Vol. 65 (1997), No. 2, pp. 163-180 Journal of Economics Zeitschrift f~r NationalSkonomie @ Springer-Verlag 1997 Printedin Austria Endogenous Information Quality: a Job-assignment Application David A. Malueg and Yongsheng Xu Received May 7, 1996; revised version received February 3, 1997 This paper investigates the optimal acquisition of information in a model of job assignment within a firm. We consider a firm with two types of jobs, skilled and unskilled. The firm draws workers randomly from the general population, and a worker is either talented or untalented. Initially, a worker's productivity in the firm is unknown to the worker and the firm. Workers are equally productive in the unskilled job, but talented workers are more productive in the skilled job than in the unskilled job, and untalented workers are more productive in the unskilled job than in the skilled job. Before assigning a worker to a job, the firm can test whether the employee is talented, and the firm is able to choose the accuracy of this test. We assume that the cost of a test is increasing and convex in test accuracy. We show that (1) the accuracy of the firm's test increases with the cost of a mismatched worker; (2) increased optimism about the worker's ability need not lead to less rigorous testing; (3) the probability that a worker is assigned to the skilled job need not increase as the gain from assigning a talented worker to a skilled job increases, or the loss from assigning an untalented worker to a skilled job decreases, or the fraction of the population that is skilled increases; and (4) a longer testing period, allowing as many as two tests of workers, leads the firm to use a less expensive, and less accurate, test initially than when there is only one opportunity to gather information. Keywords: information, tests, job assignment, skilled and unskilled jobs. JEL classification: D83, J41. 1 Introduction In m a n y situations, decision-makers can acquire information before taking an action. Moreover, decision-makers typically have some discretion over the quality of information they acquire. This paper investigates a decision-maker's choice of information quality. This problem of information acquisition arises in a variety of settings. For example, 164 D. A. Malueg and Y. Xu how thorough should be the White House's background investigation of prospective presidential nominees to Cabinet positions? Does this level of thoroughness depend on the President's prior beliefs about the candidate? A second example is given by a firm considering whether to introduce a new product. How extensive should the market research be before the decision of whether to introduce the new product? How does the extent of this market research depend on the potential loss from an unsuccessful introduction and on the potential profit from a successful introduction? A third example is provided by promotion within a firm. Consider a CEO evaluating a manager for promotion to the position of corporate vice president. How extensive should be the review of this manager for promotion? Before the review begins, is the manager more likely to be promoted if the CEO is more optimistic that the manager has the "right stuff' for the new position? This paper investigates the optimal acquisition of information in a model of job assignment within a firm, but our findings obviously apply to many more situations. MacDonald (1980, 1982a, b), Prescott and Visscher (1980), Burdett and Mortensen (1981), Lockwood (1991), and Gale (1991) analyzed labor-market models in which a firm may acquire information about a worker before deciding to which of several jobs to assign that worker. Information may be acquired through either accumulated observations of performance or through testing. MacDonald (1982b) and Prescott and Visscher (1980) provided information-accumulation models. Burdett and Mortensen (1981), MacDonald (1982a), Gale (1991), and Lockwood (1991) considered labor markets in which firms gathered information by testing. These testing models assume that the quality of information that may be gathered is exogenously specified - the firm need 9 only decide whether to test a particular worker. These authors then investigated the equilibrium characteristics of the market. Here we analyze in greater detail the firm's internal