Notes on: Personal Contacts and Earnings (Mortensen and Vishwanath 1993)

Transcription

1 Notes on: Personal Contacts and Earnings (Mortensen and Vishwanath 1993) Modigliani Group: Belen Chavez, Yan Huang, Tanya Mallavarapu, Quenhe Wang February 8, 2012 λ b α w δ F (w) G(w) π R(L) L(w) M E φ γ α ˆL π Key Notation Frequency at which offers arrive Unemployment benefit (reservation wage) Fraction of wage offers through contacts Wage earned or offered Rate at which separations take place Distribution of wage offers across firms (direct application) Distribution of wages earned by employed workers (indirect referral) Employer profit received per period Value of total revenue product when L workers are employed Labor Supply for wage w Total available work force Steady state measure of employed workers Fraction of employers offering the highest wage Fraction of workers who earn the highest wage Fraction of wage offers that yields competitive profit Common Employer size Competitive Profit 1

2 1 Introduction Purpose: to examine the effects of how different access to information sources for jobs that workers use influences their wage offers. Assumptions: 1. Monopsony: employers are sellers and employees are buyers 2. All workers are equally productive 3. Mixed distribution of sources: direct and contact 4. Unemployed workers accept any wage higher than the unemployment benefit and employed workers move only from lower to higher paying jobs. 5. All firms earn the same profit. A unique steady state distribution is reached when the search for jobs and the search for employees dissolves due to the movement of workers from being unemployed to employed or between jobs. With distributions of wages earned a steady state profit for the firms can be computed. Mortensen talks about a market clearing equilibrium which is a distribution of wages that yields the same maximal profit for all firms that offer different wages within the wage distribution. This occurs when firms participate in a one shot wage posting game, where each firm submits their wage offers at the same time without prior information on what other firms plan to offer. Each firm has a unique solution to the wage they offer since each of them have a different diminishing marginal value of labor product which is determined by the size and productivity of the firm. The diminishing marginal value of labor product is used by a firm to set the highest wage they would offer since anything higher than this value would cause loses and would not maximize profits for the firm. Mortensen also goes on to suggest that a wider range of offers would exist if there is a smaller portion of offers that go through contacts. This is because with fewer people aware of the average wage of employees at the firm, the employer would have more flexibility with setting a wage preferable to them. 2

3 Mortensen states that people earn higher wages if the offer is through personal contacts versus wage offers through direct application. Implications of the above statement: Proposition 1: Given a non-degenerate distribution of employer offers, the steady state distribution of wages earned stochastically increasing in the probability that any offer is from a contact. Proposition 2: When the equilibrium distribution of wages earned is non-degenerate, it is stochastically increasing in the probability that an offer is from a contact. The first proposition suggests that given a continuous distribution of employer offers, the steady state distribution of wages earned increases with the probability that an offer is from a contact. The second proposition implies that when the equilibrium distribution of wages earned is continuous, it increases with the probability that an offer is from a contact. The first proposition is a requirement since employees will choose employers with higher wage offers and as the portion of people coming through contacts increases the wages offered also increase as stated by Mortensen. Proposition 2 is a result since people only move from lower paying jobs to higher paying jobs and with the probability that an offer is through a contact will increase the wage offer as per proposition 1. If employers are not allowed to distinguish employees on their current wage rates and as an implication of proposition 1 workers who are less likely to receive wage information through contacts tend to get paid a lower wage. As a result of these propositions, Mortensen shows that people with different types of access to job information causes different outcomes in the wages earned. 3

4 2 Market Equilibrium In this section, we begin to consider all workers to be homogeneous with the same access to wage information, opportunity cost and equally productivity. Assumptions: 1. Homogeneous workers 2. All workers are wealth maximizers and are constantly searching for the best opportunity 3. Firms maximize steady state profits The following describes the process of wage determination. Since each employer submits their particular wage offer in the one-shot game a distribution of wage offers is created, F (w), across firms. When a workers is unemployed, they will accept the first wage offer that is higher than the unemployment benefits and an employed worker will move to a higher paying employer. As a result of the movement between firms, employment and unemployment the distribution of employment across firms will eventually converge at a steady state. The distribution of employment among firms along with the distribution of wage offers creates a steady state distribution of wages earned by those employed, G(w). An employee who applied for the position directly through the company will be offered a wage that is randomly chosen from the distribution of wage offers. However, an employee who applied for a job through a contact will be offered a wage that is drawn from the distribution of wages earned. This is because people who go through contacts will always get offered a higher wage and are looking for a firm that is paying higher than they are currently getting paid. All employers earn a profit which is equal for all and must be non-negative when all wages offered are in equilibrium. R(L) which is the revenue earned by a firm when L workers are employed is a concave curve. Since employees move only from lower to higher paying employers the relationship between labor supply and wage is strictly increasing. As a result of this condition, the employer s offer maximizes their profit as long as employment does not exceed the steady state supply at the wage offered. If employment goes 4

