Spence s model of Signaling in the job market

Transcription

1 Department of Economics, Universit of California, Davis Ecn 00C - Micro Theor - Professor Giacomo Bonanno Spence s model of Signaling in the job market I. Background. The Arrow-Debreu model of a perfectl competitive econom is based on the crucial assumption of complete markets, that is, the assumption that there exists a market (i.e. a price) for each good, for each location, for each future date and for each possible state of nature. The presence of a complete set of forward markets for future contingent deliver eliminates uncertaint since it allows complete insurance. It is sometimes claimed that the properties of the Arrow-Debreu model (in particular, the results that a competitive equilibrium exists and that all competitive equilibria are Pareto efficient) depend cruciall on the assumption of perfect information. This is not correct: it was proved b Radner in 968 that - as long as the assumption of complete markets is retained - the presence of asmmetric information (some agents are more informed than others) is not an obstacle to the proof that a competitive equilibrium exists and is Pareto efficient. The novelt of the market signaling literature, therefore, does not lie in the introduction of the hpothesis of asmmetric information, but rather in considering more complex situations where the set of signals is larger than that consisting of prices onl. Models like Spence s are ver limited in their scope and are not comparable to the Arrow-Debreu model, but their relevance lies in the fact that the show that the introduction of more complex (and closer to realit) informational assumptions ma destro one of the basic results of the Arrow-Debreu model: the Pareto efficienc of equilibria. A signaling equilibrium ma not be Pareto efficient. II. Education as a signal. The situation considered b Spence, tpified b the job market, is one where some relevant information is available to onl one side of the market (the potential emploees), while the other side of the market (the emploer) has to tr to infer this information from some observable characteristics. Spence assumes that the population is divided in two groups: those with low productivit and those with high productivit. The emploer does not know in advance whether a given applicant belongs to one group or the other and it will take some time before the true productive abilit of the emploee is revealed. It is therefore in the emploer s interest to tr to guess the applicant s productive abilit on the basis of some characteristics that the emploer himself can observe. Those observable characteristics which are under the control of the applicant (like, the wa he dresses, etc.) are called signals, whereas those characteristics which cannot be modified b the applicant (e.g. sex, race, etc.) are called indexes. It is clear that not ever observable characteristic will be considered relevant b the emploer. For example, the color of the applicant s ees (an index) or the color of his socks (a signal) are unlikel to be considered important b the emploer, while the level of is likel to be considered relevant. Therefore, within the set of potential signals the emploer will select those that - in the light of her previous experience - seem to be correlated with productive capabilities. This selection represents the beliefs of the emploer. Suppose the emploer believes that is positivel correlated with productivit. Then she will offer a higher salar to those applicants who have acquired more. We therefore have a first flow of information, Page of 9

2 from the emploer to the prospective emploees: b differentiating salaries on the basis of some characteristics the emploer reveals her beliefs. Emploer s initial beliefs Selection of signals within the set of all potential signals Emploees read the emploer s beliefs in the wage schedule she offers Let us now turn to the other side of the market: potential emploees. A potential emploee will face the problem of communicating his unobservable qualities to the emploer. The onl wa he can do this is b using some observable characteristic that can be modified b him, that is, b using a signal. In choosing a signal the potential emploee must take into account two factors: ) signaling costs, that is, the cost of acquiring that particular signal; ) the emploer s beliefs, revealed b the wage schedule (it would be a waste of mone and/or effort to acquire a signal which is known to be considered irrelevant b the emploer, i.e. a characteristic to which the emploer does not pa attention). We therefore have a second flow of information, from the applicant to the emploer: At this point the wage contract will be stipulated and after a while the emploer will be able to observe the emploee s productivit and to relate it to the signal chosen b the emploee. The emploer s beliefs can then be confirmed (if, for example, the more educated emploees are more productive) or falsified (if the more productive are the less educated emploees, or if there is no difference in productivit between more and less educated emploees). In the first case the emploer will not modif her (beliefs and) offered wage schedule, while in the second case she will. We define the first case to be a signaling equilibrium. That is, a signaling equilibrium is a situation in which emploers beliefs about the relationship between productivit (which cannot be observed at the time of hiring) and (or an other signal) are confirmed b the results of her hiring in the market. Page of 9

