and higher employment. If the binding minimum wage reaches higher levels, the monopsonist will increase capital intensity that eventually will lower employment. Therefore, as an outcome of this effect, both lower and higher wages (at given interest rates) can go along with a high capital intensity of production. (3) As a result, concurring with Piero Sraffa's re-switching hypothesis (Blumle 1975, p. 140), the monotone relationship between the factor price ratio and the capital intensity of production will vanish.

In this article, we first analyze the peculiarities of a non-discriminatory monopsony with two variable factors of production, paying special attention to the budget constraint involved: the budget constraint of a monopsonistic firm is, because of the endogeneity of the wage rate, non-linear. The issue of capital intensity chosen by the monopsonist is addressed both by making use of simple graphical instruments and by investigating a constant elasticity of substitution (CES) production function. We compare the outcomes in the absence and presence of (three) statutory minimum wages of different values. We show that only a high-level binding minimum wage rate is capable of increasing capital intensity above its initial unregulated equilibrium level. The results are exemplified by a numerical example, which is the basis for the 'minimum wage game' between government and firm in a two-period setting. Before summarizing the discourse, we address the relevance of elasticity of substitution for the optimal choice of factor intensities.

Peculiarities of Monopsony

The Simple Case: One Factor of Production

A monopsonistic entrepreneur is, almost by definition, the sole demander on the respective labour market. This means that he alone is confronted with the labour supply function. The latter, in turn, depends positively on the wage rate (w). In order to better solidity our model, we assume a linear labour supply function ([L.sup.S]): w(L) = [1/b (Z + a)]; a, b > 0, where w is the nominal wage rate and L measures the volume of labour. As a consequence, profit maximization of the monopsonist means

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [pi] stands for the profit of the firm, p is the price of the output good, Y is the output function depending on labour L and capital K (here capital [bar.K] [greater than or equal to] 0 is a constant parameter). Hence, the monopsonist maximizes profit vis-a-vis labour input I when the marginal revenues and the marginal costs of labour equate:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with MC(L) = marginal costs of procuring labour, [L.sup.D] = labour demand=marginal value of labour.

Figure 1 contains all of the important short-run solutions for the model of a nondiscriminatory monopsony: if the monopsony is not yet regulated by the introduction of minimum wages, the monopsonist will, equating marginal costs of procuring labour MC(L) and labour demand [L.sup.D], choose the wage rate [w.sup.M] and labour input of size [L.sup.M]. The four cases can be summarized as follows:

(i) The monopsonistic wage rate [w.sup.M] is always below the respective marginal costs of labour (MC(L)).

If the authorities force the monopsonist to install a minimum wage [bar.w] at the competitive wage rate [w.sup.*], the monopsonist will also choose labour input on the level of competitive employment [L.sup.*] = [bar.L]. As a consequence, the profits of the monopsonist will vanish (Cahuc and Laroque 2009, p. 7). This has some sort of a paradoxical effect: the shrinking profits will draw less potential competitors into the market as might have been the case had the minimum wage rate not been installed. In other words: minimum wages tend to stabilize existing monopsonies. Furthermore, we may conclude:

(ii) Marginal costs of labour are always above the competitive wage rate [w.sup.*].

This statement follows if we compare the marginal costs of labour of the monopsonist with the wage rate of the competitive supplier making use of the above introduced labour supply function. Notice that in equilibrium the competitive wage rate must lie both on the demand and on the supply function of labour. Hence, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[FIGURE 1 OMITTED]

This condition is obviously always fulfilled. If the government increases further the minimum wage rate (for example to a level of [??]), the new level of employment [L.sup.M] will fall below the level which is to be expected in the case of competition [L.sup.*] = [bar.L]. Such a policy would not be so unlikely nor implausible if the strategic goal of the minimum wage policy was located, more so than in the area of labour market policy, in the area of social policy and redistribution policy in favour of those currently employed at the firm of the monopsonist. In this case, the following is particularly true:

(iii) The marginal costs of labour MC(L) can in principle reach the same level as a strong binding minimum wage of amount [??]. Employment at this level of the minimum wage will then be identical with [L.sup.M] chosen by the unregulated monopsonist.

