Is Policy Coordination a Good Idea? Public Inputs and Policy Harmonization with Skilled and Unskilled Workers

Abstract

We extend the classic Zodrow–Mieszkowski model of tax competition with a public input to the case where there is skilled and unskilled labor. The policy rule governing the optimal provision of the public input contains a new term capturing an equity effect that takes into account the disparity in wages between skilled and unskilled workers. The equity effect can work in the opposite direction of efficiency. Under a coordinated policy reform across countries, total welfare improves unambiguously if the public input is underprovided prior to the reform and a concern for equity enhances the effect of improved efficiency on welfare. However, total welfare may also improve even if the public input is initially overprovided if the improvement in the unskilled wage due to the reform is large enough.

Keywords

public input heterogeneous labor equity harmonization

In the classic model of a public input due to Zodrow and Mieszkowski (1986; hereafter ZM), the government imposes a source-based tax on mobile capital and provides a public input that augments the productivity of the primary private factors. In general, both the tax and the public input will affect the capital location decision and hence the policy rule used to determine the optimal second-best amount of the public input. The tax causes capital to leave the jurisdiction and works in the direction of underprovision of the public input as a result.¹ The direct impact of the public input on the demand for capital will depend on its relationship to capital in production. If they are complements, the government can attract capital by providing more of the public input, and this works in the direction of overprovision of the public input. The subsequent literature has focused mainly on developing the efficiency effects of the policy.² For example, Matsumoto (1998) is able to rule out overprovision of the public input in the uncoordinated equilibrium as a possibility in ZM’s model.

The purpose of this article is to incorporate equity into ZM’s model of a public input by including skilled and unskilled workers who differ in their productivity. We show that the policy rule governing the optimal choice of the public input will include a new term involving equity, which is captured by the response of the skilled wage to the policy. If the policy increases the disparity in wage rates by raising the skilled wage, ceteris paribus, it will redistribute toward skilled labor, and the government should underprovide the public input relative to the first-best case as a result. On the other hand, if the policy reduces the wage disparity, it will redistribute toward unskilled workers, and the government should overprovide the public input.³

The equity effect may work in the opposite direction of the efficiency effect on the government’s policy rule. For example, suppose a single government increases its capital tax rate. This will cause some capital to leave the country since the tax distorts the location decision. It follows that a concern for efficiency will provide the government with an incentive to keep the tax rate on mobile capital low in order to attract capital, a classic result in the tax competition literature.⁴ However, if capital and skilled labor are complements, an increase in the capital tax rate will also cause the demand for skilled labor to fall relative to the demand for unskilled labor. This will reduce the disparity in wages, and equity will improve as a result. Therefore, a concern for equity may provide an incentive for the government to keep the tax rate on mobile capital high, the opposite of the efficiency effect. Indeed, we can no longer rule out the case of overprovision when the government has a concern for equity.⁵

We also study how a concern for equity can affect the impact of a classic policy reform experiment in the tax competition literature where all countries coordinate and agree to raise their capital tax rate and spend the additional proceeds on the public input. Both wages rise in response to the reform, making both skilled and unskilled workers better off; while the net return to capital may fall, making capital owners worse off. If the public input is underprovided prior to the reform, then the total welfare improves unambiguously with the reform, where total welfare is measured by the equally weighted utilitarian sum of the utility of the unskilled worker and the skilled worker. The reform improves efficiency and the impact of a concern for equity enhances the efficiency effect by raising the unskilled wage. However, total welfare may still improve even if the public input is overprovided prior to the reform if the increase in the unskilled worker’s wage due to the reform is large enough in magnitude.

The paper that is most closely related to ours is Borck (2005) who studies the provision of public consumption goods in a tax competition framework where there are skilled and unskilled workers. Several cases are studied depending on the mobility of resources. For example, the main analysis assumes skilled workers and capital are mobile, while unskilled workers are not. He shows that a government under policy competition will emphasize public consumption goods that attract skilled workers at the expense of public consumption goods that benefit unskilled workers. However, a public input is not included in the analysis.