informationacquisition problem, but we do not consider properties of the external labor-market equilibrium. We consider a firm with two types of jobs, skilled and unskilled. The firm draws workers randomly from the general population, and a worker is either talented or untalented. Initially, a worker's productivity in the firm is unknown to the worker and the firm. Workers are equally productive in the unskilled job, but talented workers are more productive in the skilled job than in the unskilled job, and untalented workers are more productive in the unskilled job than in the skilled job. Before assigning a worker to a job, the firm can test whether the employee is talented, and the firm chooses the accuracy of this test. We assume that more accurate tests are more costly. Several conjectures naturally arise in this setting. Endogenous Information Quality 165 Conjecture 1: The accuracy of the firm's test increases with the cost of a mismatched worker. Conjecture 2: Increased optimism about the worker's ability leads to less rigorous testing. Conjecture 3: The probability that a worker is assigned to the skilled job increases as a. the gain from assigning a talented worker to a skilled job increases; b. the loss from assigning an untalented worker to a skilled job decreases; c. the fraction of the population that is skilled increases. Our basic model investigates the quality of information acquired by a firm when it can observe the result of at most one test of a worker's ability. In some situations, the decision-maker may be able to spread information acquisition over time. It seems reasonable to conjecture that, if the firm has a longer horizon, consisting of several periods, during which to acquire information, convex costs of information quality will lead the firm to spread out its information acquisition. Conjecture 4: A longer testing period, allowing as many as two tests of a worker, leads the firm to use a less expensive, and less accurate, test initially than when there is only one opportunity to gather information. Section 2 of the paper formally presents the basic model described above. Section 3 then develops the model when the firm is able to observe the result of one test that may be given to an employee. Conjecture 1 is shown to be correct in this model. However, Conjectures 2, 3a, 3b, and 3c are not generally true. Section 4 allows the firm to administer at most two tests before assigning a worker to one of the jobs. We find that, relative to the case in which only one test may be given, the firm that can administer two tests will use a less accurate test initially, confirming Conjecture 4. Conclusions and extensions are discussed in Sect. 5. 2 The Basic Model A firm draws workers from a population having two types of workers: A and B, with respective fractions 3 and 1 - 3, where 3 E (0, 1). There are two types of jobs a worker can perform within the finn: skilled and 166 D. A. Malueg and Y. Xu unskilled. The productivity of a worker performing an unskilled job is v0. Without loss of generality, we assume v0 = 0. The productivity of a type A worker performing a skilled job is va, and the productivity of a type B worker performing a skilled job is - v ~ . We assume that va > 0 > --vB. We refer to type A workers as talented and type B as untalented. A worker's type is assumed to be initially unknown to all, even to that worker. Although a worker's type is initially unknown, by testing the worker, the firm may gather information about the worker's type. The test might be a formal examination or a period during which a worker's performance in a skilled job is (imperfectly) observed. If a worker is tested, then there are two mutually exclusive signals that can be observed: a and b. The outcome of a test is common knowledge. The conditional probability of observing x (x = a, b) given X (X = A, B) depends on the accuracy of the test: 0 = Pr(a[A) = Pr(b[B) and 1 - 0 = Pr(b[A) = Pr(alB). We assume 0 > 1/2, which is simply a labeling convention. It implies that observing an a signal is "good news" about the worker, leading to an upward revision in the perceived probability that the worker is type A . 