Statutory minimum wages are the result of a political decision-making process. Economic rationality usually plays a role in these types of decision-making processes, but most likely only one role in many other processes. Therefore, it cannot be ruled out that far too drastic measures are taken: in this case, the magnitude of the binding minimum wage rate could cause employment to drop to the level of a 'pure monopsony':

(iv) A strongly binding minimum wage rate of size [??] exceeds the marginal costs of labour and pushes employment down to the unprecedented low level of [??].

The Special Budget Constraint in the Case of Several Factors of Production

Similar to Barr (2009), we consider capital (K) as a second variable factor of production. It can easily be affirmed that, in such a case, the budget constraint for a representative monopsonist will become non-linear.

The budget constraint then reads C = r x K + 1/b (L + a)L = r x K + w(-a + bw), where C = total costs, K = capital input, r = interest rate, L = labour input. At a given constant C > 0, we obtain the following expression for capital input K as a function of labour input L:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The marginal rate of substitution between capital and labour is, not surprisingly, negative, which means the trend of the budget constraint curve is declining. In other words, it is no longer linear. Along with increasing inputs of labour, the steepness of the curve increases. The marginal costs of labour can be determined when the budget constraint is differentiated vis-a-vis the labour input (note that now C is no longer a constant as above):

C = K x r + 1/2 [[L.sup.2] + aL], dC/dL = 1/b (2L + a).

In the following, we assume at first a generalized production function which will be specified later on: Y(K, L). Now, we compute the profit maximum of the monopsonist. For this purpose, we make use of Lagrangian function / and maximize revenue, given the constant costs C, pY(K, L). Hence,

l(L, K, [lambda) = pY (K, L) + [lambda] [C - r x K - 1/b (L + a)L]. (1)

The first order conditions (FOC) are:

a) for capital input:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

b) for labour input:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

c) for Lagrangian multiplier:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

From the first two FOC, we may conclude:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Figure 2 shows optimal costs for the monopsonist at point A, as the wage rate is an endogenous variable in the case of a monopsony, and the isocost curve is non-linear. In any case, all isocost curves begin at the y-intercept with K = C/r. The slope of the isocost curve, different from the case of full competition or of minimum wages, following Barr (2009, pp. 505-6), now does not depend on the ratio of factor prices (w/r), but on the ratio of marginal labour costs to marginal capital costs ([MK.sub.L]/[MK.sub.K]). In a monopsony, as [MK.sub.L] will be higher than the wage rate, the slope in A (monopsony) will be greater than in B (competition), but equal to C (binding minimum wage rate), given the minimum wage rate [??]. This wage rate equates to the marginal costs of labour. C, of course, is an optimal solution, as the isocost curve is tangential to a corresponding isoquant. Only in the case of the strict binding minimum wage rate [??] (this wage rate is considerably higher than the marginal costs of labour and it causes labour demand to be at the shorter side of the market), a higher capital intensity will emerge in D in comparison to A.

The y-intercept at which the isocost curve for the monopsonist (at K = 0) intersects the x-axis, and the other parameters as shown in Fig. 2 are computed with a numerical example. The graphical analysis presented in Fig. 2 makes us presume that only a strict binding minimum wage rate w will motivate the monopsonist to increase his capital intensity of production along with his 'standard solution'. (4) Regarding the movement from A to B, from B to C, and finally from C to D, one gets an impression of the windshield wiper effect. This effect represents a special form of 're-switching', which can be observed in the case of a monopsony.

[FIGURE 2 OMITTED]

Nondiscriminatory Monopsony with Production Function of the CES-Type

Given the case of two variable factors of production, we may now generalize our results by resorting to the CES production function. As well known, this type of production function enables us to address the issue of substitutability of the factors of production via one unique parameter, [mu]. Furthermore, this type of function is homogeneous at degree 1. As already shown elsewhere, the CES-function contains the Cobb-Douglas-function (CD) as a special case. In the present case, the CES-function for L > 0, K > 0 is: (5)

Y(K, L) = E[[[delta][L.sup.-[mu]] + (1 - [delta]) [K.sup.-[mu]]].sup.-1/[mu]], 0 < [delta] < 1, [mu] > -1.

