Migration flows are determined by the the cost of moving and the utility differential between the two regions. People will only migrate if the cost of moving is less than or equal to the increase in utility from moving. Suppose that \(u_1(p_1) > u_2(p_2)\). Hence:
\[ f_{2, 1} \cdot c = u_1(p_1) - u_2(p_2) \\ \therefore \hspace{.2cm} f_{2, 1} = \frac{u_1(p_1) - u_2(p_2)}{c} \]
It has been shown that in the absence of any policy intervention, the iterative flows will arrive at a socially suboptimal equilibrium. Can this outcome be averted with a tax, and if so, what should the tax be set at to maximize social utility?
Consider a tax that does not need to be self-funding; \(t\) is paid to every individual in the lower utility region and paid by every individual in the higher utility region. Once again, assume that this period \(\ddot{u}_1(p_1, t) > \ddot{u}_2(p_2, t)\). The flow calculation thus becomes:
\[ \ddot{f}_{2, 1} = \frac{\ddot{u}_1(p_1, t) - \ddot{u}_2(p_2, t)}{c} \\ = \frac{(8 - p_1 - t) - (14 - p_2 + t)}{c} \\ = \frac{u_1(p_1) - u_2(p_2)}{c} - \frac{2t}{c} \\ = f_{2, 1} - \frac{2t}{c} \]
We see that for a given tax \(t\), flow between the lower utility and the higher utility regions is reduced, as the tax narrows the utility differential.
Social utility is then:
\[ \ddot{u}_{social} = p_1 \cdot \ddot{u}_1(p_1, t) + p_2 \cdot \ddot{u}_2(p_2, t) \\ = p_1 \cdot (8 - p_1 - t) + p_2 \cdot (14 - p_2 + t) \\ = p_1 u_1(p_1) + p_2 u_2(p_2) + (p_2 - p_1)t \]
We see from the above derivations that including a tax affects both the flow and the total social utility. The question remains how to set \(t\) such that it yields the socially optimal population levels in each region.
First, we must identify what the socially-maximizing steady state populations are. Then, using this information and the fact that the flows should be 0 between the two states in steady state, we can solve for \(t\). Finally, we verify that using this value of \(t\) does in fact lead to the socially optimal level of population in each region.
TODO
TODO
Social optimum
TODO