1 There is a cost C(O) associated with each test of quality 0. We assume that C'(O) > 0 and CI'(0) > 0 for 1/2 < 0 < 1; C ( 1 / 2 ) = C'(1/2) = 0; and lim0--,1 C'(O) = ec. Before making a job assignment, the firm chooses whether to test a worker and, if it tests, the accuracy of the test. Given the test result, the worker is assigned as follows: if signal a is observed, then the worker is assigned to perform the skilled job; if signal b is observed, then the worker is assigned to perform the unskilled job (if it is optimal to test, then this is the optimal assignment rule). Therefore, by administering a test of quality 0, the firm' s expected profit, n-, from a worker is given as zr(0[3) = VA30 -- vs(1 -- 3)(1 -- 0) -- C ( O ) . (1) 1 The information structure in our model is identical to MacDonald's (1982a), except that we allow the firm to choose the quality of the test. One might also envision other types of tests. For example, Burdett and Mortensen (1981) consider a test that reveals perfectly the worker's ability. Falling between that test and ours is Lockwood's (1991). Lockwood uses a test that workers either pass or fail; a talented worker always passes, but an untalented worker passes with (exogenous) probability q5. Endogenous Information Quality 167 3 Optimal Testing with One Test Facing the problem of assigning a worker to the proper job, the firm may assign the worker to an unskilled job without giving a test; or assign the worker to a skilled job without giving a test; or have the worker take a test, and then assign the worker to a job based on the test result. Consequently, when the firm is able to observe the result of one test that may be given to a worker, its value function, rI (6), is given by FI(3) = max{0, 7r*(3), 31)A - - (1 - 6)VB} , (2) where rr*(3) = zr(O*(3)13) = = max 0611/2,11 max a'(OI3) 0611/2,1] VA30--VB(1--3)(1--O)--C(O) (3) and 0*(3) -= argmaxzr(OI6) . 0 Here 0* denotes the quality of the optimal test, assuming a test is to be given; ~r*(3) denotes the firm's maximum expected payoff, assuming a test is to be given. Differentiating (1) with respect to 0, we find 0* is characterized by the following equation: 1)A3 + VB(1 -- 3) = C' (O*) . (4) From (3), the following remark is immediate. R e m a r k : The profit function, 7r*(3), is an increasing function of vA and a decreasing function of vs; it is a strictly increasing, continuous, and convex function of 3. Next we investigate comparative statics of test quality. The Cost o f a M i s m a t c h e d W o r k e r We begin by confirming our first conjecture about the relationship between the accuracy of the firm's test and the cost of a mismatched 168 D. A. Malueg and Y. Xu worker. The cost of a mismatched worker is either the loss due to assigning an untalented worker to a skilled job, - v B , or the foregone productivity associated with assigning a skilled worker to an unskilled job, VA. To analyze the influence of these costs on test quality, we differentiate both sides of Eq. (4) with respect to VA and vB to obtain 2 . , 00" 6 = C ( 0 ) Ov~ (5) and 00" (1 - 8) = (6) C"(O*)-- OrB Because 0 < 6 < 1 and C" > 0, it follows that O0*/OVB > 0, confirming our first conjecture. O0*/OVA > 0 and Theorem 1: The accuracy of the firm's test increases as the cost of a mismatched worker increases: O0*/Ovi > 0, i = A, B. 3 The Prior Probability of a Worker Being Talented Recall that the parameter 6 indicates the fraction of type A workers in the population of the labor force. To find the influence of 6 on the rigor of a test, we differentiate both sides of Eq. (4) with respect to 6 to obtain . VA -- ~)B = C , (0) O0* ~- . (7) Thus, if vn > vs, then 00*/08 > 0; and if v A < VB, then 00*/08 <_ O. To understand these results, consider the following three cases: (i) ve > va; (ii) VA > V~; and (iii) VA = VS. In case i, the firm is more concerned about not assigning a type B worker to the skilled job than assigning a type A worker to the skilled job. Therefore, when the firm is more confident about a worker being type B (a decrease in 6), a more rigorous test will be given so that a type B worker will more 2 Using (4), differentiability of 0* follows from the implicit-function theorem and the assumed differentiability of C'. 