The profit function of the monopsonistic entrepreneur is then given by

[pi](L, K) = pE[[[delta][L.sup.-[mu]] + (1 -[delta]) [K.sup.-[mu]].sup.-1/[mu]] Lw(L) - rK (4)

where again (6) w(L) = 1/b(L + a). In addition, the budget constraint is given for C > 0 (see (2)):

C - Lw(L) - rK = 0. (5)

This determines K [member of] [0; C/r] for L [member of] [0; [L.sub.max]] with [L.sub.max] := -a/2 + [square root of ([a.sup.2]/4 + bC)] by

K(L) = C/r - L(L + a)/br. (6)

Making use of the above introduced Lagrangian function for the profit function [pi](L, K) leads, by way of analogy to (1), (2) und (3), to two optimality conditions, (5) and (7):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

By calculating the relevant derivatives one arrives at the following equation via condition (7):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

As a result, K can be shown to depend on L as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

with the capital intensity given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

Combining this result with the constraint (6) and considering (9), we can get a solution for L:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By defining

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

optimality for labour is described by:

[phi](L) = 0. (13)

It can easily be shown by a monotonic reasoning for [phi](L) that (13) describes a unique solution [L.sup.*] which is implicitly given by:

[L.sup.*] = [[phi].sup.-1] (0). (14)

So far, we have been able to demonstrate that given the budget constraint C, the optimal labour input of the monopsonist [L.sup.*.sub.M], the optimal capital input [K.sup.*.sub.M] = K([L.sup.*.sub.M]) via (6), and the equilibrium wage rate

[w.sup.*.sub.M] = w([L.sup.*.sub.M]) = 1/b ([L.sup.*.sub.M] + a) (15)

as well as the optimal intensity of capital [[kappa].sup.*.sub.M] = [kappa] ([L.sup.*.sub.M]) = [K.sup.*.sub.M]/[L.sup.*.sub.M] can be determined completely:

[[kappa].sup.*.sub.M] = [((1 - [delta])/[delta]br (2[L.sup.*.sub.M] + a)).sup.1/[mu] + 1]. (16)

Hence, we have shown and proven that an exact solution for the equilibrium of a monopsony also exists under a far more general condition of the CES-function. However, one may usually formulate equilibrium only in an implicit way and by making use of approximative solution algorithms. Finally, the budget constraint can be written as:

C = [w.sup.*.sub.M] [L.sup.*.sub.M] + r[K.sup.*.sub.M] = ([w.sup.*.sub.M] + r[[kappa].sup.*.sub.M]) [L.sup.*.sub.M]. (17)

Non-discriminatory Monopsony with a Statutory Minimum Wage Rate

The graphical equilibrium depicted in Fig. 2 shows the case where a monopsonist, confronted with a binding statutory minimum wage, may increase his capital intensity along with the reference solution. We now analyze this case with a greater degree of precision, i.e., by using analytical tools. Upon introduction of a binding statutory minimum wage, the right hand side of the profit equation of a monopsonist changes into:

[pi](L, K) = pE[[[delta][L.sup.-[mu]] + (1 - [delta]) [K.sup.-[mu]]].sup.-1/[mu]] -Lw(L) - rK,

this time with w(L) := max([??]; 1/b(L + a)), where w is any binding minimum wage, i.e. [??] > [w.sup.*.sub.M] > 0 with [w.sup.*.sub.M] from (15).

Only if [??] > 1/b ([L.sup.*.sub.M] + a) = [w.sup.*.sub.M] respectively [L.sup.*.sub.M] < b[??] - a, the minimum wage solution will differ from the standard solution. The profit function will then read for this area of definition (budget restraint being ignored so far) as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

The wage rate [??] here is not a specific minimum wage rate (such as aforementioned [bar.w] or [??]), but is associated with a rather general, although, binding statutory minimum wage rate. The FOC for a profit maximum is now (see in comparison (3) and (8)):

[delta][L.sup.-[mu]-1]/(1 - [delta]) [K.sup.-mu]-1] = [delta]/1 - [delta] [(K/L).sup.1 + [mu]] = [??]/r.

It follows that the optimal intensity of capital [[kappa].sup.*.sub.[??]] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

Comparing this result with (16) leads us to calculate:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

This is the condition for the minimum wage to reach a level that induces the monopsonist to increase his capital intensity along with the reference solution. Therefore, at this point, the question is: what level w should have in order to motivate the monopsonist to increase his capital intensity of production in comparison to the unregulated initial equilibrium. Herein, the budget constraint can be utilized, in analogy to Eq. (17) representing the standard solution, to compute the new Z,M- and K$ for capital intensity introduced in (20). Hence, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This enables us to calculate the optimal amounts of labour and optimal capital input as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

Herewith, we have shown and proven that whenever a statutory and binding minimum wage rate is introduced, the monopsonist will, under the conditions of (20), increase his capital intensity of production according to (19). In doing so, he will reduce the amount of labour input and change the amount of capital input according to (21).

