## Concentrating economic activity in London

The Scottish Government released its paper, ‘Scotland’s economy: the case for independence‘ on Tuesday 21st May. Amongst other things, the paper argues that the union damages Scotland’s economic performance by deciding “to concentrate so much economic activity in London and the South-East of England”. Economic methodology is sometimes described as seeing something that happens in practice and checking that it can happen in theory. In this vein, can it be true that an optimising UK social planner could damage Scottish welfare? In this post I examine this in the context of an optimal investment problem, and conclude that there is a region in the parameter space of the model in which the smaller region would choose independence to enhance welfare. This is not to say that the Scottish Government are correct to claim that any economic concentration is suboptimal from the view of Scottish citizens, but it does show that this is an economic phenomenon that certainly can exist.

The decision maker(s)

Within a union, the decision maker is assumed to maximise the social welfare function across all citizens of the union, whereas with two independent states there are two decision makers who each maximise the welfare of their own citizens. The classic problem that a union can overcome is a form of the Prisoners’ Dilemma in which strategic considerations lead to a suboptimal allocation with two states, but an optimal allocation with a single decision maker. The best example of this is perhaps corporate taxation which, say, may be optimally levied at 30%, and so this is the rate chosen by the decision maker in a union. However, under independence, the best strategy may be to undercut your neighbours and so equilibrium rates may be much lower, which from an aggregate welfare perspective produces a suboptimal allocation. This is clearly an argument that favours union, can an argument be made in this framework that favours independence?

The model

If there are fixed costs to creating infrastructure in each region, then it may be more efficient to only build the infrastructure in one of the regions. If there are costs associated with not being co-located with this infrastructure, then even though aggregate welfare is maximised under union, the citizens of the smaller region may favour independence and local infrastructure.

Formally, consider a two period, two region model with populations $L_1 > L_2$. In the first period a linear technology $Y_i = L_i$ is used in both regions, and citizens consume $C_i = (1-\tau_i)Y_i$ where the tax raised (at rate $\tau_i$) is used to fund capital investment for use in the second period. Since region 1 is bigger, it will always be optimal to build there to minimise the travel costs. If capital is also built locally then region 2 consumption in the second period is $C_2 = \phi K_{2}^{\alpha} L_2^{1-\alpha}$ and if capital is only built in region 1 then region 2 second period consumption is $C_2 = \psi K_{12}^{\alpha} L_2^{1-\alpha}$ (where $\alpha$ is the output elasticity of capital, $\phi$ is the relative productivity of the new technology, and $1 - \frac{\psi}{\phi}$ is the travel cost). If capital is to be built in both regions then the quantity optimally built in each region will be $K_i = \left( \beta \alpha \phi L_i^{1-\alpha} \right)^{\frac{1}{1-\alpha}}$, and if capital is only to be built in region 1 then the quantity that is optimally built is $K_1 = K_{11} + K_{12}=\left( \beta \alpha \phi L_1^{1-\alpha} \right)^{\frac{1}{1-\alpha}}+\left( \beta \alpha \psi L_2^{1-\alpha} \right)^{\frac{1}{1-\alpha}}$ (where $\beta$ is the discount factor). The budget constraint under union is $\tau_1 = \tau_2 = \tau$ such that $\tau(L_1+L_2) = K_1 + F$ if only investing in region 1, or $\tau(L_1+L_2) = K_1 + K_2 + 2F$ if investing in both regions, where $F$ are fixed costs of creating capital. Under independence, the budget constraints are $\tau_i L_i = K_i + F$. Welfare in region 2 is always $W_2 = (1-\tau_2)L_2 + \beta C_2$.

Results

This model produces results like:

As the travel cost goes to zero, it is always best to save on fixed costs and just build in the larger location. Both parties are happier under union. As travel costs get very large, infrastructure is optimally built in both locations, and a social planner is indifferent (due to linear preferences) between independence or union. Under union in this part of the parameter space, the smaller region is likely to gain at the expense of the larger region since some of the fixed costs incurred creating the infrastructure in the smaller region will be paid for by taxpayers in the larger region (since we assume taxes are levied at a common rate). For intermediate travel costs, the union level social planner prefers union and maximises aggregate welfare by only building infrastructure in the larger region, but the citizens of the smaller region would prefer independence.

Conclusions

This simple example shows that it is possible for one region to lose out even when governed by a benevolent social planner who maximises aggregate welfare. This phenomenon could be compounded by spillover effects and agglomeration returns, as well as by complementarities between public and private investment. It could be further compounded by the decision maker not being a benevolent social planner but rather being a politically calculating agent who sees advantage in favouring one region over another – but that’s another model! This phenomenon may not pertain however (it was only manifest in a specific region of the parameter space) and it is mitigated by any curvature in the social welfare function of the benevolent social planner, as well as by the ability of unions to mitigate against other effects like the prisoners’ dilemma political economy failure discussed above.

Perhaps this model could be used in a positive manner to explain the rise in support for Scottish independence? It seems reasonable to imagine that fixed costs associated with investment have risen, and that travel costs have fallen over the past, say, century. In this case, the model would predict that previously Scotland was treated, in investment terms within the union, as if it were independent. This story is consistent with the ‘internal independence’ discussed by Michael Lynch in the ‘Moulding of modern Scotland’ chapter of his book ‘Scotland: A New History’ (1990). As fixed costs of investment have risen, and travel costs have fallen, a welfare maximising government in London would rationally choose to begin centralising investment in the populous South-East. The model then predicts that, despite this choice being the aggregate welfare maximising decision, Scots could begin to recognise that their interests would be best served under independence.

Whether this narrative holds or not, it is certainly possible for proponents of independence to (implicitly) use a mechanism like that presented in this model to make a normative case for independence on the basis that Scotland does indeed lose out under union due to the decision to concentrate economic activity in London.