اولویت های زمان ناسازگار و امنیت اجتماعی: مروری در زمان پیوسته

İmrohoroğlu et al. (2003) prove that it is impossible in a three period partial equilibrium model for social security to improve the welfare of a naive quasi-hyperbolic agent if the program has a negative net present value. This paper first generalizes their impossibility theorem to a continuous time setting and then proves analytically that no discount function exists that can rationalize a social security program with a negative net present value.

Social security is commonly justified on the grounds that it prevents poverty among elderly individuals who were too shortsighted to save adequately during the working years. While this justification may seem sensible, it is problematic in the context of basic economic theory. It is well known among economists that, under basic assumptions, the standard life cycle permanent-income (LCPI) consumer with an exponential discount function cannot benefit from a social security program that has a negative net present value (below market internal rate of return), regardless of how impatient he may be. LCPI consumers will simply annuitize the loss to lifetime wealth across the entire life cycle, so that participation in such a program leads to a reduction in consumption in every period. Hence, a basic model where shortsightedness enters strictly through the exponential discount rate cannot rationalize a program with a negative net present value. Beginning with Feldstein (1985), economists have modified the LCPI model with a variety of alternative behavioral frictions in order to understand whether a social security program with a negative net present value may be rationalized under different specifications of shortsightedness.1 One of the leading alternatives to exponential discounting is of course hyperbolic discounting. Consumers with hyperbolic discount functions will reverse their preferences with the passage of time. A naive hyperbolic consumer standing at date 0 may intend to save part of his paycheck at date 1 for retirement, but when date 1 arrives he may not follow through with this plan. Unlike the impatient exponential consumer whose poverty in old age is all part of a grand plan first concocted when young, the hyperbolic consumer may unwittingly accumulate a suboptimal nest egg for retirement. At first glance, hyperbolic discounting with such time-inconsistent behavior seems like a prime setting for demonstrating the potential welfare improving role of a social security program, even one with a negative net present value. This paper revisits Section 3 of İmrohoroğlu et al. (2003), hereafter IIJ, which shows that in partial equilibrium, naive quasi-hyperbolic consumers who do not anticipate their own time inconsistency cannot benefit from a social security program with a negative net present value.2 They show that social security can only be welfare improving if it happens to cause distortions to factor prices in just the right way. I will call this the IIJ impossibility theorem, which they proved strictly for the special case of a three period model with quasi-hyperbolic preferences. The IIJ theorem is extremely important and surprising given the conventional wisdom that social security can be rationalized as a useful commitment device for hyperbolic discounters. My purpose is to show that their impossibility theorem generalizes to a continuous time life cycle setting and the theorem holds for any discount function, not just a quasi-hyperbolic function. I also entertain other utility functions beyond log utility and other “tax and transfer” schemes beyond social security. Therefore, this paper strengthens the well-known IIJ impossibility theorem in three important ways: (i) The result in IIJ is not just an anomaly that comes from the quasi-hyperbolic approximation to a hyperbolic function. Nor is it even an anomaly relating to hyperbolic discounting in general. Discounting, in any form, cannot be used to rationalize a social security program with a negative net present value. (ii) The IIJ result does not depend on the coarseness of the time grid (three periods) since the present paper is cast in continuous time. 3 (iii) The IIJ result can be generalized to other well-known utility functions beyond log utility (CRRA, CARA, and quadratic) and it can be generalized to any tax and transfer scheme, in addition to social security, that rearranges the timing and magnitude of cash flows across the life cycle.

This paper generalizes IIJ's well-known impossibility theorem to continuous time, which shows it is impossible for naive quasi-hyperbolic consumers (in partial equilibrium) to benefit from participating in a social security program with a negative net present value. I go beyond the IIJ theorem to show that there is no discount function (be it quasi-hyperbolic, hyperbolic, or anything else) that can be used to rationalize a social security program with a negative net present value. I also generalize the IIJ theorem to a variety of common utility functions and to any tax and transfer scheme that rearranges the timing and magnitude of the disposable income flow.