Williamson defines the net governance costs of a transaction as:
NGC = A(k,¯Q) − T (k, ¯Q) where A(k, ¯Q) is the agency cost of “making”, and T (k,¯Q) is the transaction cost, of “buying” ¯Q units of an input, given the degree of asset specificity k.
Both these functions are increasing in k, but T (k,¯Q) increases faster therefore NGC is decreasing in k. Let’s assume that;
NGC(k,¯Q) = 0.1• ¯Q – (¯Q /10) • k
(a) At what value of k is NGC = 0 (given that ¯Q> 0). What does this mean?
(b) The net production costs is: NPC = I (k, ¯Q) − P(k, ¯Q), where I (k, ¯Q) is the production cost of making, and P (k, ¯Q) is the market price, i.e., the cost of buying. Let’s assume that NPC is given by the following expression;
NPC = (1000/¯Q ) • (1/k)
Assume that ¯Q= 100, plot this function in a diagram, together with the NGC -function.
(c) Add the two functions together to get the net total cost (NTC) function. Find the value of k (called k**) where NTC= 0 (given that ¯Q = 100). What does this value of k show?
(d) Assume now that the scale of the transaction increases to ¯Q= 200. Show how the three functions change in the diagram. What is now the value of k∗∗? Why has it changed in the direction it has?
Your utility for money (m) is given by the following utility function:
U (m) = √m.
(a) Assume that you have been offered the opportunity to participate in a game there you can win 1000€ with probability p1000 = 0.1, 500€ with probability p500 = 0.2, and zero with probability p0 =1 – 0.1 – 0.2. If it cost you 100€ to participate in the game are you
prepared to pay this amount to participate in the game? You have m0 = 10000€ initially.
(b) Assume now that you stand to lose 1000€ (an unavoidable risk) with probability p1000 = 0.1, to lose 500€ with probability p500 = 0,2 and zero with probability p0 = 1− 0.1 – 0.2. What is the maximum sum (insurance premium) that you’re prepared to pay to an insurance
company which offers complete insurance against these losses?
(c) If you do not buy insurance, what is the expected value of your monetary wealth? What is the certainty equivalent amount of income?
(d) If the accident probabilities are independent between potential insurance buyers the insurance companies can reduce their risk to (almost) zero by selling insurance policies to many customers. If, furthermore, the insurance market is perfectly competitve, what would the competitive insurance premium be. What would your utility be if you buy insurance at this insurance premium?
Assume that P owns a bookstore and has employed A as a seller. The number of books sold depends on A:s effort, total revenue as a function of effort (e) is, TR(e)= 50e – (1/2)•e^2
the cost of eff ort function is TR(e) = 10e.
(a) What is A:s profit maximizing effort level? Illustrate the solution in a diagram with e on the horizontal axis and the marginal revenue and marginal cost of effort on the vertical axis.
(b) If there are full information and P orders A to put in the optimal effort, and A gets a compensation equal to his marginal cost of effort, what is P:s profit? What is A:s total compensation?
(c) We now assume that it is costly for P to monitor A:s efforts, and contracts cannot be written contingent on the observed outcome. In this case P could rent the store to A in exchange for a fixed fee (F). If A has an alternative which would get him a profit of 200, what is the optimal fixed fee from P:s perspective (the fee that maximizes her profit)?
Illustrate the joint profit function, and A:s profit function, in a diagram with e on the horizontal axis and profits on the vertical axis.
(d) Instead of renting the bookstore to A, they can arrange a partnership in which they split the revenue (50% each). If A determines his own effort level, how much effort will he put in? What is their joint profit?
(e) If P can direct the number of books to be sold, what quantity would she order A to sell? Will A agree this contract?
(f) Assume now that they instead arrange a profit-sharing contract (50/50). They both agree on A:s cost of effort function. How much effort would A put in, in this case? What is the joint profit? Will A participate?
If P is the one that proposes how to split the profit and if she knows A:s outside profit opportunity, what is the optimal split of profit from her point of view?
Consider the following relationship between the realized number of sales (s) and a sale-agent’s effort (e, which could be the number of hours worked), it is:
s = 500 • e,
Effort is costly to the agent, his cost of effort function is:
(a) How much would the agent work if he worked for himself and maximized the net benefit of his sales efforts?
(b) If the agent doesn’t work for himself but is offered a contract by a firm (the principal) he will accept it if the wage is high enough given the effort required. The agent may also have an alternative wage offer which is worth $5000 per month after deducting the cost of
effort in that occupation. This amount may be considered to be the agent’ reservation utility i.e., the level of satisfaction that he/she can reach if accepting the alternative occupation. Assume that the principal offers the agent a contract with a fixed wage part (F) and
a commission rate of α% of sales. Explain how the optimal contract will look like.
(c) We now assume that sales depends on effort and a random variable, ϵ ̃
s = 500 • e + ϵ ̃
The random variable has an expected value (or average) equal to zero: E(ϵ ̃) = 0 and a variance of σ_ϵ ̃^2.
The agent’s utility increases with his expected wage but it is decreased due to the risk he faces in this case. The risk-premium can be expressed as: 1/2 ρσ_ϵ ̃^2 and the certainty equvalent wage income is therefore:
wCE= E(w) − 1/2 ρσ_ϵ ̃^2.
The agent will accept a contract if he gets at least $5000 in a certainty equivalent wage income. The agent’s realized pay in any given month is now F +α•100+ϵ ̃ ) but since ϵ ̃ has an expected value equal to zero his expected monthly pay is equal to F +α•100•e, the variance of his pay is α^2σ^2. If ρ = 2 and σ^2 = 15000 what is the agent’s certainty equivalent pay and what is his utility?
(d) Assume that the principal sets α = 1. How high must F be in order for the agent to accept this contract? Is this a feasible contract.
(e) Assume that the principal sets α = 0.50. How high must F be in order for the agent to accept this contract? Is this a feasible contract?