5 beyond the supply, there is excess demand which will lead to an increase in wages that will surpass the employers marginal product of labor leading to a loss for the firm. max{r(x) wx x L(w)} = R(L(w)) wl(w) = π x w the support of F The following are properties of an equilibrium established in the paper. Property (a): If there is a mass of firms offering wage ŵ, their common employment size, ˆL, equates the wage and marginal revenue product, i.e. R (ˆL) = ŵ, and ŵ is the largest wage offered Mathematical Derivation: Our profit function is given by R(L(w)) wl(w) = π and if we were to maximize profit with respect to ˆL we would take the derivative of π with respect to ˆL, set it equal to zero to get: Therefore, R (ˆL) = ŵ. π ˆL = 0 = R (ˆL(ŵ)) ŵ Property (b): For all employers offering less than the largest wage the marginal revenue product exceeds the wage, i.e. R (L(w)) > w for w < ŵ Property (c): If two different wage rates are both in the support of F and are both strictly less than the largest wage offer, then all wage rates between them are in the support Property (d): If there is wage dispersion, then the lowest wage offered is b. The preceding properties suggest that by equating the wage offered by firms and their marginal revenue product to obtain their common employment size. Employers offering less than the largest wage, will have a marginal revenue product that is greater than the wage.if there are two different wage rates that fall into the wage offer distribution and are both lower then the largest wage offer, all wage rates within the range of the two will be offered by firms. Finally, if different wage rates are offered, the lowest wage rate is 5

6 the unemployment benefit, b. The implications and explanations of the properties are as follows: The wage offered can never exceed the marginal product revenue as it violates the profit maximization condition. There must be excess demand for employment for all wage rates below the highest wage offered since this is what drives the wage rate to increase to the largest wage offered. The labor supply to a firm depends on the number of firms that are offering a higher wage. If a large number of firms are offering higher wages the lower paying firms would have a smaller supply of labor since employees are seeking higher paying jobs, and would gravitate towards the higher paying employers. As a result the labor supply to the lowest paying employer (being unemployed since b is the lowest offer) would be the same for all other wage offers since they are all higher than b. 6

7 3 Equilibrium Relationships In the steady state, there must be a balance between the movement into and out of jobs with respect to wage w. General Equation: λ[αg(w) + (1 α)f (w)](m E) = δeg(w) + λ(α[1 G(w)] + (1 α)[1 F (w)])eg(w) (1) The LHS of this function represents unemployed people entering employment at or below wage w; and the RHS represents employed workers leaving their jobs either to move from w to a higher wage or due to exogenous reasons. Further, we get the equation under the highest wage λ[αγ + (1 α)φ](m φˆl) = δφˆl (2) Where γ is the fraction of people who are earning the highest wage under the wage distribution G(w), while φ is the fraction of employers who are offering the highest wage under the offer distribution F (w). Of course, the various definitions require φˆl = γe (3) Which means, number of employees demanded by the highest paying firms will equal the number of workers employed at the highest wage. As all offers are necessarily acceptable in equilibrium, workers transit from unemployment to employment at the offer arrival rate λ and from employment to unemployment at the separation rate δ. Hence the steady state measure of workers employed, and that which equate flows in and out, is given by E = λm (4) δ + λ Because Eδ = λ(m E) In other words, people who separate (go from employed to unemployed) equals those who acquire jobs (go from unemployed to employed) in the 7

