Efficiency Wages and the Threat of Firing Workers: In our treatment of labor demand earlier in the text, we assumed that firms could observe the marginal revenue product of workers€”and thus would hire until wage is equal to marginal revenue product. But suppose a firm cannot observe a worker€™s productivity perfectly, and suppose further that the worker himself has some control over his productivity through his choice of whether to exert effort or €œshirk€ on the job. In part A of the exercise we will consider the sub game perfect equilibrium of a game that models this, and in part B we will see how an extension of this game results in the prediction that firms might combine €œabove market€ wages with the threat to fire the worker if he is not productive. Such wages €” known as efficiency wages€”essentially have firms employing a €œcarrot-and-stick€ approach to workers: Offer them high wages (the carrot), thus making the threat of firing more potent.
A: Suppose the firm begins the game by offering the worker a wage w. Once the worker observes the firm€™s offer, he decides to accept or decline the offer. If the worker rejects the offer, the game ends and the worker is employed elsewhere at his market wage wˆ—.
(a) Suppose the worker€™s marginal revenue product is MRP = wˆ—. What is the sub game perfect equilibrium for this game when marginal revenue product is not a function of effort?
(b) Next, suppose the game is a bit more complicated in that the worker€™s effort is correlated with the worker€™s marginal revenue product. Assuming he accepted the firm€™s wage offer, the worker can decide to exert effort e > 0 or not. The firm is unable to observe whether the worker is exerting effort €” but it does observe how well the firm is doing overall. In particular, suppose the firm€™s payoff from employing the worker is (xˆ’w) if the worker exerts effort, but if the worker shirks, the firm€™s payoff is (xˆ’w) > 0 with probability γ (c) How must wˆ— be related to γ and x in order for it to be efficient for the worker not to be employed by the firm if the worker shirks?
(d) Suppose the worker exerts effort e if hired by the firm. Since e is a cost for the worker, how must wˆ— be related to (x ˆ’e) in order for it to be efficient for non-shirking workers to be hire by the firm?
(e) Suppose wˆ— is related to γ, x and e such that it is efficient for workers to be hired by the firm only if they don€™t shirk €” i.e. if the conditions you derived in (c) and (d) hold. What is the sub game perfect equilibrium? Will the firm be able to hire workers? Is the equilibrium efficient?
(f) The sub game perfect equilibrium you just derived is inefficient. Why? What is the underlying reason for this inefficiency?
B: The problem in the game defined in part A is that we are not adequately capturing the fact that firms and workers do not typically interact just once if a worker is hired by a firm. Suppose, then, that we instead think of the relationship between worker and firm as one that can potentially be repeated infinitely. Each day the firm begins by offering a wage w to the worker; the worker accepts or rejects the offer €” walking away with a market wage w+ (and ending the relationship) if he rejects. If he accepts, the worker either exerts effort e or shirks €” and the firm observes whether it ends the day with a payoff of (x ˆ’ w) (which it gets for sure if the worker exerts effort but only with probability γ (a) Consider the following strategy for the firm: Offer w = > wˆ— on the first day; then offer w = again every day so long as all previous days have yielded a payoff of (x ˆ’ ); otherwise offer w = 0. Is this an example of a trigger strategy?
(b) Consider the following strategy for the worker: Accept any offer w so long as ‰¥ wˆ—; reject offers otherwise. Furthermore, exert effort e upon accepting an offer so long as all previous offers (including the current one) have been at least ; otherwise shirk. Is this another example of a trigger strategy?
(c) Suppose everyone values a dollar next period at δ ˆ’e) + δ P e.
(d) Use this to determine the present discounted value P e of the game (as a function of , e and δ) for the worker assuming it is optimal for the worker to exert effort when working for the firm.
(e) Suppose the firm offers w = w. Notice that the only way the firm can ever know that the worker shirked is if it€™s payoff on a given day is (ˆ’w) rather than (x ˆ’ ) €” and we have assumed that this happens with probability (1ˆ’γ) when the worker exerts no effort. Thus, a worker might decide to take a chance and shirk€”hoping that the firm will still get payoff of(x ˆ’) (which happens with probability γ). What is the worker€™s immediate payoff (today) from doing this?
(f) Suppose that the worker gets unlucky and is caught shirking the first time €” and that he therefore will not be employed at a wage other than the market wage wˆ— starting on day 2. In that case, what is the present discounted value of the game that begins on day 2?
(g) Suppose that the worker€™s expected payoff from always shirking is Ps. If the worker does not get caught the first day he shirks, he starts the second day exactly under the same conditions as he did the first €” implying that the payoff from the game beginning on the second day is again Ps. Combining this with your answer to parts (e) and (f), explain why the following equation must hold:

Derive from this the value of Ps as a function of δ, γ, , e and wˆ—.
(i) What is the highest that can get in order for the firm to best respond to workers (who play the strategy in (b)) by playing the strategy in (a)? Combining this with your answer to (h), how must (x ˆ’e) be related to wˆ—, δ, γ and e in order for the strategies in (a) and (b) to constitute a Nash equilibrium? Given your answer to A (d), will it always be the case that firms hire non-shirking workers whenever it is efficient?

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LUU 1-δ