A Numerical Solution for Optimum Monopsony: The Case of the Cobb-Douglas-Production Function

For an illustrative purpose, we assume a production technology Y of the Cobb-Douglas type (i.e., a special CES function for [mu] = 0 and [delta] = 0.3): Y(K, L) = [L.sup.0.3] [K.sup.0.7]. For an example, we further assume the parameter set as r = 0.1; a = 1; b = 1; p = 1. By calculating the marginal productivities of the respective factors and using the optimality conditions the above yields:

7L/3K = rb/(2L + a) or 7L/3K = 0, 1/2L + 1 or 14[L.sup.2] + 7L - 0.3K = 0.

This together with the budget constraint

K = 120 - 10[L.sup.2] - 10L determines the optimal solutions for K and L in the case of a monopsony. Inserting the equation for K into the first equation and a solution for L yields:

[L.sub.M] = 1.190 and hence [K.sub.M] = 120 - 10[L.sup.2] - 10L = 93.939.

We insert this result into the labour supply function and compute: [w.sub.M] = 2.190. The capital intensity amounts to 78.940. The optimal solution is depicted in Fig. 2 as point (A). Total profit amounts to [[pi].sub.M] = 13.331 and the wage sum equals [WS.sub.M] = 2.608.

Furthermore, we are now able to compute the binding minimum wage rate [??] which equals the marginal costs of labour MC(L). For [??] = [L.sub.M], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us now assume an arbitrary binding minimum wage rate [??] > [w.sub.M]. The non-monopsonistic budget constraint gives [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and from the optimisation approach we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now, in accordance with (19)), the capital intensity amounts to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and together with the budget constraint [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the marginal cost of labour for [L.sub.M], the capital intensity amounts to [??] = 70 x 3.381/3 = 78.89 in compliance with (20) and the associated optimal point with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is depicted in Fig. 2 as (C). Hence, the competitive wage rate [bar.w] lies between 2.190 (lower bound) and 3,381 (upper bound). Total profits amount to [??] = 10.655 and the wage sum equals [??]S = 3.60. The even more binding wage rate [??] hence, must lie beyond 3.381. Let us assume, [??] is at 8.50 [[euro]/h] (i.e. the amount of the statutory minimum wage suggested in 2013 in Germany by the Social Democrats Party ('SPD') and implemented nationwide in 2015. Making use of the budget constraint and the numerical values from above, we arrive at[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, the capital intensity will increase to [??] = 198.33.

The total profit amounts to [??] = 5.181 and the wage sum equals [??]S = 3.60. As a result, such a high binding minimum wage rate will significantly reduce the input of labour and capital both. The associated optimal graphical solution is depicted in Fig. 2 and labelled as (D). It is noteworthy that the existence of re-switching in conjunction with an underlying Cobb-Douglas-function, as documented by point D, is denied by the neoclassical economics.

For the full competition equilibrium, assuming [bar.w] = 2.5, we use the same analysis as for any other binding minimum wage rate and get [bar.L] = 3.6/2.5 = 1.44 and again [bar.K] = 84. Hence, the capital intensity now only amounts to [bar.[kappa]] = 58.33. Total profits amount to [bar.[pi]] = [[pi].sup.*] = 12.803. The associated optimal point is labelled (B) in Fig. 2. Table 1 summarizes the calculations we have done so far.

How [mu] Determines the Optimal Input of Factors of Production

In the following section, we would like to examine: how variation in the parameter [mu] impacts the optimal solution of a monopsony. We will deduce that, as a function of chosen [mu], any L [member of][0; [L.sub.max]] and, consequently, any K as well as n can be found as optimal solutions.

Our next exercise will reveal a strict monotonic relationship between [mu] on the one hand and labour input L, on the other hand. We will verify this outcome analytically. For this purpose, we again utilize (11), which is fulfilled in the optimum [L.sup.*.sub.M] ([mu]) determined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking logarithms and solving for the parameter [mu] gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

Hence, we conclude that, in fact, there is a strict monotonicity in the function [mu](L). The same monotonicity, therefore, will exist for the inverse function L([mu]). If we choose ascending values for L, the numerator itself will rise, whereas the denominator will decrease as long as [kappa](L) > 1, which means that K(L) > L. The marginal value [L.sub.[kappa] =1] can be calculated by using the budget constraint:

K(L) = C/r - L L + a/br [??] L.