8 steady state. Substituting (4) into (1) and solving for F (w), we get: F (w) = (δ + λ)g(w) δ + λg(w) + α 1 α λ[1 G(w)]G(w) w < ŵ (5) δ + λg(w) This represents the relationship between F (w) and G(w). Seeing as F (w) is the PDF and G(W ) is the CDF, we have the following identity: G(w) E w b L(x)F (x) This function is a general function of (3), which says we are at a steady state when workers earning at most w will be equal to the workers demanded by firms paying at most w. A differentiation of this equation gives us the following function: L(w) = L (G(w), α) dg df E = [δ + λg(w)] 2 δ[δ + λ] + [α/(1 α)]λ[δ 2δG(w) λg 2 (w)] b w < ŵ In a case of a dispersed equilibrium, the equal profit condition, and property (d) require that the equilibrium profit expressed as a function of the contact probability is the following. This is obtained by setting G(w) = G(b) = 0 which represents those being offered wage b. See Appendix for proof ( ) (1 α)δ π(α) = R(L(b)) bl(b) = R (1 α)δ + λ E (1 α)δ b (1 α)δ + λ E (7) Because of the equal profit condition, the profit of the firm who pays the highest is equal to that of the firm who pays the lowest. According to this, we can get a function of profit under a given contact rate. (6) π(α) = R(ˆL(α)) R L(ˆL(α))ˆL(α) (8) Based on property (a) and (b), if w less than the highest wage, then L( w) ˆL(α) (9) 8

9 4 Constructing the Equilibrium After finding functions F (w), G(w), π(α) in the previous sections, we can use the following steps to obtain an equilibrium: 1. The equilibrium profit candidate π is determined by (7). Since different α represent different levels of profit, so we can draw different isoprofit curves on this figure. 2. In a given w, then we can get L(w) from the function π above. Note that only the smaller of the two has the property that L is non-decreasing in w as required by (6) and the fact that G(w) is a distribution function. Because when wage w on the isoprofit curve is less than ŵ, then the horizon line of w will cross the isoprofit curve at two points. However, as we can see from the figure, only the smaller one with lower L satisfies these requirements. If we start from the higher L, we will find that the L is decreasing in w. 3. Using the function L(w) derived in the first step and function(6) to solve for the wage distribution G(w) for all w less than the largest wage ŵ. Two cases in which all employers offer the competitive clearing market wage in equilibrium: 1. R (E) = b, all employers will offer the same wage b. 9

10 2. They will pay the resulting wage of (10) if it is profitable for the firm to pay more than b. R (E) > b (10) The level of profit that would prevail in a competitive equilibrium is: φ = γ = 1, ˆ(w) = R (E) if π(α) π = R(E) R (E)E (11) Note that ˆL(α) is decreasing in α, because w is increasing in α, and this will reduce the labor force demand. The intuition about the competitive equilibrium for large values of? may be gained as follows. The information flows via personal contacts are generally based on the scale of firm. A larger firm, with a higher wage in equilibrium, will provide more opportunities for employees to get offers from contact information. If the contact probability increases, the flow of applicants to the highest wage firms will increase, which will shrink the proportion of low wage applicants. If the situation gets deteriorating, then the small firm s profit will decrease, for they cannot hire enough employees they need. Because wage dispersion equilibria occurs in two cases one is with a mass of firms offering the highest wage and another is without a mass of firms offering the highest wage depending on the value of the contact probability(α). 10

11 Throughout the section, the author tried to find an equilibrium point in dispersed equilibrium. Then he searches for an α which satisfies (12). We have discussed that when α increases, ˆL will decrease in the former section using equation (7) and (8). When G(w) = 1, L(w) can be simplified to L (1, α) = L(ŵ) = δ + λ α δ [ ]λ 1 α, for α < δ αλ,we can get δ >, L(w) is positive and strictly increasing. δ+λ 1 α When the denominator in L(w) tends to zero, L(w) tends to infinity. The following is used to prove equation (9) by way of contradiction: If α > ˆα, then there will be a mass of employers who offer the highest wage ŵ. At the same time, F ( w) = 1 along with equation (12) implies L(ŵ) = L (1, α) > ˆL(α). But this would contradict equation (9) and so it contradicts our model. The next step is to set up a coordinate axis to build a relationship between φ and γ using equations (2)-(4). There must exist a unique and positive α between ˆα and α less than unity for these variables. ˆL = L (1, ˆα) (12) Based on the fact that R(L) is strictly concave, and ˆL(α) is decreasing when α increases. We know that if α < α, ˆL(α) > E. Hence, E/ˆL(α) = φ/γ < 1, as illustrated in Figure 2. φ = ( δγ δ+λ(1 γ) αγ) (1 α) (13) We then substitute equation(4) into (3), substitute(3) into (2) to eliminate φˆl and we obtain equation(13). This is significant because we can see equation(13) is satisfied at (0,0) and (1,1) two points. Taking the derivative to γ we get equation (14). dφ dγ = ( δ(δ+λ) [δ+λ(1 γ)] 2 α ) (1 α) (14) When γ = 0, the value of E/L (1, α) < E/ˆL(α) since we assume that α > ˆα, it implies L (1, α) > ˆL). Thus we use the slope of two equations to find the equilibrium point. In the point (0,0), the slope of equation (13) is less than 11