3 Thus in equilibrium emploers beliefs are self-confirming: the initial beliefs, based on some data (past experience), generate new data which does not contradict them and hence the beliefs tend to persist. The simplest version of Spence s model is based on two main assumptions, one of which is crucial and seems to be necessar for the phenomenon of signaling to take place, whereas the second can be relaxed without affecting the qualitative nature of the results. CRUCIAL ASSUMPTION. The crucial assumption is that signaling costs are negativel correlated with productive abilit. That is, people with higher productivit incur lower costs of acquiring (or the alternative relevant signal) than those with lower productivit. B costs of acquiring we do not necessaril mean monetar costs: it could simpl be effort (e.g. measured in number of hours of stud necessar to learn a particular thing). RELAXABLE ASSUMPTION. The second assumption, which is not essential, is that does not affect productivit. Spence himself developed a model in which this assumption is relaxed. MAIN RESULTS. The two main results of the model are: () there ma be multiple equilibria (and even an infinite number of them); () equilibria ma be (and in general are) Pareto inefficient. The second result needs some explanation. First of all, it can easil be understood wh the inefficienc occurs. Since the main hpothesis is that of asmmetric information it is not legitimate to compare the signaling equilibrium with a perfect information equilibrium. So the benchmark is represented b the situation where no signaling takes place and emploers -- not being able to distinguish between more productive and less productive applicants and not having an elements on which to base a guess -- offer the same wage to ever applicant, equal to the average productivit. Call this the non-signaling equilibrium. In a signaling equilibrium (where emploers beliefs are confirmed, since less productive people do not invest in, while the more productive do) everbod ma be worse off than in the non-signaling equilibrium. This occurs if the wage offered to the non-educated is lower than the average productivit (= wage offered to everbod in the non-signaling equilibrium) and that offered to the educated people is higher, but becomes lower (than the average productivit) once the costs of acquiring are subtracted. The possible Pareto inefficienc of signaling equilibria is a strong result and a worring one: it means that societ is wasting resources in the production of. However, it is not per se enough to conclude that (i.e. the signaling activit) should be eliminated. The result is not that, in general, elimination of the signaling activit leads to a Pareto improvement: Spence simpl pointed out that this is a possibilit. What is a general result is that within the set of possible equilibria (recall that multiple equilibria represent the rule and not the exception) some are Pareto superior to others. That is, a Pareto ranking of the signaling equilibria is normall possible. This is a strong result, because it goes against one of the main results of the Arrow-Debreu model. In the Arrow-Debreu model multiple equilibria are possible, but each equilibrium is Pareto efficient and therefore equilibria are not Pareto comparable. Page 3 of 9

4 THE MODEL Suppose that there are two groups of individuals (where 0 < q < ): Group L Group H Marginal productivit = Marginal productivit = Proportion in population: q Proportion in population: q Note the implicit assumption that a person s productivit is constant, in particular it does not depend on the level of acquired b the person. Suppose that the labor market is perfectl competitive, so that in a situation of perfect information (where emploers were able to distinguish low-productivit from high-productivit individuals) each worker would be paid a wage equal to his/her marginal productivit. The average productivit over the entire population is: ( q) ( q) q (> since 0 < q < ). Suppose that workers of both tpes are able to bu, at a cost. Assume that the amount of is a continuous variable and that it is full verifiable (e.g. through a certificate). Suppose also that tpe-l workers face a higher cost of acquiring than tpe-h workers. The cost is not necessaril a monetar cost: it could be a disutilit cost (tpe-l individuals have to stud harder and longer hours than tpe H individuals in order to obtain the same level). Let Cost of for Group L individuals: C L = Cost of for Group H individuals: C H = Cost of acquiring C L = C H = / amount of Page 4 of 9

5 With the presence of as a signal, emploers will form beliefs about the correlation between and productivit. For example, suppose that emploers believe that anbod with a level of less than has a productivit of (and thus are offered a wage of ) while everbod with a level of greater than or equal to has a productivit of (and thus are offered a wage of ). The emploers beliefs are thus reflected in the following wage schedule: wage w level of Given this wage schedule, it is clear that nobod will want to acquire a level of > or a level 0 < <. Ever individual will limit himself to choosing between = 0 and =. For a GROUP L individual If choose = 0 If choose = get w = pa C = 0 net wage = get w = pa C = net wage = For a GROUP H individual If choose = 0 If choose = get w = pa C = 0 net wage = get w = pa C = net wage = Page 5 of 9