3 A similar result was obtained by MacDonald (1980) in a setting of person-specific information in the labor market, where, when the productivity of performing a task increases, workers tend to choose a higher quality of information if workers' utility functions exhibit declining absolute risk aversion. Endogenous Information Quality 169 likely fail the test and consequently not be assigned to the skilled job. In case ii, the firm is more concerned about assigning a type A worker to the skilled job than not assigning a type B worker to the skilled job. Therefore, when the firm is more confident about a worker being type A (an increase in 3), a more rigorous test will be given so that type A workers will more likely pass the test and consequently be assigned to the skilled job. This second case shows that Conjecture 2 is not generally true. In case iii, the firm is equally concerned about assigning type A workers to the skilled job and not assigning type B workers to the skilled job. On balance, even with greater optimism about the worker's ability, the firm will not change the accuracy of its test. These results are summarized in the following theorem. Theorem 2: If UA -~- UB, then the quality of the test does not depend on 3; if va < VB, then the increased prior probability of a worker being talented leads to less rigorous testing; and if va > VB, then the increased prior probability of a worker being talented leads to more rigorous testing: sgn(O0*/O3) = s g n ( v a - - VB). Assignment to the Skilled Job Let s be the probability that a worker is assigned to the skilled job. Then (8) s:30"+(1-3)(1-0"). Thus, Os O0* , (9) Os O0* OrB = (23 -- 1)OV----B ' (10) O1)a 0_s o3 = (23 - 1 ) - - OVA (11) = (20* -- 1) + (23 - 1) v a Z I ) B c"(o*) ' where (11) relies on (7). Recall that the firm's loss from misassigning a talented (untalented) worker is VA (--vB). Therefore, (9), (10), and Theorem 2 imply the following theorem. Theorem 3: If 3 = 1/2, then the probability of assigning a tested worker to the skilled job is independent of the cost of a mismatched worker; 170 D. A. Malueg and Y. Xu if 8 < skilled if 8 > skilled 1/2, then the probability of assigning a tested worker to the job decreases as the cost of a mismatched worker increases; 1/2, then the probability of assigning a tested worker to the job increases as the cost of a mismatched worker increases: sgn(Os/Ovi) = sgn(28 - 1), i = A, B. The intuition behind Theorem 3 is straightforward. Low values of 8 (8 < 1/2) reflect the firm's pessimism about the likelihood that a randomly drawn worker is talented. As discussed earlier, an increase in either VA or vB will induce the firm to administer a more accurate test to the worker. Given the firm's pessimistic beliefs, this more rigorous test is even more likely to result in an observation of b, making the firm more likely to assign the worker to the unskilled job. Similarly, if the firm is optimistic about the worker (8 > 1/2), then an increase in VA or v~ results in a more accurate test, which in turn is more likely to identify the worker as talented. This reasoning shows why Conjectures 3a and 3b are invalid. Although (11) leaves the sign of Os/08 ambiguous, Os/08 can be understood by considering separately the two effects represented in (11). If, after an increase in 8, the firm did not alter the test given to workers, then a tested worker's chance of being assigned to the skilled job would increase at the rate 20* - 1 (consistent with Conjecture 3c). However, the induced change in test accuracy also affects s, an effect represented by the second term on the right-hand side of (11). If the firm is optimistic about the worker (8 > 1/2), and if an increase in 8 leads to a more accurate test (Va > re), then this second effect will reinforce the first, ensuring that s increases. Analogous reasoning applies when the firm is pessimistic about the worker's ability. Thus, we have the following sufficient conditions for an increase in 8 to lead to an increase in the probability that a worker is assigned to the skilled job. Theorem 4: If (i) 8 > 1/2 and then Os/08 > O. VA > VB or (ii) 8 < 1/2 and VA < IJB, Theorem 4 shows that under certain conditions, Conjecture 3c is true. However, in general, it is false. 