12 equation (3). Therefore,(13) will increase until it intersects the ray(3) once within this interval. Therefore, the unique solution to (3) and (13) is the point when (0,0) (1,1), the equilibrium has no mass point in this case. Which means no mass firms pay the highest wage in this case. 12

13 5 Personal Contacts and Wage Outcomes If the contact probability is high enough, then the previous section tells us that employers will offer the competitive wage as the wage offer. Seeing as competitive wage is higher than any wage with dispersed equilibrium then a high enough contact probability leads to higher wage outcomes at equilibrium. The following is a result of equation (5) and it tells us higher contact probability is beneficial to obtaining greater job offers. Proposition1. Given a dispersed offer distribution F (w), the associated steady state earning distribution G(w) is stochastically increasing in the contact probability α The intuition behind this is that people only move from low paying to high paying jobs, and the probability of getting a value from wages earned distribution, G(w), is higher than the probability of getting the same value from the wage offer distribution, F (w).since the larger value is more likely to be obtained from G than from F, the probability of it being obtained through a contact is higher. Thus, the probability of receiving a higher offer increases with α. It is obvious then that the number of workers earning a higher wage in the steady state increases because of the increase in probability of receiving a higher wage offer. In other words, higher wage offers lead to higher wages accepted and higher wages in steady state. The following proposition helps us understand that even though there is an adjustment in the equilibrium offer distribution through contact probability, α, it does not affect the partial equilibrium effect we just discussed through the first proposition. Proposition 2. The equilibrium earning distribution G(w) is stochastically increasing in α Proof. By using equation (6) and knowing that there is equal profit in our model, G(w) solves: R(L (0, α)) bl (0, α) = R(L (G(w), α)) wl (G(w), α) (15) for all w offered less than ŵ. Completely differentiating (15) with respect 13

14 to α for every such fixed value of w gives us: L 1 (G(w), α) dg(w) dα = R (L(b)) b R (L(w)) w L 2(0, α) L 2 (G(w), α) (16) where the subscripts denote partial derivatives. Because R (L) is decreasing, w > b, L(w) > L(b), and Property (b) holds, R (L(b)) b > R (L(w)) w > 0 w < ŵ (17) Given G positive and α in the unit interval, equation (6) implies: L 2(G, α) L(w) = λ(δ 2δγ λg 2 ) (1 α) 2 [δ(δ + λ) + [α/(1 α)]λ(δ 2δG λg 2 )] λδ < (1 α) 2 [δ(δ + λ) + [α/(1 α)]λδ] = L 2(0, α) L(b) (18) Hence, we know L(w) L(b) and by equation (6) L 2 (G(w), α) > L 2 (0, α) (19) Given (17) and (19) together with the fact that L 1 (G, α) > 0 and L 2 (0, α) < 0 we get: dg(w) dα < L 2(0, α) L 2 (G(w), α) < 0 L 1 (G(w), α) by (16). QED. It makes sense that profit decreases while the distribution of wages offered/earned increase with an increase in unemployment benefit b. Employers would have to pay higher than b to incentivize people to come work for them and they would have to pay higher wages if this b were to increase. Similarly, if b increases then we would require for the wage distributions to increase in a stochastic dominance sense. This is similar to an increase in the revenue function, where it shifts out and gives us the same results. Meanwhile, an increase in the supply, M, will have opposite effects. It will increase profit and employed people in each firm while decreasing the wages offered/earned. Recall if supply increases, prices fall. Using proposition (2) and equation (5) we can see that wage earnings/offers fall in the sense that F and G are stochastically decreasing in M. 14

15 One important thing to take away from this is that an increase in the arrival rate (λ) or a reduction of the separation rate (δ) (same effect) will have two opposing reactions. First, there will be an increase in the labor force size as more people are now employed. Second, the increased labor force size of higher wage employers increases relative to those who pay less. The net effect is ambiguous as it is unclear which effect dominates the other. 15

