IB Principal-Agent Model PDF

Behavioural Economics of Organisations The Principal-Agent Problem Winter 2024-25 Introduction So, where should we start? We’ll begin our study of employees’ motivation, and by doing some theory Specifically, we’ll study the simplest possible theoretical model of the optimal design of financ...

Behavioural Economics of Organisations The Principal-Agent Problem Winter 2024-25 Introduction So, where should we start? We’ll begin our study of employees’ motivation, and by doing some theory Specifically, we’ll study the simplest possible theoretical model of the optimal design of financial incentives—i.e. the principal-agent model To motivate this model, imagine you have just been seriously injured in an accident in one of the big box stores. You need to hire a lawyer to sue the store for damages. How should you pay this person? 2 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Structure of the Principal-Agent Problem Timeline of Actions in the Principal–Agent Problem (Figure 1.1, Personnel Economics, 1e, Peter Kuhn Copyright © 2018 Oxford University Press) 3 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Structure of the Principal-Agent Problem Essentially, there are two key stages: 1. The principal (or the principal and agent together) set the “rules of the game” or contract. These rules must be acceptable to the agent, or they won’t work for the principal. 2. The agent maximises his own utility, taking the contract/rules as given, and payments are made. Solution Method (Backwards Induction): 1. Figure out what the agent is going to do under a set of rules: (i.e., find the agent’s “reaction function”). 2. Solve for the optimal contract/rules given the agent’s expected response. 4 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Structure of the Principal-Agent Problem Our Example: One principal, one agent, no uncertainty One output ( ), observed by principal and agent. is dollars of net revenue. Principal can’t observe effort ( ) (so we can’t base the contract directly on it). The production function: is a productivity parameter that can capture ability or technology differences 5 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Structure of the Principal-Agent Problem Our Example: Much of the time, we’ll use our baseline production function : When we use this function, we are measuring effort in terms of the amount of output it yields, i.e. one unit of effort is ‘what it takes’ to produce one unit of output. 6 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Structure of the Principal-Agent Problem The agent’s utility is: where is income and is the cost of effort. We assume increasing marginal costs of effort, i.e. and 7 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Structure of the Principal-Agent Problem Cost of effort function: Much of the time, we will use this baseline cost-of-effort function: (Figure 1.2, Personnel Economics, 1e, Peter Kuhn Copyright © 2018 Oxford University Press) 8 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Structure of the Principal-Agent Problem The utility function (1) can be represented by an indifference map, which looks like this: Re-arranging equation (1), the equation for an indifference curve is just , or. For any given effort level, , the indifference curve tells us how much we must pay the agent ( ) to give them a utility level of. The slope of every indifference curve is , (Figure 1.3, Personnel Economics, 1e, Peter Kuhn Copyright © 2018 Oxford University Press) 9 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Structure of the Principal-Agent Problem In general, the contract between the principal and agent can be any relationship between what the principal observes (in this case ) and what the agent is paid ( ). In today’s example, we will restrict our attention to linear piece rate contracts, i.e. to contracts of the form :  The “contract” is an ordered pair, , where is base pay and is the “piece rate” (per dollar of net revenue generated by the agent’s effort). (From Instructors’ resources, Personnel Economics, 1e, Peter Kuhn Copyright © 2018 Oxford University Press) 10 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Structure of the Principal-Agent Problem Finally, we assume the agent’s alternative utility is. Thus, the agent’s participation constraint can be expressed as: This participation constraint can be interpreted in two ways: 1. Literally: In order to maximise profits, an employer knows and wants to offer the agent the lowest possible utility that induces the agent to accept the contract, i.e. a contract that leaves the agent indifferent between accepting or not. 2. As a mathematical device for finding any “optimal” contract: First, choose how well off you want the agent to be. Then make yourself as well off as possible. 