Let's suppose that I want to maximize total welfare (Social Planner Problem) using lump sum transfers to individuals.

It's known then that the price ratio changes relative to when there are no lump sum transfers. Total income and total endowments are fixed in the model, only distribution of initial endowments is changed between the individuals by planner.

My Question is :

Why do lump sum transfers affect the price ratio in the social planners problem?


Prices (and hence price ratios) carry information about what people want, and what is supplied.

You're only changing initial endowments, hence you're not affecting what is supplied. However, as you change the initial distribution, you change what each individual wants, and possibly (not necessarily) also what the economy as an aggregate wants. This change in wants and haves is reflected in prices.


I find that @FooBar already gave the meaningful answer, all mine aims to do is provide a more technical perspective.

Short answer:
Because given the equilibrium price the income offer curves of the consumers do not perfectly complement each other. If the income of some increase while those of the others decrease the total change in demand is not zero. Given such a change no equilibrium belongs to old equilibrium price ratio.

Long answer:
Before addressing your question let us think about homothetic preferences a bit. A property of homothetic preferences is that if you move radially out of the origin all indifference curves that you intersect will have the same slope at the point of intersection. This means that if a consumer has homothetic preferences then any change in her income/value of her initial endowment will result in a proportional change in her consumption if prices are fixed. Another way of saying this is that the income offer curve is linear. Linear and Cobb-Douglas preferences are homothetic, quasilinear preferences are not.

Now to your question:
Suppose all consumers in your model had homothetic preferences. We want to test whether the current equilibrium price would also provide an equilibrium if a lump sum transfer was made. By making a transfer the social planner increases the wealth of some consumers and decreases the wealth of others. Because of homothetic preferences he also makes a proportional change in their consumption. The price ratio will only result in equilibrium if these proportional changes balance each other out perfectly so there is no change in aggregate demand. This is needed because the quantity of goods to be consumed is fixed. (Even if you allow for production because the price ratio is the same.) In terms of an Edgeworth box this would mean that the set of Pareto-optimal points is exactly the diagonal and the two consumers have the same income offer curve, which is again the diagonal.

Now that we have seen what the problem is let us lose homothetic preferences. Still considering an Edgeworth box, lump sum transfers would not change the price ratio if and only if given the current price ratio both consumers have an income offer curve that coincides with the contract curve. Without homothetic preferences this can be non-linear and since the consumers are in different corners they are likely not identical: the income offer curve coincide with the contract curve in each consumer's own perspective. This condition is not fulfilled given generic utility functions (even if we severly restrict the class of utility functions considered).


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