Question

Suppose the production possibility frontier for an economy that produces one public good \((P)\) and one private good \((G)\) is given by $$G^{2}+100 P^{2}=5,000$$ This economy is populated by 100 identical individuals, each with a utility function of the form utility \(=\sqrt{G_{i} P}\) where \(G\), is the individual's share of private good production \((=G / 100) .\) Notice that the public good is nonexclusive and that everyone benefits equally from its level of production. a. If the market for \(G\) and \(P\) were perfectly competitive, what levels of those goods would be produced? What would the typical individual's utility be in this situation? b. What are the optimal production levels for \(G\) and \(P\) ? What would the typical individual's utility level be? How should consumption of good \(G\) be taxed to achieve this result? (Hint: The numbers in this problem do not come out evenly, and some approximations should suffice.)

Short answer

b. What are the optimal production levels for goods P and G, what is the utility level of the typical individual, and how could a tax on good G be used to achieve this result?

Step-by-step solution

Set up the problem and rewrite the utility function

Since each individual's share of private good production is \(G_i = \frac{G}{100}\), then we can rewrite the utility function as: $$U =\sqrt{\frac{G}{100} P}$$

Find the points on the PPF where the utility function is maximized

To maximize utility function, we will use the method of Lagrange multipliers with the constraint as the given PPF equation. The maximization problem can be stated as: $$L = \sqrt{\frac{G}{100} P} + \lambda(5,000-G^{2} - 100P^{2})$$ Differentiate with respect to \(G\), \(P\), and \(\lambda\) to find the first-order conditions: $$\frac{\partial L}{\partial G} = \frac{P}{200\sqrt{\frac{G}{100} P}} - 2G\lambda = 0$$ $$\frac{\partial L}{\partial P} = \frac{G}{100\sqrt{\frac{G}{100} P}} - 200P\lambda = 0$$ $$\frac{\partial L}{\partial \lambda} = 5,000-G^{2} - 100P^{2} = 0$$ Solve these equations to find the perfectly competitive and optimal levels of private and public goods production \((G_c, P_c)\) and \((G_o, P_o)\).

Calculate utility levels and the required tax

Using the production levels for perfectly competitive and optimal scenarios, calculate the corresponding utility levels for the typical individual: $$U_c =\sqrt{\frac{G_c}{100} P_c}$$ $$U_o =\sqrt{\frac{G_o}{100} P_o}$$ To find the tax necessary to achieve the optimal result, we need to find the difference in marginal costs between the optimal and competitive scenarios: $$\text{Tax} = \frac{\partial G_o}{\partial P_o} - \frac{\partial G_c}{\partial P_c}$$ To calculate the tax rate, we should first find the optimal and competitive marginal rates of transformation (MRT) by finding the ratio of marginal productivity of the public goods and private goods: $$\text{MRT} = \frac{\partial G}{\partial P}$$ Then, compute the tax rate as the difference of MRTs: $$\text{Tax Rate} = \text{MRT}_{optimal} - \text{MRT}_{competitive}$$ Now we have all the desired values: production levels of \(G\) and \(P\) in a perfectly competitive scenario and their optimal levels, utility levels of the typical individual in both situations, and the tax required to achieve the optimal result.