11 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Agent’s Problem (Assuming a linear contract , and using our baseline production- and cost-of-effort functions) Given , , , subject to: , and to 12 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Agent’s Problem Substituting the two constraints (and the baseline effort-cost function) into the maximand, this is equivalent to: The first-order condition (FOC) for a maximum is: Solving this for the effort level the agent will choose, yields: ∗ So, a higher commission rate ( ) induces higher effort. Changing base pay ( ) has no effect on effort. 13 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Agent’s Problem Agent’s Problem with Graphs – part 1: By definition, E* maximises the difference between Y and V(E) (a). At E*, the marginal benefit of effort, b, just equals the marginal cost of effort, V′(E) (b). (Remember, we set d=1) You can also see this in part a, where the slope of the V(E) curve just equals the slope of the pay schedule (b) at E*. (Figure 2.1, Personnel Economics, 1e, Peter Kuhn Copyright © 2018 Oxford University Press) 14 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Agent’s Problem Agent’s Problem with Graphs – part 2: The optimal effort level E* is where the budget constraint is just tangent to the highest indifference curve attainable. At that point the slope of both curves equals (and therefore ). (Remember, we set d=1) (Figure 2.3, Personnel Economics, 1e, Peter Kuhn Copyright © 2018 Oxford University Press) 15 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Agent’s Problem Summing up our results for the agent’s problem: Agents’ reactions to changes in the employment contract and productivity (d) [Result 2.1 in textbook]: 1. For any given contract, more productive agents (with higher ) will work harder than less productive agents. 2. Raising the slope parameter ( ) of the employment contract will make the agent work harder. 3. Changing the intercept parameter ( ) of the employment contract will have no effect on the agent’s optimal effort level. 16 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Agent’s Problem Summing up our results for the agent’s problem: Agents’ reactions to changes in the employment contract with the baseline 𝑬𝟐 production (Q = E) and cost-of-effort functions ( 𝟐 ) and d = 1 [Result 2.2 in textbook] : 1. The agent’s optimal effort, , now equals exactly. Therefore, it’s still true that: 2. Raising the slope parameter ( ) of the employment contract will make the agent work harder. 3. Changing the intercept parameter ( ) of the employment contract will have no effect on the agent’s optimal effort. 17 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem The Principal’s problem is to figure out: how high a commission rate ( ) to pay the agent, and how much to pay the agent to show up ( ), so as to maximise profits. 18 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem Before doing the full-blown principal’s problem, we’ll do a simpler, “warm-up” problem first. As it turns out, this “warm-up” problem is the way most people think about the P-A problem when they first encounter it. It gives the wrong answer to the P-A problem. But we’ll learn some useful things from this mistake. Note: We’ll stick to the baseline production and utility functions for the rest of this chapter. 19 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem The Warm-up Problem: Maximising profits when (This seems like the right thing to do, since we’ve already proved that raising doesn’t elicit any additional effort.) Mathematically, we want to solve: Subject to the agent’s incentive-compatibility constraint,. (We’ll ignore the agent’s participation constraint in this warm-up problem by assuming that the principal’s most preferred contract turns out to be acceptable to the agent). 20 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem Substituting the agent’s response to the contract, , into the definition of profits yields: The first-order condition (FOC) for a maximum is: (SOC for a maximum is fulfilled, too) Thus, ∗ a 50% (of net revenues generated by the agent) commission rate maximises profits in this situation. 21 KING’S BUSINESS SCHOL | kcl.ac.uk/business Solving the Principal’s Problem To see why, consider a graphical solution: At the profit-maximising contract: Agent’s effort level will be: (Using the production function) output will be: Profits will be:. Agent’s utility will be: (Figure 3.1, Personnel Economics, 1e, Peter Kuhn Copyright © 2018 Oxford University Press) 22 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem More generally: More generally (beyond the baseline production and utility functions): the profit-maximising is always strictly between zero and 1 for any production and utility function. Why: yields no profits because the agent does nothing yields no profits (despite high effort) because it gives all the profits to the agent. Thus, in this warm-up problem, the profit-maximising commission rate trades off two competing goals: -incentives (higher raises effort) -distribution (higher redistributes income from the principal to the agent). 