We present our model in the next section. After that, we discuss the policy rule and relate it to the results of the literature. Next, we discuss the policy reform and welfare analysis. Finally, we briefly conclude the article in the last section by answering the question posed in the title of this article.

Analytical Framework

Our starting point is the ZM model with a public input extended to allow for two types of labor. There are J identical countries in the world economy, where J is large enough so each country is small relative to the rest of the world. Capital is perfectly mobile across countries, while labor is not. There are two types of labor in each country, skilled and unskilled, that are separate factors of production that differ in their productivity. There is a measure of size one of each type and agents within a type are identical.

There are two goods produced in each country, a private consumption good and a public input. The private good is produced by a neoclassical technology using capital, the two types of labor, and the public input. One unit of the private good can be converted to one unit of the public input for simplicity. There are no spillover effects of the public input across countries. The private good is the numeraire good. Governments impose a source-based tax on capital invested in the local economy and use the proceeds to finance the public input. Each government takes aggregate variables like the net return to capital and the policies of the other governments as given when it chooses its policy. All agents, including governments, optimize, markets are competitive, and all markets clear in equilibrium.

There is a measure of size one of identical firms that produce the private good. Each firm uses a technology that exhibits constant returns to scale (CRS) in private inputs. The production function in levels is Y = F(K, H, L, G), where Y is the total output of the private consumption good, K is the capital, H is the skilled labor, L is the unskilled labor, and G is the public input, and where we have omitted country-specific superscripts for brevity. The public input augments the private inputs.⁶ The production function is strictly concave, twice differentiable, and strictly increasing in each argument. We will assume that capital and skilled labor are complements to one another and that all private inputs are complements to the public input. Under CRS in private inputs, the production function in intensive form becomes y = f(k, h, G), where y is the output per unskilled worker, k is the capital per unskilled worker, and h is the skilled labor per unskilled worker.

The firm’s profit per unskilled worker is $π = f (k, h, G) - (r + t) k - vh - w$ , where r is the net return on capital, t is the source-based tax imposed on capital, v is the skilled wage, and w is the unskilled wage. The firm chooses inputs to maximize profit, π. The first-order necessary conditions for an interior solution are given by, $f_{k} (k, h, G) \equiv \frac{\partial f}{\partial k} = r + t \equiv r_{n}$ and $f_{h} (k, h, G) \equiv \frac{\partial f}{\partial h} = v$ , where a subscript of a function denotes a derivative. Solve the first-order conditions to obtain the demand function for capital per unskilled worker and the demand function for skilled labor per unskilled worker, $k (r + t, v, G)$ and $h (r + t, v, G)$ , respectively. Totally differentiate the two first-order conditions of the firm’s decision problem and solve to obtain the response of the factor demands to the parameters (r + t, v, G):

k_{r} = \frac{f_{hh}}{D} < 0,

k_{v} = - \frac{f_{kh}}{D} = h_{r},

h_{v} = \frac{f_{kk}}{D} < 0,

k_{G} = \frac{f_{h G} f_{kh} - f_{k G} f_{hh}}{D},

h_{G} = \frac{f_{kG} f_{hk} - f_{hG} f_{kk}}{D},

where D = f_k kf_hh − f_k h f_hk > 0.⁷ Under our assumptions, the own price effects are negative, k_r, h_v < 0, the cross-price effect is negative, k_v = h_r < 0, and capital per unskilled worker and skilled labor per unskilled worker are both increasing in the public input, k_G, h_G > 0.

Since economic profits are zero when there is CRS in private inputs, the unskilled wage rate is determined by the following wage function:

W (r_{n}, v, G) = f (k (r_{n}, v, G), h (r_{n}, v, G), G) - r_{n} k (r_{n}, v, G) - vh (r_{n}, v, G),

where we have substituted the capital and skilled labor input demand functions into the profit equation. The partial equilibrium responses of the unskilled wage to the parameters are given as:

W_{r} \equiv \partial W / \partial r_{n} = - k, W_{v} \equiv \partial W / \partial v = - h, and W_{G} \equiv \partial W / \partial G = f_{G},

after using the envelope theorem. We also will assume that v > w which reflects the difference in skill.