16 6 Differential Access Through Contacts In the entire paper we have assumed that people have equal access to information, however in real life this is not the case. This section extends the preceding results for multiple worker types who have different contact probabilities. A strong motivation for this paper is the prevalence and degree of social segregation along demographic lines. Suppose we have workers who are of type i, M i represents the measure of workers per employer and α i represents the probability that an offer is from a contact type i. As stated before, our assumptions are that all workers are otherwise identical in their reservation wage, rate of separation, rate of offer arrival, and level of productivity. Suppose all workers of type i have the same offer distribution F. In other words, employers are not permitted to discriminate and give offers which are type contingent. Equations (2) through (6) will hold for each type with M = M i and α = α i. The total labor force available at each wage is given as the sum: L(w) = L i (w) (20) Note that a dispersed equilibrium will exist if the profit equation below exceeds the profit that results in competitive equilibrium for the entire vector of contact probabilities α = (α 1, α 2,...), π(α) = R ( δ(1 αi ) δ(1 α i ) + λ E ) δ(1 αi ) i b δ(1 α i ) + λ E i, E i = λm i δ + λ (21) As we established before, in this equilibrium, those with greater access to jobs through contacts are more likely to earn more as a corollary of proposition (2). Employers generally have an incentive to discriminate, however, against one or the other type since access to information through contacts is bound to affect the elasticity of the available labor supply. Let F i (w) represent the distribution of wages offered to type i. We know that we have a dispersed equilibrium if the vector of contact probabilities α is small enough. The equal profit condition (1) requires the following analogue of (15): 16

17 R ( Li (0, α i ) ) b L i (0, α i ) = R ( Li (G i (w i ), α i ) ) w i L i (G i (w i ), α i ) (22) for all w i less than the largest offer to type i where from eq. (6) L i (w) = L i (G i (w), α i ) [δ + λg i (w)] 2 δ[δ + λ] + [α i /(1 α i )]λ[δ 2δG i (w) λg 2 i (w)] λm i δ + λ (23) Our previous proposition (2) holds once more in this case given that the argument used to prove it holds given (22) and (23). Proposition 3. Given a dispersed equilibrium with two or more types denoted as i and j, α i > α j implies G i (w) < G j (w) for all w between the lowest and the highest offer In sum, Propositions 1 and 3 imply that those who have less contacts can expect to earn less whether or not discrimination is present. Therefore, it s all about who you know. 17

18 7 Summary The model in this paper is consistent with empirical findings. First, the distribution of wage rates received by those who obtained information through contacts is stochastically greater than the distribution received by those who applied directly to their employer. Second, for job to job movers, the wage is also higher in the same sense for those who find the new job through contacts. Finally, because wages earned are higher, a subsequent quit is less likely for those who found their job through contacts. According to everything that has been proposed in this paper, if men have a higher contact probability than women, say because members of each sex network primarily with their own and men have a higher employment ratio than women, then the model predicts that men earn more in the sense that the male distribution of wages earned stochastically dominates that for females both within occupations composed of men and women and across occupations that are exclusively male and female. 7.1 Extensions and Applications The Caste system came about in India to create division of labor, however it has now caused segregation instead of providing economic benefits. In regards to inequality this paper can help further explain the Caste system. People stay within their caste because they are less likely to know people outside their caste. This then causes a decrease in the ability to move up in the caste system and thus reduces the opportunity for economic mobility for individuals who may of the same productivity. Moreover, this paper has implications with regards to us Duke students. We are at a private institution with probably more contact probabilities than other people of our age at public institutions or elsewhere. Everything being equal in the model, we are more likely to earn higher wages due to the fact that we have higher contact probabilities. In the end this causes an inequality and some form of discrimination in the way that we know our contacts which causes higher wages. In the current economic condition, a considerable amount of people do not have jobs right now, and the people who do are probably not being paid 18

