1. Find the efficient amount of public good G*. 2. Now suppose agents are privately deciding how much to contribute to the public good. Find the reaction functions of the agents, R... 1. Find the efficient amount of public good G*. 2. Now suppose agents are privately deciding how much to contribute to the public good. Find the reaction functions of the agents, Ri(gj). That is, for any contribution gj, find the optimal contribution of agent i. 3. What are the (Nash) equilibrium contributions to the public good? Is the private provision of the public good efficient? 4. In a graph, show the reaction functions and the equilibrium point.
Understand the Problem
The question outlines a scenario involving two agents contributing to a public good and presents multiple tasks related to finding the efficient amount of the public good, the reaction functions, and equilibrium contributions. Each task requires analysis of mathematical relationships and economic concepts.
Answer
The efficient amount is \(G^*\), with Nash equilibrium contributions determined by reaction functions \(R_1\) and \(R_2\).
Answer for screen readers
The efficient amount of public good is (G^*). The Nash equilibrium contributions are determined by solving the simultaneous equations for (g_1) and (g_2).
Steps to Solve
- Find the Efficient Amount of Public Good (G^*)
To find the efficient provision of the public good, we need to maximize the total utility of both agents. The utility functions are given by:
$$ u_1(G, x_1) = b_1 \ln(G) + x_1 $$
$$ u_2(G, x_2) = b_2 \ln(G) + x_2 $$
The total utility (U) is:
$$ U = u_1(G, x_1) + u_2(G, x_2) = b_1 \ln(G) + b_2 \ln(G) + x_1 + x_2 $$
Since (G = g_1 + g_2), we set (x_1 = \omega_1 - g_1) and (x_2 = \omega_2 - g_2) for the goods consumed:
To find the efficient amount (G^*), we maximize (U) with respect to (G):
- First-Order Condition for Efficiency
Taking the derivative of (U) with respect to (G) and setting it to zero gives:
$$ \frac{\partial U}{\partial G} = \frac{b_1}{G} + \frac{b_2}{G} = 0 $$
This implies:
$$ G^* = \frac{b_1 + b_2}{b_1 + b_2} \Rightarrow G^* \text{ is efficient when the sum of the marginal contributions equals zero.} $$
- Finding Reaction Functions (R_i(g_j))
To find the reaction functions, we need to maximize each agent's utility given the contribution of the other agent. For agent 1, the maximization problem is:
$$ \max_{g_1} U_1 = b_1 \ln(G) + (\omega_1 - g_1) $$
Substituting (G = g_1 + g_2):
$$ U_1 = b_1 \ln(g_1 + g_2) + \omega_1 - g_1 $$
Taking the derivative and setting to zero:
$$ \frac{b_1}{g_1 + g_2} - 1 = 0 \Rightarrow R_1(g_2) = \frac{b_1}{1} - g_2 $$
Similarly, for agent 2:
$$ U_2 = b_2 \ln(g_1 + g_2) + \omega_2 - g_2 $$
Taking the derivative:
$$ \frac{b_2}{g_1 + g_2} - 1 = 0 \Rightarrow R_2(g_1) = \frac{b_2}{1} - g_1 $$
- Equilibrium Contributions
The Nash equilibrium occurs where each agent's reaction functions intersect. Set (R_1(g_2) = g_1) and (R_2(g_1) = g_2) and solve the simultaneous equations.
- Graphing Reaction Functions
In a graph, plot (g_1) on the x-axis and (g_2) on the y-axis. The reaction functions will show the optimal contributions of each agent, and the intersection point represents the Nash equilibrium.
The efficient amount of public good is (G^*). The Nash equilibrium contributions are determined by solving the simultaneous equations for (g_1) and (g_2).
More Information
Public goods have unique characteristics, particularly that they are non-excludable and non-rivalrous, leading to underprovision in private markets. The equilibrium reflects this challenge.
Tips
- Confusing utility maximization with profit maximization.
- Forgetting to substitute (G = g_1 + g_2) correctly in utility functions.
- Not checking the second-order conditions to ensure maximum utility.
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