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Lecture 6Contracts This lecture studies how those who create and administer organizations design the incentives and institutional rules that best serve their ends. We focus on schemes that are designed to maximize the manager’s objectives by creating the appropriate incentives for the people he deals with at minimal cost to the organization he manages. We analyze upstream contracts with suppliers, employment contracts for workers, service contracts for consumers, and incentive schemes within the organization.
Designing the bargaining rules An implication of our studies on bargaining is the manifest value from setting the rules and conventions that determine how bargaining proceeds. Almost by definition managers are placed in a strong position to set the rules of bargaining games they play. We look at upstream supply contracts, downstream consumer agreements, and employment contracts with labor.
A rent extraction problem • Employers seek to minimize their wage bill, or in the case of sole proprietors loss in expected utility, subject to two constraints: • They must attract workers they wish to hire. This is called the participation constraint. • The workers must perform the tasks to which they are assigned. This constraint is called incentive compatibility.
Full information principal agent problem A firm wishes to build a new factory, and will hire a builder. How should it structure the contract? Firm:RL-wL Builder: wL-uL RH-wH wH-uH
Constraints facing the firm • We can use backwards induction to solve the problem: • The incentive compatibility constraint is: wH – uH wL – uL if H wL – uL wH – uH if L • The participation constraint is: wH - uH 0 if H wL- uL 0 if L
Theconstraintsillustrated wL wH =uH wH-wL=uH-uL uH -uL wH uH uL-uH (IC)
Minimum cost of achieving L • The minimum cost of achieving L is found by minimizing wL such that: • wL uL • wL – uL wH – uH • The first constraint bounds wL from below by uL. • Since uL uH the second constraint is satisfied by not making the wage depend on effort. • Therefore the minimum cost of achieving L is found by setting w* = u*L
Minimum cost of achieving H • The minimum cost of achieving H is found by minimizing wL such that: • wH uH • wH – uH wL – uL • The first constraint bounds wH from below by uH. • Since uL uH we must penalize the worker to deter him from choosing L, by setting: wL < wH – uH + uL • Therefore the minimum cost of achieving H is: w*H = u*H w*L = wH – uH + uL - Penalty
Profit maximization • The net profits from achieving L are RL – uL* • The net profits from achieving H are RH – uH* • Therefore the firm hires a worker to achieve H if RH – uH* > RL – uL* and hires a worker to achieve only L otherwise.
Service provider Multipart pricing schemes are commonly found in the telecommunications industry, amusement parks. sport clubs, and time sharing vacation houses and small jets. In this example a provider incurs a fixed cost of c0 to connect the consumer to the facility, and a marginal cost of c1 for every unit provided. It follows that if the consumer purchases x units the total cost to the provider is: c0 + c1x. We assume the monetary benefit to the consumer from a service level of x is: x1/2. How should the provider contract with the consumer?
Optimal contracting • To derive the optimal contract, we proceed in two steps: • derive the optimal level of service, by asking how much the consumer would use if she controlled the facility herself. • calculate the equivalent monetary benefit of providing the optimal level of service to the consumer, and sell it to the consumer if this covers the total cost to the provider. • The equivalent monetary benefit can be extracted two ways, as membership fee with rights to consume up to a maximal level, or in a two part pricing scheme, where the consumer pays for use at marginal cost, plus a joining fee.
A parameterization In our example we maximize x1/2 - c0 - c1x with respect to x to obtain interior solution x = (2c1)-2 It follows that the costs from an interior solution are: c0 + 1/4c1 and the monetary equivalent from consuming the optimal level of service is 1/2c1. Therefore the provider extracts 1/2c1 if: 4c0c1 <1
Charging a uniform price If the service provider charges per unit instead, the consumer would respond by purchasing a level of service a a function of price. Anticipating the consumer’s demand, the provider constructs the consumer’s demand curve, and sets price where marginal revenue equals marginal cost. The provider serves the consumer if and only if the revenue from providing the service at this price exceeds the total cost. Since lower levels of service are provided, and since the consumer achieves a greater level of utility, than in the two part contract, the provider charging a unit price realizes less rent than in the two part contract.
The parameterization revisited In our example the consumer demands x = (2p)-2 where p is the uniform unit price of the service. The service provider maximizes: x1/2/2 - c0 - c1x with respect to x to obtain the interior solution x = (4c1)-2 which is the optimal choice if: 16c0c1 <1
Comparing multipart with uniform pricing schemes Since lower levels of service are provided, and since the consumer achieves a greater level of utility, than in the two part contract, the provider charging a unit price realizes less rent than in the two part contract.
Service provider Multipart pricing schemes are commonly found in the telecommunications industry, amusement parks. sport clubs, and time sharing vacation houses and small jets. In this example a provider incurs a fixed cost of c to connect the consumer to the facility, and a marginal cost of 1 for every unit provided. It follows that if the consumer purchases x units the total cost to the provider is: c + x. We assume the monetary benefit to the consumer from a service level of x is: 8x1/2. How should the provider contract with the consumer?