Solving for [L.sub.[kappa]=1] gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

Furthermore, it must be guaranteed that for [mu] > -1, the numerator of the quotient in (22) is not negative. Therefore, it must be valid for the argument of the logarithm, given the non-negativity condition for labour input L,

As the denominator will strictly rise in the interval [[L.sub.min]; [L.sub.[kappa]=1]), and, at the same time, the numerator will strictly fall, it is shown and proven that with rising L, the parameter [mu] will strictly rise in a monotonic form. Hence, the same monotonicity applies to the optimal solution [L.sup.*.sub.M] ([mu]) in the case of a monopsony. Let [[L.sub.min]; [L.sub.[kappa]=1]) be the interval of monotonicity for L, so that L = [L.sub.min] or K[[L.sub.[kappa]=1]) = [L.sub.[kappa]=1]. Then it can easily be shown that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Accordingly, it holds for situations past the pole [L.sub.[kappa]=1] in the open interval ([L.sub.[kappa]=1]; [L.sub.max]) that [mu] increases strictly with L in a monotonic form, where:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Such values for [mu], however, were excluded from the parameters of the CES-function. In conclusion, for any solution of the three parameters ([mu], L, K) in the optimization process of the monopsonist, we may state the following:

(1) with [mu] [member of] (-1; [infinity]) L [member of] ([L.sub.min.]; [L.sub.[kappa]=1]) increases and [kappa] [member of] (1; [infinity]) decreases;

(2) with [mu] [member of] (-[infinity]; -1) L [member of] ([L.sub.[kappa]=1]; [L.sub.max]) increases and [kappa] [member of] (0; 1) decreases;

(3) with increasing L [member of]([L.sub.min]; [L.sub.max]), the elasticity of substitution, [sigma](L) = 1/[mu](L) + 1], decreases within the interval (-[infinity]; [infinity]).

What might all this mean and possibly imply for the challenge of installing statutory minimum wages in the presence of monopsonies? One point must be clear that before one starts to implement such minimum wages over a huge variety of branches, empirical estimates of the elasticities of substitution must be evaluated. Only then will it be possible to assess the likely effects on labour input and the way they affect the choice of capital intensities.

The Minimum Wage Game

A game theoretic approach is now used to investigate the interaction between the government setting minimum wage and the firm choosing the level of labour and capital. Assuming that government (G) and monopsonistic firm (F) find themselves in a two-period strategic interaction situation while deciding whether to install a minimum wage at all and if so, at what level, seems to be a fair approximation to reality. The government has two choices for the first period: to announce the future increment of a minimum wage or to announce that it renounces implementation of any minimum wage in the foreseeable future. In the second period, the government will either choose a specific minimum wage as designed in Fig. 2, or let the monopsonists choose their 'first-best' solution with no minimum wage around whatsoever. The government's payoffs can be calculated as the wage sum which accrues at each alternative situation. The firm, in turn, can optimize its profits (which may nicely symbolize its pay-off as well) in the second period according to the four regimes depicted in Fig. 2, or give up production. The latter alternative has severe consequences for both players: while the government is confronted with a wage sum of zero, the plant with the employees, including the management, is dismissed and the wage costs 'disappear.' If the capital of the firm is totally owned by shareholders, the latter could now lend the capital stock to a third party, earning an interest income of r[K.sub.M]. Notice that the respective capital stock corresponds to the 'pure monopsony,' that is to the reference situation (the first period) before minimum wages were implemented. In Fig. 3, we have depicted the two-period minimum wage game. It is obvious that the government will always prefer a minimum wage over the status quo, provided that the firm does not shut down its business. As in two cases of minimum wages, the firm's pay-off for production exceeds the pay-off for closing down. The government will have to choose between the minimum wage rate [??] = 3.381 and the alternative minimum wage rate [bar.w] = 2.5. As only the latter promises higher profits for the firm (12.803 > 10.655) which translate into higher expected corporate taxes or higher private investment, the government will choose [bar.w] = 2.5. Hence G (3.600), F (12.803) is the equilibrium of the game. (7) It is noteworthy that this outcome is not sensitive to the assumption made with regard to the capital owners of the firm: suppose that the equity owners may only claim 50 % of the interest income of 9.3921, which is 4.696. In this case, the government anticipates that the firm will not close down the business even if the minimum wage [??] = 8.50. But, here again, the government will choose [bar.w] = [w.sup.*].

