At the same time, we are undertaking a 'race to the top' to imply that too many public goods are being provided and taxes are being set too high. Necessary and sufficient conditions for the model primitives for the efficient level of public services can be identified in the usual way. In this case, the government is 'too big'.5 And if mpgv is equal to mcpg at the efficient level of public services, then there is no incentive to deviate and the equilibrium result is efficient.
In the current section, where we examine efficiency, we assume that the level of public good provision is centrally chosen by a planner for both jurisdictions. The next section decentralizes the provision of public goods through taxation by a government in each jurisdiction. The general functional form represents a production technology that depends on the level of provision of public goods yi and of capital ki.
The Efficiency of Production
Because the main feasibility condition kEi +kjE =k is used to replace forkjE, solving the planner's problem for an efficient plan involves solving three first-order conditions for the three unknowns, ki, yi, and yj. The first condition, (4), states the well-known requirement that with an efficient plan the marginal unit of capital in each jurisdiction is equally productive. Conditions (5)-(6) determine, for jurisdictions i and j respectively, that the marginal cost of forgoing a unit of the consumer good to produce the marginal unit of the public good must be equal to the marginal product of the public goods in production.
In this paper, we will maintain the focus of the previous literature on the situation where the efficient point is interior, that is, where production occurs in both jurisdictions.
3 Tax Competition in the Z-M Framework
- The Firm’s View of Production
- Capital Market Clearing for Given Public Good Levels
- Welfare and the Feasibility of Consumption
- The Government’s View of Production
- Definition of Equilibrium
We now have a complete characterization of the solution to the company's problem at a fixed level of the common good. But in the current context, the government would always want to use all the revenue it collects to provide the common good. subject to feasibility conditions) the government must raise revenue through taxation to produce the public good. Although we showed that a solution to the firm's problem k∗i can always be found, it is not obvious that we can always find a C1 solution ki∗ that conforms to conditions 1 and 2 of the government problem.
Given the importance of the term for this assumption, we define the total marginal productivity of capital at i (ompki) as. Equation (11) reflects the responsiveness (reduction) of a change in taxes with an accompanying change in the provision of the public good. Thus, the negativity of ompki means that the derivative of the production function with respect to capital, which (from A1) is a continuous function, is downward sloping.
For given r > 0, ti > 0 in the feasible set Fyi, there always exists at least one solution k∗i that satisfies both conditions 1 and 2 of the government problem. While companies take the level of the public good as given, governments take full account of the impact of providing the public good when making decisions about the level of provision and the level of tax required. Holding r constant, the government problem in jurisdiction i is then solved using the following first order condition;.
The assumption A5 introduced below ensures that this problem has a unique maximum.23 Implicit in the formulation of the government problem is the assumption that lump sum transfers are not available as a policy instrument. The uniqueness of the solution to the government problem is evident from the proof of Proposition 2 for the under- and over-provisioning cases.
4 Our Alternative Assumption and Equilibrium
Main Results and Intuition
Let policymakers be driven by an effective plan. given by tEi =yiE/kiE and tEj =yEj /kjE. denote taxes and the induced provision of public goods, given that ki∗ = kj∗ = k2 in a decentralized equilibrium. Symmetry in the following result means a solution where ki =kj, yi =yj, ti =tj, etc. The induced supply of the public good is yiE, yjE. The assumption that ˜ki < k/2 implies that diminishing returns to capital cause relatively low levels of capital utilization.
To exhaust the full range of possibilities, we must consider the situation where yiI < yiE and where yiE < yIi as well. Then there exists a decentralized symmetric internal equilibrium with t∗i =t∗j =t∗, k∗i =k∗j =k/2 and r∗, and the induced level of provision of the public good yi∗. Equilibrium can be characterized as follows: The intuition behind both of these results can be easily seen by using (13) to determine the incentive to deviate from the efficient solution.26 By assumption, the government predicts that in equilibrium∂f(ki∗, yi)/∂ki =r+ ti.