23 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem A final thing to notice about our ‘warm-up’ solution to the P-A problem: Imagine we implement the suggested solution by setting , then ask: How would the principal and agent feel about raising a little bit? (Figures 3.1 and 3.2, Personnel Economics, 1e, Peter Kuhn Copyright © 2018 Oxford University Press) 24 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem A final thing to notice about our ‘warm-up’ solution to the P-A problem: Principal wouldn’t mind very much (because profits are a relatively flat function of b at ) Agent would like it a lot, (utility is convexly increasing in ) Thus, starting at , a small increase in benefits the worker but doesn’t really hurt the firm. This suggests that efficiency might be improved by raising beyond …. 25 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem The ACTUAL Principal’s Problem Mathematically, we want to solve: , Subject to the agent’s incentive-compatibility constraint, and to the agent’s participation constraint, (we’ll assume this is satisfied with equality, ). 26 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem STEP 1: Characterise the participation constraint, i.e. find the level of needed to reach a utility level of (assuming the agent maximises his utility subject to any contract they faces). To do this, recall that: Rearranging and setting (1) 27 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem STEP 1: (1) If the agent always chooses their effort to maximise their utility (taking as given), equation (1) tells us how much base pay ( ) we need to give the agent so they’ll attain the ‘target’ utility of exactly. The better off we want the agent to be (i.e. the higher a we want to achieve), the higher we need to set. However, since and are alternative ways to make the agent better off, when is higher we don’t need to give the agent as much to attain the same level of utility. 28 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem STEP 2: Subject to: (incentive compatibility constraint) and to: (participation constraint) Profit function then is Taking the derivative with respect to b and setting that equal to zero yields: Therefore, ∗ 29 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem Graphically: (Figure 3.3, Personnel Economics, 1e, Peter Kuhn Copyright © 2018 Oxford University Press) 30 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem Thus, in the profit-maximising contract, the principal should set a commission rate of 100%; i.e. the agent’s pay should rise by one dollar for every dollar the agent contributes to net revenue. Notice that this result doesn’t depend on , i.e. the level at which we choose to set the agent’s utility. It follows that a 100% commission rate is profit-maximising, regardless of how well off we want the agent to be! 31 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem To sum up, let’s list the agent’s effort, output, utility, and the firm’s profits, etc. at two different commission rates (50% and 100%), with set in both cases to guarantee the worker a utility level ( 𝒂𝒍𝒕 ) of (remember ) : 32 KING’S BUSINESS SCHOOL | kcl.ac.uk/business (Table 3.1, Personnel Economics, 1e, Peter Kuhn Copyright © 2018 Oxford University Press) Solving the Principal’s Problem To sum up: We can make the principal better off without hurting the agent by switching from the (a,b) = (1.25,.5) contract to the (a,b) = (-.25, 1) contract. 33 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem Lessons: 1. “Put the rewards where the decisions are made”. When the agent controls a decision that affects the utilities of a larger group (in this case himself plus the principal), it is profit maximising to have the agent bear all the costs, as well as all the returns of his actions. This result does not depend on the level of —i.e on how well-off we want the agent to be. 2. When agents receive 100% of the fruits of their labour at the margin ( ), principals’ only source of profits is from setting a negative level of base pay, (in other words by “selling the job to the worker”). Compared to the ‘warm-up’ problem: we can set b =1 to fully incentivise the agent, while a can be used to achieve any feasible distributional outcome you like. 34 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem Is it Crazy to “sell the job to the worker”? While this might seem strange at first, there are at least three distinct ways it actually occurs in today’s economy. 