Each consumer’s tastes are represented by a well-defined utility function, Uⁱ (c_i), for i = s, u, for skilled and unskilled, respectively, where c_i is the consumption of the private good. The utility function is strictly increasing, strictly concave, and twice differentiable. We will assume tastes are the same across types for simplicity. The marginal utility of consumption is denoted as $U_{c}^{i} \equiv d U^{i} / d c_{i}$ for i = s, u. Each high-wage skilled worker is endowed with one unit of skilled labor and a > 0 units of capital, while low-wage unskilled workers are only endowed with one unit of unskilled labor. Both types of labor are supplied to the labor market in a perfectly inelastic manner. The budget constraint of a skilled worker is v + ra = c_s, and the budget constraint of an unskilled worker is w = c_u . It follows that c_s > c_u. This implies that the ratio of the skilled worker’s marginal utility of consumption to the unskilled worker’s marginal utility of consumption is less than one in magnitude, that is, $φ \equiv U_{c}^{s} / U_{c}^{u} < 1$ , since utility is strictly increasing in consumption and strictly concave.

Equilibrium in the local skilled labor market requires:

h^{j} (r + t^{j}, v^{j}, G^{j}) = 1,

for each country j. The world capital market must clear:

\sum_{j} (k^{j} (r + t^{j}, v^{j}, G^{j}) - a) = 0.

The government’s budget constraint must also hold with equality:

t^{j} k^{j} (r + t^{j}, v^{j}, G^{j}) = G^{j}

in each country j. And the unskilled wage satisfies, $w^{j} = W^{j} (r + t^{j}, v^{j}, G^{j})$ .

The social welfare function for the representative government is the equally weighted utilitarian sum of utilities of the unskilled and skilled worker, respectively, $U^{u} (w) + U^{s} (v + r a)$ . Each government imposes a source-based tax on capital to finance local government spending on the public input, as in the ZM model. And each government chooses policy to maximize social welfare subject to its budget constraint, equation (6), the demand for capital per unskilled worker, k(r = t, v, G), and the impact of its policy on the local labor markets, equations (2) and (4), while taking the net return on capital and the policies of the other governments as given. The outcome of this decision process is a second-best policy rule.

We will focus on a symmetric uncoordinated equilibrium where each government takes the policies of the other governments as given and consider the choices of the representative government in that equilibrium in studying the optimal policy rule. A symmetric uncoordinated equilibrium is a tax rate, a public input level, and factor prices, $(t, G, r, v, w)$ , that satisfy the following system of equations, $w = W (r + t, v, G), h (r + t, v, G) = 1, k (r + t, v, G) = a,$ and $t k (r + t, v, G) = G$ , and the policy rule chosen by the representative government, which we turn to in the next section.

The Optimal Policy Rule in the Uncoordinated Equilibrium

In order to characterize the government’s policy rule, we must first consider the government’s budget constraint and how it responds to policy. Solve the skilled labor market equilibrium condition, equation (4), for the skilled wage, $v = V (r + t, G)$ , with derivatives:

\frac{\partial v}{\partial t} = - \frac{h_{r}}{h_{v}} = \frac{f_{k h}}{f_{k k}} < 0,

\frac{\partial v}{\partial G} = - \frac{h_{G}}{h_{v}} = \frac{f_{h G} f_{k k} - f_{k G} f_{h k}}{f_{k k}} > 0,

where we have used equations (1) to simplify.⁸ The skilled wage falls with an increase in the tax rate since skilled labor and capital are complements. And the skilled wage rises with an increase in spending on the public input since skilled labor is a complement to both capital and the public input.

Substitute the skilled wage function, V(r + t, G), into the government’s budget constraint, equation (6):

t k (r + t, V (r + t, G), G) = G,

and consider a balanced budget change in the government’s policy variables:

(k + t (k_{r} + k_{v} \frac{\partial v}{\partial t})) d t = (1 - t (k_{G} + k_{v} \frac{\partial v}{\partial G})) d G .

Using equations (1) and (7) and simplifying, equation (9) becomes:⁹

k (1 + \frac{t}{k f_{k k}}) d t = (1 + \frac{t f_{k G}}{f_{k k}}) d G .