4 4 One can use the cost function C(O) = (20-1)2/(1 --0), for 0 6 [1/2, 1), to construct explicit examples showing Conjecture 3c is not generally true. Endogenous Information Quality 171 4 Optimal Testing with Multiple Tests This section first discusses the optimal testing when at most two tests may be given and then generalizes one of the results to the n-test case. 5 The firm's problem when n tests are allowed is that it can either assign the worker to the unskilled job or to the skilled job without testing, or give a test, recognizing that on the basis of the test result it can make an assignment or test again, with at most n - 1 more tests being allowed. In the case of at-most-two tests and a test being given, this reduces to the plans represented by S.1, S.2, and S.3 below. The firm may, S. 1, have the worker take the first test, and if signal b is observed, then assign the worker to the unskilled job; if signal a is observed, then have the worker take the second test and assign the worker to a job based on the second test result; S.2, have the worker take a test, and if signal a is observed, then assign the worker to the skilled job; if signal b is observed, then assign the worker to the unskilled job; S.3, have the worker take the first test, and if signal a is observed, then assign the worker to the skilled job; if signal b is observed, then have the worker take the second test and assign the worker to a job based on the second test result. Let 7~i (8) denote the firm's maximum expected profit for the strategy S.i, i = 1, 2, 3, given initial beliefs 8, and let 1-I2(8) denote the value function for the case of at-most-two tests. Note that the expected profits assigning the worker to the unskilled and skilled job without giving a test are, respectively, 0 and v a 8 -- VB (1 -- 8). Therefore, FI2(8) = max{~l (8), Jr2(8), 7r3(8), O, VA8 -- UB(1 -- 8)} . The first concern of this section is the comparison of the optimal test quality for the one-test case and for the initial test in the at-mosttwo-tests case. Equivalently, we compare 0* and Oi, i = 1, 2, 3, where Oi is equal to the optimal test quality in the at-most-two-tests case for the firm following strategy Si, given initial beliefs 8. The following 5 The reason that we only consider finite n is the following. If an unlimited number of tests is allowed, then with C(1/2) = C'(1/2) = 0, the firm may be able to obtain almost perfect information at almost zero cost by taking a very large number of very inexpensive, but still informative, tests. To ensure a solution to the problem, we impose a bound on the number of tests allowed. 172 D. A. Malueg and Y. Xu theorem confirms Conjecture 4, showing that if testing occurs in the one-test model, then it is at least as accurate as the initial test when as many as two tests are allowed. The proof may be found in the appendix. Theorem 5: Suppose 1)A > 0, VB > 0, C tl > 0, and 0 < 6 < 1. ff IlZ(3) = r0(6), then Oj(3) < 0"(6), j = 1, 2, 3, with strict inequality i f j c{1,3}. The intuition of Theorem 5 can be explained as follows. The quality of the assignment decision derives from reducing the chance of misallocation. If test quality were free, the firm would always use test quality 0 = 1. In the one-test situation, the firm chooses test quality so that the marginal cost of test quality just equals the expected marginal improved quality of the assignment decision that says if the test result is a, assign the worker to the skilled job, and if the result is b, assign the worker to the unskilled job. Now consider the case in which strategy S. 1 is optimal. Here the initial chance for misassignment occurs only in the case where the test result is b, for otherwise a second test is given. Consequently, ignoring costs, the marginal benefit of the first test is strictly lower in the case where strategy S. 1 is optimal than in the case where only one test is given. Therefore, given that the marginal cost of test quality is strictly increasing, if strategy S. 1 is optimal, then the quality of the first test in the at-most-two-tests case is strictly less than the optimal test quality when at most one test can be given. Similar reasoning applies when strategy S.3 is optimal. Let [_3, 6] denote the test region when at most one test may be given. Similarly, let [_32, ~2] denote the set of initial beliefs for which at least one test is given when at most two tests of a worker are allowed. The following theorem illustrates the relationship between the testing region for the one-test case and the testing region for the at-most-twotests case. The proof can be found in the appendix. Theorem 6: The set of parameters 6 for