[FIGURE 3 OMITTED]

Summary and Outlook

A monopsonist tends to produce with an extremely high capital intensity, which in principle can be reduced when he is confronted with a minimum wage rate. However, only a high-level binding minimum wage rate such as [??] is capable of increasing the capital intensity above its initial unregulated equilibrium level ('windshield wiper' effect as a special case of 're-switching'). The choice of capital intensity, in turn, is intimately linked to the existing elasticities of substitution in production. Therefore, we have analyzed the factor input for a monopsonist using the rather general production technology of the CES-function. We could show that under the quite general conditions of the CES-function, there also exists an exact solution for a monopsony. However, the conditions for the solution, in general, can only be provided implicitly or by making use of approximative techniques. Still in the framework of a CES function, we have addressed the question of what level a specific minimum wage rate [??] (a rather generally \constructed, albeit binding, statutory minimum wage rate) should have in order that a monopsonist chooses a higher capital intensity than would have been chosen under an unregulated labour market. Finally, we have investigated how variation in the elasticity of substitution affects the choice of production (capital) intensity. In the subsequent paragraphs, we presented a numerical example for optimal solutions under a regulated and an unregulated monopsony. The results helped us to define the payoffs for a two-period game between the government and the monopsonistic firm. We demonstrate that the minimum wage rate [bar.w] = [w.sup.*] is the equilibrium of the game.

DOI 10.1007/s11293-016-9484-8

Acknowledgments We thank Beate Sauer, Thomas Werner, Anne Oeking, Reinhard Neck, Maks Tajnikar, Petra Dosenovic Bonca and further participants of the session, 'European Economics' at the 77th International Atlantic Economic Conference held in Madrid, April 2-5, 2014 and two anonymous referees for their most valuable comments and suggestions. All remaining errors are ours.

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(1) An Internet search revealed that the article of Maurice (1974) has been cited since then only seven times (!), and adding our own makes eight.

(2) Assuming an upward sloped labour supply function. We do not consider the case of a non-linear function.

(3) The latter is the neo-classical wording for what Sraffa labels 'techniques'. See Schefold (2005, p. 443).

(4) In the relevant literature (Barr 2009), one finds the argument that a monopsonist, once confronted with minimum wages at a level of a competitive wage rate or even with higher levels, will in principle react like a competitive firm. This statement is correct, but does not alter the fact that as such, the monopsonist remains the sole demander of labour and thereby continues to be a monopsonist.

5 Notice that for [mu] [right arrow] 0, the CES-function converges against Cobb-Douglas-function Y(K,L) = [EL.sup.[delta]] [K.sup.1 - [delta]], therefore, [mu] = 0 is included in the parameter space.

6 One may use this method of calculation for every wage function w(L) which is continuously differentiable. For reasons of simplicity, we make use of the above introduced linear wage function.

(7) As the original pay-offs are identical for the government when establishing a minimum wage regime (3.600), one has to look for further discriminatory criteria such as expected corporate tax income or expected private investment.

Friedrich L. Sell [1] (iD) * Ernst K. Ruf [2]

Published online: 26 February 2016

Ernst K. Ruf ernst.ruf@unicredit.de

[mail] Friedrich L. Sell

friedrich.sell@unibw.de

[1] Macroeconomics and Economic Policy, Universitat der Bundeswehr Munchen, Neubiberg, Germany

[2] UniCredit Group, Munich, Germany
Table 1 Results of numerical example illustrating the
re-switching effect

Label      L       w       K         K        Y       [pi]    Wage sum

A        1.191   2.191   93.921   78.891    25.331   13.331    2.608
B        1.440   2.500   84.000   58.333    24.803   12.803    3.600
C        1.065   3.381   84.000   78.891    22.655   10.655    3.600
D        0.424   8.500   84.000   198.333   17.181   5.181     3.600

Source: own calculations
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Author:Sell, Friedrich L.; Ruf, Ernst K.
Publication:Atlantic Economic Journal
Article Type:Statistical data
Geographic Code:1USA
Date:Mar 1, 2016
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