# Why solving the model for a benevolent social planner gives an efficient allocation?

Probably this question is really dumb - and I'm sorry if it is - but why solving the model for a benevolent social planner, gives an efficient allocation?

What's the criteria for this efficiency?

I'm asking this because in several NK models in macro, when we're comparing the 'desirability' of simple rules, we take as efficient the allocation obtained from solving the model as stated above.

Any help would be appreciated.

I think the general proof is in this paper.

Thinking in a simple exchange Edgeworth Box, a competitive economy involves prices, which mediate the relationship between consumers relative valuation of goods - the marginal rate of substitution. A (benevolent) social planner does not need those prices, as goods will not be traded in the market, but simply allocated to whom it "corresponds", based on the same optimality conditions. Such equilibrium is simply found by equalising consumers marginal rate of substitution.

In other words:

• equilibrium in competitive economy through exchange/markets:

$$MRS^1_{X,Y} = \frac{P_x}{P_y} = MRS^2_{X,Y}$$

• equilibrium via allocation from benevolent social planner:

$$MRS^1_{X,Y} = MRS^2_{X,Y}$$

(the formulas above can be trivially extended to $n$ consumers and goods)

• @Anoldmaninthesea. You may also want to have a look at MWG Section 16.E and F for a textbook treatment of this. – Theoretical Economist Oct 20 '17 at 11:19
• @TheoreticalEconomist Thanks for the references – An old man in the sea. Oct 24 '17 at 22:48
• This is good information, but isn't an answer per se, because you can go from a benevolent social planner framework to a perfectly competitive framework with prices and achieve the same distributional outcomes. – heh Nov 14 '19 at 16:30

At least in environmental and resource economics it is because it internalizes all externalities. If there is only a single person to consider (and the payoff of this person is "made up" of the sum of individual welfare) than by maximizing the sum all externalities get accounted for.

Actually, the answer is even more simple than what's been given. The social planner problem is just an optimization problem. It's just math. Under some mathy conditions, it has a solution. The key question isn't "why is this optimal", it's "what am I optimizing, and why?"

The benevolent social planner problem is characterized by a particular social welfare function that is optimized in the NK framework, partly because of its mathematical tractability but also (I think) because of some implied value judgments in the field. But it's not the only social welfare function out there, and much of politics in society could be understood as a debate over what the "right" social welfare function actually is.