which testing is optimal is strictly larger when up to two tests may be administered than when only one test may be administered: _32 < 6 and ~ < ~2. Theorem 6 can be generalized to the case where the firm is allowed to administer n (n > 2) tests: The set of parameters 6 for which testing is optimal is strictly larger when up to n § 1 tests may be administered than when only n tests may be administered. A sketch of the proof of this generalization can be found in the appendix. Endogenous InformafionQuNi~ 173 The intuition behind Theorem 6 and its generalization may be explained as follows. In the case where the firm can have n tests, it always has the option of not giving the n-th test if (n - 1) tests are enough. The implication of this is that the testing region should not shrink at least. As for the actual expansion of the test regions from the (n - D-tests case to the n-tests case, let us consider the neighborhood of, say, the lower bound of the testing region in case of at most (n - 1) tests. We know that in the neighborhood of an e to the right of the lower bound, at least one test is given. Then, if the firm has the option of testing one more time, given the convex cost and C ( 1 / 2 ) = C ( t / 2 ) = 0, it costs the firm almost nothing to administer a test whose quality is just above 1/2, and because the value function with at most (n - 1) tests is convex and kinked at this lower bound for testing in the at-most(n - D-tests case, this small test is strictly valuable. Hence, when at most n tests are allowed, for 3 equal to the lower bound for testing in the at-most-(n - 1)-tests case the firm strictly prefers to test the worker. This pushes down the lower bound for testing in the n-test case below that in the (n - 1)-tests case. Thus, by having the option of giving one more test to the worker, the initial lower bound will be expanded to the left. Similar reasoning can be applied to the upper bound as well. 5 Conclusion In the context of a job-assignment model, this paper has considered a firm's information-acquisition problem when the quality of information is a choice variable for the firm. We assumed that workers were of two types, and, at a cost, the firm could test a worker. The cost function was assumed to be increasing and convex in test accuracy. Two of our initial conjectures were confirmed: the firm chooses a more accurate test when the cost of a mismatched worker increases; and when the firm may administer two tests, its first test will be less accurate (and less expensive) than when only one test is allowed. In addition, we uncovered several surprising possibilities. In particular, it was shown that increasing the probability that a worker is talented may lead the firm to use a more accurate test; and the probability that a worker is assigned to the skilled job may decrease as the productivity of a skilled worker in the skilled job increases, increase as the loss from assigning an unskilled worker to a skilled job increases, or decrease as the fraction of the population that is skilled increases. 6 These somewhat paradoxical effects arise from the endogeneity of test quality in our 6 We can re-interpret our firm as having only skilled jobs, in which case, 174 D. A. Malueg and Y. Xu model. As each of the parameters mentioned changes, the firm adjusts test quality, sometimes with unexpected consequences. Our job-assignment model can be applied to other economic problems, such as quality control in a firm. Consider a firm that produces widgets with a technology that yields an exogenous fraction (3) of defective widgets. The firm can test the widgets before shipping them to clients. With what accuracy should the widgets be tested? Given that the test is not perfectly accurate, should more than one test be given? If so, what should be the accuracy of the second test? These questions are analogous to those in our job-assignment model, and the answers will depend on the costs of testing, the value (e.g., in terms of money and reputation) to the firm of selling a good widget (VA) and the loss from selling a defective widget ( - v ~ ) . Our analysis might be carried in several directions. For example, if a worker knows his or her probability of being talented, but the firm does not, then the firm might use testing as part of a screening process in designing the optimal wage contract between a worker and the firm. Also, the labor-market equilibrium with and without asymmetric information might be explored. These two extensions would generalize Burdett and Mortensen (1981) to the case o f endogenous test quality. Our analysis could also be extended to have the firm choose not only the quality of the test, but also the number of workers to be tested. 