Optimal contracting • To derive the optimal contract, we proceed in two steps: • derive the optimal level of service, by asking how much the consumer would use if she controlled the facility herself. • calculate the equivalent monetary benefit of providing the optimal level of service to the consumer, and sell it to the consumer if this covers the total cost to the provider. • The equivalent monetary benefit can be extracted two ways, as membership fee with rights to consume up to a maximal level, or in a two part pricing scheme, where the consumer pays for use at marginal cost, plus a joining fee.
A parameterization In our example we maximize 8x1/2 – c – x with respect to x to obtain interior solution 4x-1/2 = 1 or x = 16 Hence the costs from an interior solution are c + 16, and the monetary equivalent from consuming the optimal level of service is 32. Therefore the provider can extract 16 – c if c < 16. A two part pricing scheme that achieves this goal is to charge a joining fee of 16 and a unit price of 1, achieving profits of 16 – c.
Charging a uniform price If the service provider charges per unit instead, the consumer would respond by purchasing a level of service a a function of price. Anticipating the consumer’s demand, the provider constructs the consumer’s demand curve, and sets price where marginal revenue equals marginal cost. The provider serves the consumer if and only if the revenue from providing the service at this price exceeds the total cost.
The parameterization revisited Supposing p is the price charged for a service unit, the consumer maximizes: 8x1/2 – px The first order condition yields the consumer demand 4x-1/2 = p = p(x) or x = 16p-2 The service provider maximizes: p(x)x – c – x = 4x1/2 – c – x with respect to x to obtain the interior solution 2x-1/2 = 1 or x = 4 and p = 2 In this case the firm extracts a rent of 4 – c if c < 4.
Comparing multipart withuniform pricing schemes In a two part contract rents are 16 – c but with a uniform price the rent is only 4 – c if c < 4. Furthermore if 4 < c < 16, a uniform price scheme cannot yield a profit but a two part price scheme can. Since lower levels of service are provided in the uniform price case, and since the consumer achieves a greater level of utility than in the two part contract, the provider charging a unit price realizes less rent than in the two part contract.
Terms of employment • The same principles apply to hiring a worker. For example let y denote the income the worker receives for her labor, in other words her wage earnings. • Let h denote her hours of labor supplies to the firm if she is employed by the firm. • Assume the worker’s utility function takes the form y + k log(16 - h) where k is a positive constant that measures her willingness to trade off goods for leisure and 16 is the maximum number of hours she would consider working. • We also assume that if she is not employed with the firm her income equivalent is v.
Firm value Suppose firm profits are : ph - y where p is the output price (or value of the worker’s product). The firm chooses h and y to maximize profits subject to the participation constraint that the worker chooses to be employed.
Optimization • If the firm offered more than v, then it could always reduce y by so that hours remains unchanged. • Therefore the participation constraint is met with equality and we set: y = v – k log(16 - h) • The firm maximizes: ph+ k log(16 - h) – v • The first order condition for this problem can be written as h = 16 – k/ p
Solution • Substituting this equation for hours into the profit function we obtain: 16p – k+ k log(k/p) – v • Therefore the firm sets h = 16 - k/ p if profits are positive, meaning 16p – k+ k log(k/p) > v and otherwise h = 0.
Outsource A second type of work contract is for the worker to approach the firm, and propose an arrangement to the firm, which the firm can either accept or reject. This is quite close to outsourcing tasks that might have been undertaken within the firm. In this case the worker chooses both the payment y and hours or output h to maximize her utility y + k log(16 - h) subject to the constraint that the firm accepts her proposal (does not make losses): y6 ph
Solution to Outsourcing The solution is almost identical to the employment contract problem, except that all the rent accrues to the worker instead of the firm. The outsourcer sets a contract so that the firm only just breaks even, meaning y = ph. Hours are now chosen by the outsourcer to maximize ph + k log(16 - h) yielding the same choice of hours as in the original problem.
Sales commission: the worker chooses her hours An alternative method of payment is for the firm to pay its employee a commission, denoted by s, on her output. In this case the worker chooses h to maximize sh + k[log(16 – h)]. Analogous to the previous problem, the solution to this maximization problem is h = 16 – k/s if 16s – k + klog(k/s) > v and h = 0 otherwise.
Sales commission: the firm chooses the commission Upon solving for h(s), the worker’s supply of hours as function her commission, the firm chooses s to maximize: (p – s)h(s) = (p – s)(16 – k/s) = 16p – 16s – pk/s + k This solution to this maximization problem is found by solving the first order condition to the firm’s optimization problem: 16 = pk/s2 Solving for s gives s = (pk)1/2/4.
