21.23 Principal-Agent Models and Salesforce Compensation
Key Question:
- Ensuring agents exert effort
- Design compensation plans such that workers exert high effort?
Designing contracts:
Effort can be monitored
Monitoring costs are too high
Timing
- Manger designs the construct
- manager offers the construct and worker chooses to accept
- Worker decides the extent of effort
- Outcome is observed and wage is given to the worker
Scenario 1: Certainty
e = effort put in by worker
2 levels of e
- 2 if he works hard
- 0 if he shirks
Reservation utility = 10 (other alternative: can work somewhere else, or private money allows them not to work)
Agent’s Utility
\[ U = \begin{cases} w - e & \text{if he exerts effort e} \\ 10 & \text{if he works somewhere else} \end{cases} \]
Revenue is a function of effort
\[ R(e) = \begin{cases} H & \text{if } e = 2 \\ L & \text{if } e = 0 \end{cases} \]
Contract
\(w^H\) = wage if \(R(e) = H\)
\(w^L\) = wage if \(R(e) = L\)
Constraints:
Worker has to participate in this labor market - participation constraint \(w^H - 2 \ge 10\)
Incentive compatibility constraint (ensure that the works always put in the effort and the manager always pay for the higher wage): \(w^H - 2 \ge w^L -0\)
Hence,
\[ w^H = 12 \\ w^L = 10 \]
Thus, contract is simple because of monitoring
Scenario 2: Under uncertainty
\[ R(2) = \begin{cases} H & \text{w/ prob 0.8} \\ L & \text{w/ prob 0.2} \end{cases} \\ R(0) = \begin{cases} H & \text{w/ prob 0.4} \\ L & \text{w/ prob 0.6} \end{cases} \]
Agent Utility
\[ U = \begin{cases} E(w) - e & \text{if effort e is put} \\ 10 & \text{if they choose outside option} \end{cases} \]
Constraints:
Participation Constraint: \(0.8w^H + 0.2w^L -2 \ge 10\)
Incentive compatibility constraint: \(0.8w^H + 0.2w^L - 2 \ge 0.4 w^H + 0.6w^L - 0\)
Thus,
\[ w^H = 13 \\ w^L = 8 \]
Expected wage bill that the manager has to pay:
\[ 13\times 0.8 + 8 \times 0.2 = 12 \]
Hence, the expected money the manager has to pay is the same for both cases (certainty vs. uncertainty)
Scenario 3: Asymmetric Information
Degrees of risk aversion
Manger perspective
\[ R(2) = \begin{cases} H & \text{w/ prob 0.8} \\ L & \text{w/ prob 0.2} \end{cases} \]
Worker perspective (the number for worker is always lower, because they are more risk averse, managers are more risk neural) (the manager also knows this).
\[ R(2) = \begin{cases} H & \text{w/ prob 0.7} \\ L & \text{w/ prob 0.3} \end{cases} \]
Participation Constraint
\[ 0.7w^H + 0.3w^L - 2 \ge 10 \]
Incentive Compatibility Constraint
\[ 0.6 w^H + 0.3 w^L - 2 \ge 0.4 w^H + 0.6 w^L - 0 \]
(take R(0) from scenario 2)
\[ 0.7 w^H + 0.3 w^L = 12 \\ 0.3w^H - 0.3w^L = 2 \]
Hence,
\[ w^H = 14 \\ w^L = 22/3 \]
Expected wage bill for the manager is
\[ 14 * 0.8 + 22/3*0.2 = 12.66 \]
Hence, expected wage bill is higher than scenario 2
Risk aversion from the worker forces the manager to pay higher wage
| Salesperson | |||
|---|---|---|---|
| Risk neutral | Risk averse | ||
| Effort | Observable | Any mix desired effort |
All salary Desired effort |
| Not observable | All commission Desired effort |
Specific mix (S+C) Salesperson shirks |
Grossman and Hart (1986)
- landmark paper for principal agent model
21.23.1 Basu et al. (1985)
Types of compensation plan:
Independent of salesperson’s performance (e.g., salary only)
Partly dependent on output (e.g., salary with commissions)
In comparison to others (e.g., sales contests)
Options for salesperson to choose the compensation plan
In the first 2 categories, the 3 major schemes:
- Straight salary
- Straight commissions
- Combination of base salary and commission
| Compensation Type | Best when | Limitation |
|---|---|---|
| Straight salary | Long-term objective Hard to measure performance |
Less effort |
| Straight commission | Easy-to-measure performance | Effort-reward ratio is emphasized High risk (uncertainty) to the salesperson |
| Combination |
Dimensions that affect the proportion of salary tot total pay (p. 270, table 1)
Previous research assumes deterministic relationship between sales and effort, but this study says otherwise (stochastic relationship between sales and effort).
Assumptions:
Firm: Risk neutral: maximize expected profits
Salesperson: Risk averse . Hence, diminishing marginal utility for income \(U(s) \ge 0; U'(s) >0, U''(s) <0\)
Expected utility of the salesperson for this job > alternative
Utility function of the salesperson: additively separable: \(U(s) - V(t)\) where \(s\) = salary, and \(t\) = effort (time)
Marginal disutility for effort increases with effort \(V(t) \ge 0, V'(t)>0, V''(t) >0\)
Constant marginal cost of production and distribution \(c\)
Known utility function and sales-effort response function (both principal and agent)
dollar sales \(x \sim Gamma, Binom\)
Expected profit for the firm
\[ \pi = \int[(1-c)x - s(x)]f(x|t)dx \]
Objective of the firm is to
\[ \underset{s(x)}{\operatorname{max}} \int[(1-c)x - s(x)]f(x|t)dx \]
subject to (agent’s best alternative e.g., other job offer - \(m\))
\[ \int [U(s(x))]f(x|t) dx - V(t) \ge m \]
and the agent wants to maximize the the utility
\[ \underset{t}{\operatorname{max}} \int [U(s(x))]f(x|t)dx - V(t) \]
21.23.2 Lal and Staelin (1986)
21.23.3 Raju and Srinivasan (1996)
Compare quota-based compensation with (Basu et al. 1985) curvilinear compensation, the basic quota plan is simpler, and only in specical cases (about 1% in simulation) that differs from (Basu et al. 1985). And it’s easier to adapt to changes in moving salesperson and changing territory, unlike (Basu et al. 1985)’s plan where the whole commission rate structure needs to be changed.
Heterogeneity stems from:
Salesperson: disutility effort level, risk level, effectiveness, alternative opportunity
Territory: Sales potential an volatility
Adjusting the quota (per territory) can accommodate the heterogeneity
Quota-based < BLSS (in terms of profits)
Constraints:
- quota-based from curve (between total compensation and sales) (i.e., shape-induced nonoptimality)
- common salary and commission rate across salesforce (i.e., heterogeneity-induced nonoptimality)
To assess the shape-induced nonoptimality following
21.23.4 Joseph and Thevaranjan (1998)
21.23.5 Simester and Zhang (2010)
- Tradeoff: Motivating manager effort and info sharing.