Notice that if we differentiate the first-order condition of the firm’s decision problem for capital, we obtain $\frac{\partial k}{\partial t} = \frac{1}{f_{k k}} < 0$ and $\frac{\partial k}{\partial G} = - \frac{f_{k G}}{f_{k k}} > 0$ . Use this in equation (10) to obtain:

k (1 + (\frac{t}{k}) (\frac{\partial k}{\partial t})) d t = (1 - t (\frac{\partial k}{\partial G})) d G .

We will assume the following:

(\frac{t}{k}) (\frac{\partial k}{\partial t}) > - 1,

12a

t (\frac{\partial k}{\partial G}) < 1.

12b

Under assumptions (12), an increase in the tax rate produces more tax revenue that can be spent on the public input, and an increase in spending on the public input requires an increase in the tax rate to finance it, from equation (11). It will be convenient to define the efficiency effects of the policy in the following manner, from equations (10) or (11), for simplicity,

Ω \equiv \frac{(1 + \frac{t f_{k G}}{f_{k k}})}{(1 + \frac{t}{k f_{k k}})} = k (\frac{d t}{d G}) > 0.

Next, note that the Lagrangean for the j government’s decision problem is:

U^{u j} (w^{j}) + U^{s j} (v^{j} + r a) + λ^{j} [t^{j} k^{j} (r + t^{j}, v^{j}, G^{j}) - G^{j}],

where $w^{j} = W^{j} (r + t^{j}, v^{j}, G^{j})$ , $v^{j} = V^{j} (r + t^{j}, G^{j})$ , and λ ^j are the multipliers for the budget constraint of the government. The first-order conditions are given as:

U_{c}^{u} f_{G} + (U_{c}^{s} - h U_{c}^{u}) (\frac{\partial v}{\partial G}) - λ (1 - t (k_{G} + k_{v} (\frac{\partial v}{\partial G}))) = 0,

- k U_{c}^{u} + (U_{c}^{s} - h U_{c}^{u}) (\frac{\partial v}{\partial t}) + λ (k + t (k_{r} + k_{v} (\frac{\partial v}{\partial t}))) = 0,

where we have used the derivative properties of the unskilled wage function, equations (3), and where we have omitted country-specific superscripts. Equation (14) governs the optimal choice of the public input, while equation (15) governs the optimal choice of the tax rate. Combine equations (14) and (15) by eliminating the multiplier and use equations (1), (7), and (10) to obtain the following policy rule:

f_{G} = Ω + (1 - φ) (\frac{d v}{d G}),

which is evaluated at a symmetric uncoordinated equilibrium, where $\frac{d v}{d G} = \frac{\partial v}{\partial G} + Ω (\frac{1}{k}) \frac{\partial v}{\partial t}$ , $\frac{\partial v}{\partial G} > 0$ , and $\frac{\partial v}{\partial t} < 0$ , from equations (7). The first term in equation (16), Ω > 0, captures the efficiency effects of the policy, while the second term, $(1 - φ) (d v / d G)$ , represents the equity effects.

First, consider the special case where there is only one type of worker so equity is not a concern $(φ = 1)$ . In that case, we obtain the classic result due to ZM, f_G = Ω, which represents their policy rule.¹⁰ The distortion caused by the tax rate works in the direction of underprovision, f_G = Ω > 1, while the positive impact of the public input on the demand for capital, and hence tax revenue, works in the direction of overprovision, f_G = Ω < 1. From this, it would appear that the sign of Ω − 1 is ambiguous in ZM’s model. However, Matsumoto (1998) proves that the impact of the tax distortion dominates the effect of the public input in the ZM model, so the net effect of the policy is underprovision, f_G = Ω > 1.

Unfortunately, Matsumoto’s argument does not go through when there is more than one type of labor and the government has a concern for equity since the first-order conditions of the government’s decision problem are used in constructing the argument. In our model, these conditions include a term that captures the government’s concern for equity, which may work in the opposite direction of efficiency. It follows that the sign of Ω − 1 is ambiguous in our model because of the government’s concern for equity, in contrast to the literature, which emphasizes the efficiency aspects of the policy (see Appendix A for a more detailed discussion of this issue.). In particular, we cannot rule out overprovision, F_G < 1, as a result of the efficiency effects.