6 If is such that < and, that is, if then group L individuals will choose = 0 and group H individuals will choose = and therefore the emploers beliefs will be confirmed: people with low will turn out to be of low productivit and people with high will turn out to have high productivit. This is called a separating signaling equilibrium. REMARKS. () There is an infinit of such equilibria (one for each value of strictl between and ). () At an such equilibrium, Group L workers get a wage of and have zero costs, while Group H workers get a wage of but face costs equal to. Since in this model does not affect productivit and is of no value to the worker (it onl involves a cost and no benefit), the lower the value of the better off Group H workers are. Hence the equilibria can be Pareto ranked. (3) Can a separating signaling equilibrium be Pareto efficient? Let us compare an arbitrar separating signaling equilibrium with the non-signaling equilibrium in which everbod gets paid a wage equal to the average productivit of the entire population, namel w = q. Group L workers would be better off, since q > (because q < ), so that the strictl prefer the non-signaling equilibrium. Group H workers prefer the nonsignaling equilibrium if and onl if q, that is, if and onl if q. For example, if =.5 and q = this condition is satisfied and therefore everbod prefers the non-signaling equilibrium: if the government intervened and shut down all the schools, everbod would be better off! Thus for ever such that () < < and () > q we have a signaling equilibrium where Group L workers choose = 0 and Group H workers choose = and the equilibrium is strictl Pareto dominated b the non-signaling equilibrium where everbod is paid the average wage w = q. III. The informational impact of indexes. So far we onl considered signals, that is, those observable characteristics that can be modified b the agents concerned. The observable characteristics which cannot be modified, like sex, race, nationalit, etc., are called indexes. The question that the presence of indexes gives rise to is: can the informational structure of the market have the effect of persistentl and consistentl discriminating between objectivel identical individuals? We can imagine that the population is divided into two groups of equal size: Whites and Blacks. Within each group we have an equal proportion of low-productivit and high-productivit individuals. Signaling () costs are different for low- and high-productivit people, but people with the same level of productivit have the same signaling costs, no matter whether the are white or black. Assume also that people with the same productivit have the same preferences and the same objective: to maximize their income net of signaling costs. Therefore the index (race, sa) should be absolutel irrelevant: it is a general principle of economics that people with the same opportunit sets and the same preferences will make similar decisions and end up in similar situations. In our Page 6 of 9

7 model people with the same level of productivit face the same maximization problem, with the same data, and therefore should make the same decisions concerning, no matter whether the are white or black. The informational structure of the market, however, can destro this principle. If the emploer believes that race (besides ) is correlated with productivit, she might offer a wage schedule which is differentiated on the basis of race and. Her beliefs ma force high-productivit black people to invest in more than their white counterpart, that is, more than the high-productivit Whites. The reason wh this situation can persist is that emploers will interpret the incoming data separatel for the two groups of Whites and Blacks because the believe that race is a splitting factor. If different levels of were associated with the same observed level of productivit within the same group, emploers would be forced to revise their beliefs. That is, if within the group of Whites different levels of were accompanied b the same observed productivit, then emploers would conclude that, at least above a certain level, no longer increases productivit. But since Blacks and Whites are judged separatel and independentl (data on Whites is not used to classif Blacks and vice versa) emploers can consistentl think that Blacks need to acquire more than their white counterpart in order to compensate for a genetic handicap. As before, emploers beliefs ma force Blacks to invest more in than Whites, thereb confirming emploers beliefs, despite the fact that those beliefs have no objective grounds. Thus Blacks end up being over-qualified for their jobs as compared to Whites. EXAMPLE. Suppose that the population is 50% men and 50% women. Within the male population, 50% are low-productivit (= ) and 50% are high productivit (= ). Similarl, within the female population, 50% are low-productivit (= ) and 50% are high productivit (= ). Productivit is innate and is not influenced b either or gender. Women, L Women, H Men, L Men, H productivit proportion Cost of acquiring ears of Suppose that emploers mistakenl believe, not onl that affects productivit, but also that women are somewhat geneticall handicapped and therefore need to acquire more in order to become more productive. Accordingl, the emploers offer the following wage schedules, with w > m : If applicant is male If applicant is female Education Wage Education Wage < m < w m w Page 7 of 9

8 Graphicall: wage schedule for men m w wage schedule for women m w What choices will men and women make? If choose = 0 For a MAN of tpe L If choose = m get w = pa C = 0 net wage = get w = pa C = m net wage = m For a MAN of tpe H If choose = 0 If choose = m get w = pa C = 0 net wage = get w = pa C = m net wage = m Page 8 of 9

9 A separating equilibrium occurs if m is such that m < and, that is, if m then men of tpe L will choose = 0 and men of tpe H will choose = m and therefore the emploers beliefs will be confirmed: men with low will turn out to be of low productivit and men with high will turn out to have high productivit. For a WOMAN of tpe L If choose = 0 If choose = w m get w = pa C = 0 net wage = get w = pa C = w net wage = w For a WOMAN of tpe H If choose = 0 If choose = w get w = pa C = 0 net wage = get w = pa C = w net wage = w A separating equilibrium occurs if w is such that w < and, that is, if w then women of tpe L will choose = 0 and women of tpe H will choose = w and therefore the emploers beliefs will be confirmed: women with low will turn out to be of low productivit and women with high will turn out to have high productivit. Now, if both conditions are satisfied, that is, if, then emploers beliefs will be confirmed in ever respect: in order for women to become more productive the need to spend more time in school. A man acquires a productivit of b spending m ears in school, while women acquire a productivit of b spending more time in school: w ears rather than m. m w w Page 9 of 9