1. Explicit payment for jobs: Can you think of any examples? 35 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem Here are some links: Angrist, Joshua, Sydnee Caldwell and Jonathan Hall, Uber vs. Taxi: A Driver's Eye View Gentile, Marie. “How Does Commission Work at a Hair Salon?” Chron.com, 2016. Nir Sarah Maslin.“The Price of Nice Nails” New York Times, May 7, 2015 Rooney, Ben. “The FedEx driver who sued and won” CNN Money, November 21, 2014: Weintraub, Elizabeth. Desk Fees for Real Estate Agents thebalance.com, updated June 27, 2016 Kimmons, James. Methods of Compensating Real Estate Agents - Commissions and Splits. thebalance.com, updated 19 September, 2022 Stock traders at proprietary (prop) firms may pay a desk fee, which can range from $200 to $4000 per month. 36 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem Is it Crazy to “sell the job to the worker”? Many economists refer to the ( ) solution to the principal-agent problem as the “franchise solution”. Today, there are about 800,000 franchisees in the United States: www. statista.com, Number of franchises has more than doubled in the last 25 years to 48,000 in the United Kingdom: www.franchise-uk.co.uk. 37 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem Is it Crazy to “sell the job to the worker”? 2. “Buy, don’t make.” Imagine that King’s wants students to have options to buy tasty lunches on campus. One way to provide those lunches would be simply to sell the equipment (or simply the right to operate their cart on campus!) to the individual vendors, and let them keep everything they earn (b = 1) More generally, for agency problems that are this simple, the best solution may not be to hire a worker to provide a product or service, but to buy the product or service from someone else. Because that ‘someone’ is their own business, they bear 100% of the costs and rewards associated with producing their product. 38 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem Is it Crazy to “sell the job to the worker”? 3. Implicit payment for jobs In many cases, workers may be reluctant (for liquidity or trust reasons) to make a large up-front payment for a job, even when that might be socially optimal. For many workers receiving commission or other forms of performance-based pay, a solution is to “build the entry fee into the worker’s pay schedule”: 39 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem To see how this can work let’s imagine a car salesperson who is paid on the basis of their monthly net sales,. The profit-maximising solution to their P-A problem is shown here: at ∗ , the worker’s indifference curve is tangent to their budget constraint, there, the income is less than they produce (by , which is the fee for the job) profits are , which is the vertical distance shown. (From Instructors’ resources, Personnel Economics, 1e, Peter Kuhn Copyright © 2018 Oxford University Press) 40 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem But suppose our worker can’t afford to pay up front for the job. Why not ‘take the entry fee in kind’ by just not paying them for the first n cars they sell each month? More precisely, let’s replace the previous contract, by the contract in bold (see next slide): 41 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem The new schedule sets when Q Q0. 43 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem More precisely, the new contract works like this: If 0 If 0 1 (fixed base pay). If 1 1 where. So the worker still collects their positive ‘draw’, and earns a 100% commission only on units sold above Q1. 44 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem Diagrammatically: This new schedule also yields exactly the same output, profits and utility as the original one, provided that D is not too generous. What happens when D is too high? If D is too high (or if Q0 is too low) point n will lie above the indifference curve through point m. What will the agent do then? (From Instructors’ resources, Personnel Economics, 1e, Peter Kuhn Copyright © 2018 Oxford University Press) 45 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Solving the Principal’s Problem Summing up: Incentive pay plans where: you get a fixed positive base pay as long as you keep your job but you have to sell a minimum amount to qualify for incentive pay can deliver the exact same results as the superficially ‘extreme’ pay plans predicted by our first principal-agent model. -to accomplish this, the draw, D can’t be ‘too’ generous relative to output needed to keep your job, Q0. 46 KING’S BUSINESS SCHOOL | kcl.ac.uk/business Literature Kuhn, P. (2018). Personnel economics. Oxford University Press. Preface Chapters 1-3 47 KING’S BUSINESS SCHOOL | kcl.ac.uk/business