The impact of incorporating a concern for equity into the government’s decision-making will depend on the weight 1 − φ, where it is recalled that $φ \equiv \frac{U^{s} (c_{s})}{U^{u} (c_{u})} < 1$ , and on the response of the skilled wage to the policy, dv/dG. First, consider the weight. The weight will be larger, the greater the disparity in wages, or the larger the skilled worker’s capital income. So a concern for equity will have a greater impact at the margin in an economy where wages are more disparate across labor types than in an economy where wages are less disparate. The same will be true, the larger the skilled worker’s capital income.

Next, consider how the policy affects the demand for skilled labor and hence the skilled wage. An increase in the tax rate on capital will cause both the demand for capital and the demand for skilled labor to fall relative to unskilled labor, and this will reduce the skilled wage. Since this reduces the disparity in wages across types, it will redistribute toward unskilled workers and improve equity. The improvement in equity will work in the direction of overprovision, f_G < 1. However, the public input will have the opposite effect. An increase in the public input will increase the demand for capital and for skilled labor relative to unskilled labor, and this will increase the skilled wage and hence increase the disparity in wages. Since this redistributes away from unskilled workers, it will harm equity and will work in the direction of underprovision, f_G > 1.

There are a number of interesting possibilities that extend the literature when we combine these effects. First, it is certainly possible for the equity and efficiency effects to work in opposite directions on the policy rule, making the outcome ambiguous. Second, it is also possible that the two effects can work in the same direction and imply that overprovision of the public input is optimal. This may occur because the provision of the public good increases tax revenue and the tax rate imposed on capital may reduce the demand for skilled labor, which would in turn reduce the skilled wage and improve equity.

It is instructive to consider the special case where the efficiency effects exactly cancel so that Ω = 1, which is equivalent to $k \frac{\partial k}{\partial G} = - \frac{\partial k}{\partial t}$ or $1 = k f_{k G}$ , from equation (10). In that case, the equity effect simplifies to the following:

\frac{d v}{d G} = \frac{\partial v}{\partial G} + (\frac{1}{k}) (\frac{\partial v}{\partial t}) = f_{h G} > 0,

after using equations (1) and (7) (see Appendix B). Substitute equation (17) into equation (16) and set Ω = 1, to obtain the following special case of the policy rule:

f_{G} = 1 + (1 - φ) f_{h G} > 1.

Underprovision is optimal when the efficiency effects cancel. The impact of the public input on the skilled wage outweighs the impact of the tax rate causing the disparity in wages to be larger than would otherwise have been the case, which works in the direction of underprovision. This is an interesting result since one would be tempted to advocate for the first-best level of the public input if the focus is solely on efficiency. This result also provides an example where the government’s concern for equity keeps spending on the public input low because its provision increases the skilled wage and harms equity.

Policy Reform Analysis

Under the classic policy reform experiment in the tax competition literature, it is imagined that the economy is in an uncoordinated equilibrium prior to the reform and the collection of countries competing with one another agrees to increase the capital tax rate in order to increase spending on the public input. Under symmetry, we can study the impact of such a reform on the representative country and find conditions under which the reform is beneficial.

The two equations governing the equilibrium in the capital and skilled labor markets and the government’s budget constraint, equations (4), (5), and (6) under symmetry, can be solved so r, v, and G can be written as functions of the tax rate, which is the control variable under the reform. The impact of the reform on the unskilled wage can be derived from equation (2) in a recursive manner once the response of r, v, and G is determined. The general equilibrium response of relative prices to the policy reform is derived in Appendix C and is given by:

\frac{d r}{d t} = - (1 - K F_{k G}),

\frac{d G}{d t} = K,

\frac{d v}{d t} = K H F_{h G},

\frac{d w}{d t} = K L F_{L G} .

Both wages increase and government spending on the public input increases. However, the net return to capital falls, if 1 > KF_KG.