The incentive to deviate can then be determined directly from the relationship between yEi and yiI. IfyiE < yiI, because mpgvi decreases inyi, this implies that mpgvi >1 at yiE. 26We are grateful to an anonymous referee for suggesting the following presentation of our results.
Clearly, by the same logic, if mpgvi <1 at the efficient solution, then there is an incentive to deviate by lowering taxes, which would generate a race to the bottom. And if mpgvi = 1 at the efficient solution, there is no incentive to deviate from the efficient solution, so the equilibrium is efficient.
5 Examples
Then by Theorem 2 a race to the bottom occurs under Cobb-Douglas with diminishing returns to scale.30. The derivation, for Cobb-Douglas, ofyiEandyiI is carried out in exactly the same way, but is omitted as it is more straightforward. First, it is worth noting that the public good has no impact on the level of output for ki ≤ k.
The first step in characterizing efficiency is to solve for yiE, which can be done using (5). Since the analysis of theorems 1 and 2 essentially requires a comparison of symmetric solutions where ki =kj =k/2 = 1, it will be useful to solve for the value of pi where ki∗ = 1. We can now solve for a symmetric decentralized capital market equilibrium where k∗i = k∗j = 1 gives pi = (yi)12.
It will prove most convenient to solve for the tax level using the government's budget constraint; ten =yi/ki. To fully characterize Theorems 1 and 2 for this example, we must also solve for yIi. It is easy to see from (15) that yiE is decreasing in k, and from (17) that yiI is increasing in k.
As k is increased, this has the effect of increasing ∂f(ki, yi)/∂ki at any given value of ki >k, and therefore also increasing ∂2f(ki, yi)/∂ki∂yi. This is illustrated in Figure 6, which shows that 1−ki∂2f(ki, yi)/∂ki∂yi increases in yi and decreases in k.
6 Conclusions
For k > 12 we see that yIi > y∗i > yiE, the second case considered in Theorem 2; that of over-provision of the public good in equilibrium. This in turn requires a shift of resources away from public goods production at all levels of production, including efficiency. Intuitively, an increase in k has the effect of increasing the complementarity between the public good and capital, and consequently the incentive to deviate from efficiency is increased.
We use a version of the standard model where the public good enters the production function of firms. In the past literature, it has been assumed that the additional output gained by providing a public good through taxation is never as large as the opportunity cost in terms of tax revenue. Therefore, in the conventional setup, there is always a one-sided incentive to deviate from the efficient level of public goods provision by lowering taxes.
Under our alternative assumption, there may be a one-sided incentive to deviate upward, downward, or not at all from the efficient level of public goods. The analysis of this present paper suggests that in a situation where consumers value the public good provided in parallel with the valuation of public goods by companies, the conventional free rider problem may disappear completely under certain circumstances. In the state where the marginal cost to a jurisdiction of a unit of the public good equals its marginal value, another set of first-order conditions is satisfied.
Comparing these two points (and the intermediate value theorem) indicates whether the equilibrium outcome is efficient or inefficient by over- or under-provision of the public good. In our model, the relationship between these two sets of first-order conditions is determined by the degree of complementarity between capital and the public good, a feature that is most clearly illustrated in the example we present.
A Appendix
Then there exists a symmetric internal ~ decentralized equilibrium t∗i = t∗j = t∗, ki∗ = kj∗ = k/2 and r∗, and induced level of public good provision yi∗. If yIi < yiE then yiI < yi∗ < yiE and if yEi < yiI then yiE < yi∗ < yiI so the equilibrium outcome is inefficient, with under- (over-) provision of the public good. Proof of Theorem 2: The firm's profits are non-negative atki =k/2 since the production function is concave atki >˜k and, by assumption, k/2>˜k.
We will now show that while at yEi it is the case that dci/dti < 0, we have that at yiI it is the case that dci/dti > 0, so there must exist (by continuity) a point at which dci/dti = 0. Therefore, there must exist at least one point at which dci/dti = 0, and if there are more than one such point, then the one for which ci is larger is the global maximum.
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