7 Appendix Let 3(a) = Pr(Ala) and 3(b) = Pr(AIb). Thus, 3(x) describes the firm' s beliefs about the worker, given test outcome x has been observed, x c {a, b}. Of course, 3(x) also depends on 0, the accuracy of the test. We have 6(a) = Pr(a JA) Pr(A) = Pr(a) Pr(aIB) Pr(B) ~(b) = Pr(b) 03 06 + (1 - 0)(1 - S) ' (1 - 0)6 = (1 - 0)~ + 0(1 - 3) those workers that would have been assigned to the unskilled job cannot be interpreted as not being hired by the firm. Then our finding that Os/O6 < 0 is possible has the interpretation that government interventions to increase the overall skill level in the workforce do not necessarily lead to more employment. We thank an anonymous referee for this interpretation. 7 We thank an anonymous referee for suggesting this extension. Endogenous Information Quality 175 Let ~1(816) = P r ( a ) z r * ( 6 ( a ) ) - C ( 0 ) , (12) :rv2(SI3) = P r ( a ) [ 6 ( a ) v A -- (1 -- 6(a))VB] -- C ( 0 ) , (13) ~3(816) = P r ( a ) [ 6 ( a ) v a -- (1 -- 6(a))vB] + Pr(b)Jr*(a(b)) - C ( 8 ) , (14) where P r ( a ) z r * ( a ( a ) ) = UA8288 -- VB(1 -- 8~)(1 - 0)(1 - 3) - (88 + (1 - 8)(1 - S ) ) C ( 8 ~ ) , P r ( a ) [ 8 ( a ) v A -- (1 -- 6(a))VB] = VAO6 -- VB(1 -- 0)(1 -- 3) , P r ( b ) J r * ( 6 ( b ) ) = VAO;(1 -- 0)6 -- VB(1 -- 8 ; ) 8 ( 1 -- 3) - ((1 - 0)6 + 0(1 - 6))C(8;). Let 8i(6) = a r g m a x r c i ( 8 t 6 ) , i = 1, 2, 3. Then 81, 82, and 03 are char- 0~[1/2,11 acterized by the following first-order conditions: 81: C ' ( 0 ) = VAS*6 + VB(I -- 8a*)(1 - 3) - (23 - 1)C(8"), 82: C ' ( 8 ) = v a 6 + va(1 - 6) , 83" C ' ( 8 ) = UA6 -}- VB(1 -- 6) - - [VZ38; -]- I)B(1 (15) (16) -- 6)(1 -- 8;) -- (26 - 1 ) C ( 8 ; ) 1 , (17) where 0x* = 8 " ( 6 ( x ) ) , x = a, b. Note that 7ri(3) = 7ri(8i(6)16), i = 1, 2, 3. F r o m the definition o f FI2(6), it follows that if FI2(3) = Jrj(6), then 7rj(6) = 7rj(Oj(6)16) >_ 7ri (Sj (6)16). Proof of Theorem 5 Note that n-2(016) is simply the function zr(0[8) considered in the onetest model. Therefore, 02 (6) = 0" (8). The remainder o f the p r o o f shows that if FI2(8) = zrj(6), then 8j(8) < 8*(8), j E {1, 3}. Case 1: 1-I2(8) = Jrl(8). Case 1 . 1 : 8 > 1/2. Letting rhs denote "right-hand side," in this 176 D.A. Malueg and Y. Xu case, as 1 > 0~ > 1/2, we see that rhs of (15) < v a 0 2 3 q-- vB(1 - 0")(1 - 8) < 8va + (1 -- 3 ) v , . Therefore, because marginal cost is strictly increasing, 0~ (3) < 0 * ( 8 ) . Case 1 . 2 : 8 < 1/2. Because 7r1(3) > n-2(02(8)]6), it follows that yr*(3(a)) _> 6 ( a ) V A 0")(1 8(a))vB - -- (1 - 8 ( a ) ) v B , implying O*8(a)VA -- (] -- - C(O*) > 8(a)v A - (1 - 8 ( a ) ) v B ; equivalently, C ( O * ) < 0"(1 - 8 ( a ) ) v B -- (1 -- O * ) 8 ( a ) V A . Therefore, applying this inequality to (15), we have rhs of (15) < V A 0 2 3 q- V B ( I - - 0 2 ) ( 1 -- 8) - (28 - 1)(Oa*(1 - 8 ( a ) ) v ~ - (1 - O * ) 8 ( a ) v A ) = OA[Oa8 "q- (26 -- 1)(1 -- 0*)8(a)] + VB[(1 -- 0~)(1 -- 8) -- (26 -- 1)0a*(1 -- 8(a))] < UAO*8 -4- V B [ I - - 3 -- 0 2 { 1 - - 3 "4- ( 2 8 - - ] ) ( 1 - - 8 ( a ) ) } ] = v a O * 6 + v~[1 -- 8 -- 0*{6 -- 2 6 3 ( a ) + 6(a))] = vAO*3 + v ~ [ 1 - - 6 - - 0 ~ ( ~ ( 1 - - ~ ( a ) ) + 3 ( a ) ( 1 - - 6))~ < VA6 "-~ I)B(1 - - 8 ) , f r o m which it follows, by increasing marginal cost, that 0~ (8) < 0 * ( 8 ) . Case 2: FI2(8) = 3r3(8). C o m p a r i n g (4) and (17), to show 04(8) < 0"(3), it suffices, because marginal cost is strictly increasing, to show that VATO~ + VB(1 -- 8)(1 - 0~) - (28 - 1)C(0~) > 0 . (18) Case 2 . 