The welfare implications of the reform for total welfare are governed by the following equation, which is derived in Appendix D:

\frac{d U^{u}}{d t} + \frac{d U^{s}}{d t} = K U_{c}^{u} [φ (F_{G} - 1) + (1 - φ) L F_{L G}],

starting from the uncoordinated equilibrium. The first term in equation (23) captures the efficiency effect of the reform, while the second term captures the impact of the public input on the marginal product of unskilled labor. The ZM result emerges as a special case, where φ = 1 and F_G = Ω > 1 by Matsumoto’s (1998) result. The reform only improves total welfare in the ZM model if efficiency improves, for example, which is the standard result.

In the more general case, total welfare can improve under the reform not just when efficiency improves but also because the unskilled worker’s wage increases with the reform. This is true when the public input has already been provided at a first-best level, where its marginal product is equal to its marginal cost, F_G = 1, in which case the first term in equation (23) drops out altogether. And it is also true when the efficiency effects cancel, that is, when Ω = 1 or 1 = KF_KG. In that case, equation (23) becomes:

\frac{d U^{u}}{d t} + \frac{d U^{s}}{d t} = k (1 - φ) U_{c}^{u} [φ H F_{H G} + L F_{L G}] > 0,

after substituting equation (18) into equation (23). The reform increases both wages, which unambiguously improves total welfare, while the net return to capital is unaffected by the reform in this special case, from equation (19), when 1 = KF_KG. So the reform is a Pareto improvement, despite the fact that the efficiency effects cancel. It is even possible for the public input to be overprovided, FG < 1, and have the reform improve welfare if the impact of the reform on the unskilled wage is large enough in magnitude. This clearly goes against the standard result.

A second point involves the weights φ and 1 − φ in equation (23). They depend on the income of the skilled worker relative to the unskilled worker. The more disparate wages are, or the larger the skilled worker’s capital income, the larger the weight $1 - φ = 1 - U_{c}^{s} (v + r a) / U_{c}^{u} (w)$ is, and the greater the importance of the unskilled wage effect in determining the outcome of the reform on total welfare. It is certainly possible that 1/2 > φ, in which case the equity term may dominate the efficiency term in determining the impact on total welfare at the margin because of the relative weights. This also goes against the standard result.

What are the welfare implications of the reform for individuals? The key to these effects is the ownership of capital. If the low-wage unskilled workers do not own any of the capital, as we have assumed, they are unambiguously better off with the reform since the unskilled wage increases as a result of the reform. The welfare effect of the reform for the high-wage skilled workers is ambiguous because the skilled wage rises, while the net return to capital falls. An exception to the importance of the ownership of capital in determining the welfare effects of the reform occurs in the special case where the efficiency effects cancel. In that case, the ownership of capital does not affect the welfare results because the net return to capital is not affected by the reform, while both wages increase, so the reform is a Pareto improvement. If, however, the unskilled workers share in the ownership of the capital stock, then their welfare response to the reform also becomes ambiguous in general. The less capital they own, the more likely it is they will benefit from the reform.

Conclusion

The answer to the question posed in the title of this article is yes. Harmonization of public input policy across countries can be beneficial and improve total welfare when there is skill heterogeneity in the workforce and the government has a concern for equity. In general, the reform improves the welfare of the unskilled worker by increasing their wage. It follows that a concern for equity will tend to work in the direction of enhancing the beneficial efficiency effects of the reform. If the skilled worker is better off, the reform is a Pareto improvement. If they are made worse off by the reform because their capital income has fallen, but efficiency has improved under the reform, then total utility rises as the distribution of welfare narrows. A caveat to this conclusion occurs if the unskilled workers own some of the capital since the response of their welfare to the reform then becomes ambiguous.

Footnotes

Appendix A

Appendix B

Appendix C

Appendix D

Acknowledgments

The author is grateful to three very helpful reviewers who provided numerous helpful comments. Thanks also to Toshihiro Ihori, who commented on an early draft. We also thank Mark Gibson and Christopher Clarke, who provided useful comments. All errors that remain are the sole responsibility of the author.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

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