In this exercise we explore some technical aspects of general equilibrium theory in exchange economies and Robinson Crusoe economies. Unlike in other problems, parts A and B are applicable to both those focused on A-Section material and those focused on B-Section material. Although the insights are developed in simple examples, they apply more generally in much more complex models.
A: The role of convexity in Exchange Economies: In each part below, suppose you and I are the only individuals in the economy, and pick some arbitrary allocation E in the Edgeworth Box as our initial endowment. Assume throughout that your tastes are convex and that the contract curve is equal to the line connecting the lower left and upper right corners of the box.
(a) Begin with a depiction of an equilibrium. Can you introduce a non-convexity into my tastes such that the equilibrium disappears (despite the fact that the contract curve remains unchanged?)
(b) True or False: Existence of a competitive equilibrium in an exchange economy cannot be guaranteed if tastes are allowed to be non-convex.
(c) Suppose an equilibrium does exist even though my tastes exhibit some non-convexity. True or False: The first welfare theorem holds even when tastes have non-convexities.
(d) True or False: The second welfare theorem holds even when tastes have non-convexities.
B: The role of convexity in Robinson Crusoe Economies: Consider aRobinson Crusoe economy. Suppose throughout that there is a tangency between the worker’s indifference curve and the production technology at some bundle A.
(a) Suppose first that the production technology gives rise to a convex production choice set. Illustrate an equilibrium when tastes are convex. Then show that A may no longer be an equilibrium if you allow tastes to have non-convexities even if the indifference curve is still tangent to the production choice set at A.
(b) Next, suppose again that tastes are convex but now let the production choice set have non convexities. Show again that A might no longer be an equilibrium (even though the indifference curve and production choice set are tangent at A).
(c) True or False: A competitive equilibrium may not exist in a Robinson Crusoe economy that has non-convexities in either tastes or production.
(d) True or False: The first welfare theorem holds even if there are non-convexities in tastes and/or production technologies.
(e) True or False: The second welfare theorem holds regardless of whether there are non-convexities in tastes or production.
(f) Based on what you have done in parts A and B, evaluate the following: “Non-convexities may cause a non-existence of competitive equilibria in general equilibrium economies, but if an equilibrium exists, it results in an efficient allocation of resources. However, only in the absence of non-convexities can we conclude that there always exists some lump-sum redistribution such that any efficient allocation can also be an equilibrium allocation.”