19 as high as they should be. However, if someone is looking for a job abd if they have high contact probability they will earn a higher wage offer than their counterparts who do not have contacts. 7.2 Pitfalls While this paper brings up inequality in wages and discrimination, the model is not fully applicable due to several pitfalls. 1. It is unlikely for all people to have the same reservation wage. Moreover, this model is unlikely to hold in a country with no unemployment benefits. 2. People are not equally productive. 3. Although number of contacts is important, so is quality of contacts to obtaining higher wages. However, this would be difficult to model. First, what would constitute a higher quality contact? For example, if I were to know 20 workers at Wall Street this does not necessarily mean that I am more likely to obtain a higher paying job than someone who does not know someone working on Wall Street, because it could be the case that I know 20 janitors but applying for a managerial position. 19

20 8 Appendix 8.1 Derivation of Equation (5) We start with equation (1): λ[αg(w)+(1 α)f (w)](m E) = δeg(w)+λ(α[1 G(w)]+(1 α)[1 F (w)])eg(w) Now we try to solve for F (w) λαg(w)(m E) + λ(1 α)f (w)(m E) = δeg(w) + λα[1 G(w)]EG(w) + λ(1 α)[1 F (w)]eg(w) F (w)(m E) [1 F (w)]eg(w) = δeg(w) + λα[1 G(w)]EG(w) (M E)λαG(w) λ(1 α) Now we can substitute equation (4) into E in the above equation s LHS: F (w)(m λm λm ) [1 F (w)] δ + λ δ + λ G(w) = F (w)m F (w) λm δ + λ λm λm G(w) + F (w) δ + λ δ + λ G(w) = Now working with the entire equation: F (w)m F (w) λm λm + F (w) δ + λ δ + λ G(w) δeg(w) + λα[1 G(w)]EG(w) (M E)λαG(w) = λ(1 α) + λm G(w) (24) δ + λ So we can see that we added λm G(w) to both sides of the equation, leaving δ+λ us with the following on the LHS: F (w)δm δ + λ + λm G(w)F (w) = δ + λ [ (δ + λg(w))m ] F (w) δ + λ 20

21 Finally, we can isolate F (w): F (w) = [ δ + λ δeg(w) G(w)]EG(w) (M E)λαG(w) +λα[1 + δmg(w) ] (δ + λg(w))m λ(1 α) λ(1 α) λ(1 α) λ(1 α) δg(w) = (1 α)(δ + λg(w) αδg(w) (1 α)(δ + λg(w) + α λ[1 G(w)]G(w) + λg(w) (1 α (δ + λg(w) δ + λg(w) = (1 α)δg(w) (1 α)(δ + λg(w) + λg(w) (δ + λg(w) + α (1 α) λ[1 G(w)]G(w) [δ + λg(w)] (25) After terms cancel-out,this simplifies the above equation and we are left with equation (5): F (w) = (δ + λ)g(w) δ + λg(w) + α λ[1 G(w)]G(w) (1 α) [δ + λg(w)] (26) 8.2 Derivation of Equation (6) If we treat G as a function of F, then the equation (5) will be an implicit function. Because, we will write y as a function of x in the form of y = f(x), but equation (5) is in the form of 0 = f(g, F ). In this case, when take differentiation of (5), we should differentiate both sides of the equation with respect to F. F (w) = (δ + λ)g(w) δ + λg(w) + α λ[1 G(w)]G(w) 1 α δ + λg(w) (27) Differentiate both sides of the equation with respect to F: 1 = dg dg (δ + λ) (δ + λg) (δ + λgλ df df + α λ[1 2G] dg(δ + λg) λ(g df G2 )λ dg df [δ + λg] 2 1 α [δ + λg] 2 (28) Then, the whole equation can be rewritten as: dg df = [δ + λg] 2 δ(δ + λ) + α (1 α)λ[δ 2δG λg 2 ] (29) which is the first term on the RHS of equation (6). 21

22 8.3 Derivation of Equation (7) To derive equation 7 we find L(b) by plugging in G(w)=0 into equation (6) because no one will be working for less than the reservation wage. Thus, Simplify with the zeros: [δ + λ(0)] 2 L(b) = L (0, α) α δ[δ + λ] λ[δ 2δ(0) 1 α λ(0)2 ] E (30) L(b) = [δ] 2 δ[δ + λ] + α (31) λ[δ]e 1 α The δs in the denominator cancel out, leaving us with: L(b) = Now we can use algebra to simplify the denominator: δ δ + λ + α (32) λe 1 α L(b) = δ (δ+λ)(1 α)+λα 1 α E (33) L(b) = L(b) = δ δ+λ δα αλ+λα (1 α) δ(1 α) λ + δ(1 α) E (34) (35) 22