1 : 3 < 1/2. It is clear by inspection that (18) is satisfied. Endogenous Information Quality Case 2.2:3 177 > 1/2. B e c a u s e rr3(3) > 7r2(03(3)[3), it f o l l o w s that 7r*(3(b)) = O~3(b)vA -- (1 -- 0~)(1 -- 3(b))VB -- C(O~) > O, implying C(OD) < O~3(b)vA -- (1 -- 0~)(1 -- 3(b))ve . W e use this inequality in the left-hand side o f (18) to obtain "VA30 ~ -}- V B ( 1 -- 3 ) ( 1 - - 0 ~ ) - - ( 2 3 - - 1 ) C ( 0 ~ ) > VA30[~ + vB(1 -- 3)(1 - 0~) - (23 - 1)[O~,3(b)vA -- (1 -- 0~)(1 -- 3 ( b ) ) v e ] = VAO~[3 -- (23 - 1)3(b)] + v s ( 1 - Off)[1 - 8 + (23 - 1)(1 - 3(b))] > UAO~[3(1 -- 3 ( b ) ) At- 3(b)(1 - 3)1 + VB(1 -- 0~)[(1 -- 3)3(b) -I- 3(1 - 3(b))] >0, s h o w i n g that (18) is satisfied. [] Proof of Theorem 6 F r o m the r e m a r k it f o l l o w s that dzr*(3) - > 0 d3 for any 3 6 (0, 1) . (19) F r o m the e x p r e s s i o n s for 3(a) and 3(b) it is easily verified that d3(a) (1 - 3)3 - dO (Pr(a))2 and d3(b) dO (1 - 3)3 (Pr(b)) 2 " (20) To p r o v e _~2 < _~ and $ < ~2, it suffices to s h o w that 7r.l(~) > yt-2Q~) and zr3(6) > rrz(g), respectively. First w e p r o v e rrl (_~) > rr2(~). O b s e r v e that ~1(01~)1o=1/2 = ( 1 / 2 ) ~ * ~ ) = 0. 178 D.A. Malueg and Y. Xu Therefore, to show zq (_~) > zr;(_6), we show that dYrldO (013) - 0=1/2 > 0 . (21) Showing (21) suffices because it implies that by following strategy S. 1, corresponding to zrl, and administering the first test of quality slightly above 1/2, the firm can obtain a strictly positive expected payoff, which exceeds zr2 (~), as the latter is equal to zero. Differentiating (12) we have dJr 1(013) dzr * d3 (a) d Pr(a) - - Pr(a) -+ zr*(8(a))-dO d8 dO dO C'(O) . (22) Evaluating (22) at 0 = 1/2 and 8 = 8, and using C'(1/2) = 0, (20), and the definition of 8, we have dTrl (0[~) 0=1/2 =23(1_ -_8)--~dzr* > 0 , dO where the inequality follows because 6 E (0, 1) and by (19). Next we prove 7~3(8) > 7r2(8). Observe that Yr2(0[~)10=l/2 = 1(~1) a "~- (1 - 8)VB) + 89 = yr2(~ ) , (23) using the definition of 6. Therefore, in order to show Jr3 (6) > Yr2 (~), it SUffices to show that dzr3(0[~)d0 0=1/2 > 0 . From (14) we have dzr3 (0 I3) dO dPr(a) - - dO [3(a)vA -- (1 -- 3(a))vB] d Pr(b) + Pr(a)(vA + vB)dSd~ + ~ z r + Pr(b) drc*(8(b)) d3(b) d8 dO CI(0). , (8(b)) (24) Evaluating (24) at 0 = 1/2 and 8 = 6, and using CI(1/2) = 0, (20), Endogenous Information Quality 179 and the definition of ~, we have dYr3(Ol~)dO0=1/2 = (2g - 1)[gvA - (1 - g)ve - 7c*(g)] ~ * ( 6 ) ] 6(1 - 6) dyr*(6) >0, where the inequality follows from the fact that 6 6 (0, 1) and that at 6, as a function of 5, re* is strictly less steep than is the function 6VA - (1 - 6)VB. [] Proof of the Generalization of Theorem 6 Since the proof of this generalization is similar to that of Theorem 6, we shall only give a sketch. Let r p ( 3 ) be the profit function when at most n tests may be given. Then the maximum value of administering a test of quality 0, given initial beliefs a and given that up to n additional tests may be given, is V(Ol~) = P ~ W ( a a ) § P b W ( a b ) -- C(O) , where Pa = Pr(a), Pb = Pr(b), aa = a(a), and 3b = 6(b). To prove our assertion, it suffices to show that (dV(1/2[~_n))/dO > 0 and (dV(1/2[6n))/dO > 0. Note that dPa Fin((~a)@ pa~m((~a)daa dV(Ola) dO dO + dO I-In(6b) + PbFI'n(ab) dab d---O-_ C' (O) " Now consider 0 = 1/2, and ~ = a n. Then aa = ab = 6 n, and dV(1/216 n) dO -- 2(1 -~n)[II~(_6 n) - I-I'2(~n)], where the indices " + " and " - " indicate the fight and the left derivates respectively. Note that the profit function is increasing and convex, and 180 Malueg and Xu: Endogenous Information Quality In n is "kinked" at6 n, so l-I+(~ ) > II~(~n). Hence, (dV(1/216_~))/dO > O. Similarly, it can be shown that (dV(1/2[~n))/dO > O. [] References Burdett, K., and Mortensen, D. (1981): "Testing for Ability in a Competitive Labor Market." Journal of Economic Theory 25: 42-66. Gale, D. (1991): "Incomplete Mechanisms and Efficient Allocation in Labour Markets." Review of Economic Studies 58: 823-851. Lockwood, B. (1991): "Information Externalities in the Labour Market and the Duration of Unemployment." Review of Economic Studies 58: 733-753. MacDonald, G. (1980): "Person-specific Information in the Labor Market." Journal of Political Economy 88: 578-597. (1982a): "Information in Production." Econometrica 50:1143-1162. (1982b): "A Market Equilibrium Theory of Job Assignment and Sequential Accumulation of Information." American Economic Review 72: 1038-1055. Prescott, E., and Visscher, M. (1980): "Organization Capital." Journal of Political Economy 88: 446-461. - - - - Addresses of authors: David A. Malueg, Department of Economics and A. B. Freeman School of Business, Tulane University, New Orleans, LA 70118, USA; - Yongsheng Xu, Department of Economics, University of Nottingham, Nottingham, NG7 2RD, UK.