Government policies to achieve efficient resource allocation and correct market failure — practice questions

(a) Free-Market Equilibrium:

• Step 1: Find the equilibrium quantity by setting MPC = MPB. $10 + Q = 70 - Q$ $2Q = 60$ $Q_{market} = 30$ units.

• Step 2: Find the market price. Substitute $Q=30$ into the MPB equation: $P_{market} = 70 - 30 = \text{\textdollar}40$.

(b) Socially Optimal Equilibrium:

• Step 1: Find the Marginal Social Cost (MSC). $MSC = MPC + MEC$. $MSC = (10 + Q) + 20 = 30 + Q$.

• Step 2: Find the optimal quantity by setting MSC = MPB. (Assuming no external benefits, MPB = MSB). $30 + Q = 70 - Q$ $2Q = 40$ $Q_{optimal} = 20$ units.

• Step 3: Find the socially optimal price. This is the price on the demand curve (MPB) at the optimal quantity: $P_{optimal} = 70 - 20 = \text{\textdollar}50$.

(c) Pigouvian Tax:

• Step 1: Determine the tax rate. The tax should equal the MEC, which is a constant $20 per unit.

• Step 2: Verify the outcome. The tax shifts the MPC curve up by $20. The new private cost is $MPC_{taxed} = MPC + Tax = (10 + Q) + 20 = 30 + Q$. This is identical to the MSC curve. The new equilibrium where $MPC_{taxed} = MPB$ correctly yields $Q = 20$.

• Step 3: Calculate total tax revenue. $\text{Tax Revenue} = \text{Tax per unit} \times Q_{optimal}$ $\text{Tax Revenue} = \text{\textdollar}20 \times 20 = \text{\textdollar}400$.

(a) Finding Equilibria:

• Step 1: Free-market equilibrium (MPB = MSC). We set the private benefit equal to the cost: $120 - Q = 40$ $Q_{market} = 80$ vaccinations.

• Step 2: Socially optimal equilibrium (MSB = MSC). First, find the marginal social benefit: $MSB = MPB + MEB$. $MSB = (120 - Q) + 20 = 140 - Q$.

• Now set social benefit equal to social cost: $140 - Q = 40$ $Q_{optimal} = 100$ vaccinations. The market under-provides by 20 units.

(b) Designing the Subsidy:

• Step 1: Determine the subsidy amount. The optimal per-unit subsidy equals the MEB at the socially optimal quantity. Since MEB is constant at $20, the subsidy should be $20 per vaccination.

• Step 2: Verify the outcome. A $20 subsidy given to consumers shifts their MPB curve up by $20, making it identical to the MSB curve ($MPB_{subsidised} = 140 - Q$). The new equilibrium is where $MPB_{subsidised} = MSC$, so $140 - Q = 40$, which gives $Q = 100$. The policy achieves the optimal quantity.

• Step 3: Calculate the total cost. The total government expenditure on the subsidy is the subsidy per unit multiplied by the new, socially optimal quantity. $\text{Total Cost} = \text{Subsidy per unit} \times Q_{optimal}$ $\text{Total Cost} = \text{\textdollar}20 \times 100 = \text{\textdollar}2000$.