Comparing the schemes Total profit under the sales commission is: 16p + k – 8 (pk)1/2 Total profit under the optimal wage contract is: 16p – k + klog(k/p) – v Noting p > s, the participation constraints imply there are (k,p) parameter combinations where participation occurs under the wage contract but not the sales commission. Give participation in both schemes, the worker is better off under the commission system than under the contract. Since the contract extracts all the gains from trade, it is more profitable than the commission.
Contracting with specialists Often managers know less than their own workers about the value employees contribute to and take from the firm. More generally, medical doctors and specialists diagnose the illnesses for patients, strategic consultants evaluate firm performance for shareholders, and building contractors tell property owners what needs to be done. This leads us to investigate how principals (like managers) should design contracts for agents (such as workers) when the information on their employees is incomplete. Consider a game between company headquarters and its research division, which is seeking to increase its budget so that it can proceed with “product development”.
Research and product development There are two types of discoveries, minor and major, denoted by j = 1, 2. The probability it is minor (j = 1) is p, and the probability it is major one (j = 2) is 1 - p. It costs cjx to develop a commercial product with appeal of x, where c1 > c2, which in turn produces a present value of log(1+x) to the firm. A budget of bi is allocated to the research division to develop the product up to a consumer appeal level of xi when the research division announces a discovery of type i = 1,2. If the research division does not announce any discovery, it gets its standard budget r.
Research funding policy Headquarters forms a policy on funding product development, by announcing (b1,x1) and (b2, x2). After the policy formulation stage at headquarters, the division announces whether it has made a major discovery (i=2), a minor (i=1), or none at all (i=0). If i = 0, then shareholders net 0 and the research division nets r to sustain continued operations. Otherwise shareholders net: log(1+xi) – bi and the research division nets: bi – cjxi where cj is the true discovery.
Full information solution In this case headquarters directly sees the discovery, and sets the budget just high enough to motivate optimal development. Thus : bj = r + cjxj Substituting for bj into headquarters’ objective function, it chooses xj to maximize log(1+xj) – bj= log(1+xj) – r – cjxj Taking the first order condition and solving we obtain xj = 1/cj – 1 Funding is undertaken only when cj < 1 and profits, as defined below, are positive – log(ci) – r – 1 + cj
Participation and incentive compatibilitywhen there is incomplete information • Suppose headquarters does not directly observe the discovery, but relies exclusively on the divisional report . • The division will truthfully report the outcome of its activities if the following two constraints are met: • The participation constraint requires for each j: bj – cjxj r • The incentive compatibility constraint requires: b2 – c2x2 b1 – c2x1 and vice versa. Note that both inequalities cannot be satisfied by strict equality since c1 < c2.
Solving for the budgets • The participation constraint binds for the minor discovery (j = 1), but not for major ones. That is: b1 – c1x1= r b2 – c2x2 b1 – c2x1 > b1 – c1x1 = r • Substituting for b1 in the incentive compatibility constraint yields : b2 b1 + c2x2 – c2x1 = r + c1(x1 – x2) – c2x1 • Minimizing b2 we conclude the incentive compatibility constraint binds with strict equality for major discoveries (j = 2), but not for minor ones.
Optimal product development • Having derived the optimal budget as a function of product development, we choose x1 and x2 to maximize: p[log(1+x1) – b1] + (1 – p) [log(1+x2) – b2] = p[log(1+x1) – r – c1x1] + (1 – p) [log(1+x2) – r – c2x2 + c2x1 – c1x1] • = p[log(1+x1) – cx1] + (1 – p) [log(1+x2) – c2x2] – r • In the third line, c is called the virtual cost of x1 and is defined by the equation: c = c1 + (c1 – c2) (1 – p)/p
Solution to the full disclosure policy Mathematically this is almost the same problem as the full information case. Taking the first order condition and solving, we obtain: x1 = 1/c – 1 x2 = 1/c2 – 1 Substituting for x1 and x2 into the profit equation derived on the previous slide, we obtain: p[log(1+x1) – cx1] + (1 – p) [log(1+x2) – c2x2] – r = p[c– log(c)] + (1 – p)[c2 – log(c2)] – r – 1
Comparing the policy options on research disclosure To summarize, the profits from having a full disclosure policy are: p[c– log(c)] + (1 – p)[c2 – log(c2)] – r – 1 Alternatively if all discoveries are treated as minor discoveries, the profits are: c1 – log(c1) – r – 1 Finally if only major discoveries are reported then shareholder profits are: (1 – p)[c2 – 1 – log(c2)] – r
Lecture Summary Optimal contracting provides an opportunity for the contractor to extract rents from his business partners, employees, customers and clients. Extracting maximal rent may require relatively complicated contracts, which if written incorrectly, carry the prospect of loss. If the rent opportunities are too meager, surrendering the rent, and using the market, may provide a better solution. These factors form the basis for defining where firm boundaries should be relative to the market.
