Problem 15 - Advanced Macroeconomics

Question

The common-pool problem in government spending. (Weingast, Shepsle, and Johnsen, \(1981 .\) ) Suppose the economy consists of \(M>1\) congressional districts. The utility of the representative person living in district \(i\) is \(E+\) \(V\left(G_{i}\right)-C(T) . E\) is the endowment, \(G_{i}\) is the level of a local public good in district \(i,\) and \(T\) is taxes (which are assumed to be the same in all districts). Assume \(V^{\prime}(\bullet)>0, V^{\prime \prime}(\bullet)<0, C^{\prime}(\bullet)>0,\) and \(C^{\prime \prime}(\bullet)>0 .\) The government budget constraint is \(\sum_{i=1}^{N} G_{i}=M T\). The representative from each district dictates the values of \(G\) in his or her district. Each representative maximizes the utility of the representative person living in his or her district. (a) Find the first-order condition for the value of \(G_{j}\) chosen by the representative from district \(j,\) given the values of \(G_{i}\) chosen by the other representatives and the government budget constraint (which implies \(T=\left(\sum_{l=1}^{M} G_{i}\right) / M\). CNote: Throughout, assume interior solutions.) (b) Find the condition for the Nash equilibrium value of \(G\). That is, find the condition for the value of \(G\) such that if all other representatives choose that value for their \(G_{i},\) a given representative wants to choose that value. (c) Is the Nash equilibrium Pareto-efficient? Explain. What is the intuition for this result?

Step-by-Step Solution

Answer

(a) \(V'(G_j) = \frac{1}{M} C'(T)\). (b) \(V'(G) = \frac{1}{M} C'(G)\). (c) No, Nash equilibrium is not Pareto-efficient as it results in under-provision of public goods.

Step 1: Understand the Utility Function

The utility function for a representative person is given by \(U(E, G_i, T) = E + V(G_i) - C(T)\). Here, \(E\) is a constant (endowment), \(V(G_i)\) is the utility from local public goods, and \(C(T)\) is the disutility from taxes. The function has the following properties: \(V'(G_i) > 0, V''(G_i) < 0, C'(T) > 0, C''(T) > 0\). This means that utility from public goods has diminishing returns, and disutility from taxes increases at an increasing rate.

Step 2: Government Budget Constraint

The government budget constraint requires that the sum of public goods across all districts equals the total taxes collected, i.e., \(\sum_{i=1}^{M} G_i = M T\). Therefore, taxes can be expressed as \(T = \frac{1}{M} \sum_{i=1}^{M} G_i\).

Step 3: Maximizing Utility for District j

The representative from district \(j\) chooses \(G_j\) to maximize \(U = E + V(G_j) - C\left( \frac{1}{M} \sum_{i=1}^{M} G_i \right)\). Differentiating with respect to \(G_j\), and considering the effect of \(G_j\) on taxes, we get the first-order condition as \(V'(G_j) = \frac{1}{M} C'(T)\).

Step 4: Nash Equilibrium Condition

In Nash equilibrium, each representative chooses a level of \(G_j\) such that no representative has an incentive to deviate, given that all other representatives choose the same level \(G = G_i\) for their district. Thus, the Nash equilibrium condition is \(V'(G) = \frac{1}{M} C'(\frac{M}{M}G)\). Simplifying, the condition becomes \(V'(G) = \frac{1}{M} C'(G)\).

Step 5: Assessing Pareto Efficiency

Pareto efficiency requires maximizing the overall welfare, considering the utility of all districts equally. In a Pareto-efficient allocation, \(V'(G) = C'(G)\) would hold. However, the Nash equilibrium condition \(V'(G) = \frac{1}{M} C'(G)\), implies under-provision of public goods compared to the Pareto-efficient level because the marginal cost is distributed across all districts, leading to lower provision of public goods.

Key Concepts

Government Budget Constraint

The government budget constraint is a crucial concept in understanding how public spending is balanced against tax revenue. In this model, we look at an economy with multiple congressional districts, each having its local public goods. The constraint requires the total level of these goods across all districts to sum up to the total collected taxes. This constraint can be mathematically represented as

• \( \sum_{i=1}^{M} G_i = M T \)

Here, \(G_i\) is the public good in district \(i\), and \(T\) represents taxes collected, shared equally among districts.
As much as districts may desire to maximize their local benefits, this constraint ensures a balanced national budget. Taxes are calculated as

• \( T = \frac{1}{M} \sum_{i=1}^{M} G_i \)

This implies that any increase in public goods in one district affects the tax shared across all districts. Keeping this balance is crucial for sustainable development and public approval.

Nash Equilibrium

Nash Equilibrium is a fundamental concept in game theory that applies to this problem. It describes a situation where each representative of a district optimizes their local public good, given the decisions of others. No representative gains by unilaterally changing their decision.
In the exercise, Nash Equilibrium is achieved when:

• \( V'(G) = \frac{1}{M} C'(G) \)

This condition ensures that each district's representative is choosing a level of public good that aligns with others’ choices, balancing the marginal utility and the shared tax cost.
This equilibrium often leads to under-provision of public goods because representatives account for only a portion of the overall tax burden. The structure incentivizes districts to benefit from public goods while sharing costs with others, a classic demonstration of the 'tragedy of the commons'.

Pareto Efficiency

Pareto Efficiency is about optimizing resources without making others worse off. It's the ideal state where no one can be better off without someone else being worse off.
In the context of this exercise, Pareto efficiency would mean that

• \( V'(G) = C'(G) \)

This would ensure that the marginal benefit of public goods equals their marginal cost. However, the Nash Equilibrium does not guarantee Pareto efficiency. Instead, it produces

• \( V'(G) = \frac{1}{M} C'(G) \)

indicating under-provision of goods. Since each district considers only a fraction of the cost, it skews the overall welfare optimization. This illustrates a critical outcome of decentralized decision-making in public goods allocation.

Local Public Goods

Local public goods are goods provided within a specific geographical area, benefiting only residents of that area.
Each district in the problem is responsible for its local public good, \(G_i\), which directly affects the utility of its residents. Local public goods are essential for providing services that cater to the specific needs of a community, such as parks, roads, and schools.
Such goods demonstrate characteristics of non-excludability and non-rivalry limited to the local population. They provide utility to the residents and are paid for through shared district taxes, spreading the burden of financing them across an entire economy.

• Each district aims to balance their local needs with the acknowledgment of a shared national budget.

This responsibility often leads to tension between maximizing local utility and adhering to national fiscal responsibilities.