Download 3. This question allows you to focus on profit sharing. Pissarides

Macroeconomics
William Scarth
Chapter 8 Questions
1. (a) Write a short essay explaining why Keynesian economists find
efficiency-wage theory an appealing analytical underpinning for
thinking about unemployment.
(b) In the text, we considered two versions of efficiency wages (due to
Solow and Summers) that are based on workers shirking while on the
job. Other versions of this theory are based on labour turnover, and we
consider one such specification (due to Salop) in this question. In this
model, firms’ profits are equal to Y  tqN  wN, that is, output minus
training costs minus the wage bill. Training costs involve a payment of t
for each worker who quits and needs to be replaced each period, and
proportion q of the N workers quit each period. There are two
constraints that firms respect while maximizing profits: that their
input–output function is Y  F (N ), and that the quit rate depends
inversely on the after-sales-tax (rate s) purchasing power of the wage
that the firm offers its workers: q  H(w /(1  s)). Specifically, the
assumptions about these functions are: F   0, F   0, H   0, and H   0.
Derive the firm’s optimizing rules (for hiring and wage setting) by
differentiating the objective function with respect to these two choice
variables (and setting each to zero). Then use the resulting relationships
to determine the effect of an increase in the sales tax rate on the level of
employment. Explain your reasoning.
2. Consider the following efficiency-wage analysis of minimum wage
laws.
Y  F (qN)
effective labour)
q  ((w  x) / x) a
x  [ fw * (1  f )m]
P  Y  wN
production function (output depends on
worker effort function
worker outside option
definition of profits, P
The notation is as defined in the text except for the following: f is the
probability that a separated worker gets another ‘good’ job (which pays
a wage equal to w*), while (1 − f) is the probability that the separated
worker has to accept a minimum wage job (which pays m < w*). There is
no unemployment insurance or any taxes. Both f and the shirking
probability parameter, a, are fractions. Assume also that (f + a) < 1. The
production function has the usual properties: F   0; F   0. There are six
endogenous variables (Y, N, P, x, q and w) and only four equations. You
need to get the two additional equations by setting the derivatives of the
profit function with respect to N and w equal to zero. (For this part of
your answer, use only the first, second and fourth of the given equations
(since each individual firm takes x as given). Then, use the full model,
along with the assumption that wages across the economy are equal (w
= w*), to determine the effect on the level of employment of an increase
in the minimum wage. Explain your reasoning.
3. This question allows you to focus on profit sharing. Pissarides built a
model involving partial cooperation between each union–firm pair in
the economy. This question involves this model, extended to allow for
workers to receive part of their remuneration via a standard wage, and
part via a share in the company’s profits. Each union and firm cooperate
by having an arbitrator choose the wage that maximizes the following
function:
A  [ N (w  (1  u)w * uw)   (Y  wN )] [(1   )(Y  wN )](1 )
where w is the real wage paid at this firm, w* is the wage individuals
would receive if they became employed at other firms, and w is the
unemployment insurance payment individuals would receive if they
became unemployed. The square-bracketed terms indicate the gain each
party achieves by reaching an agreement.  is the profit share that goes
to labour, and you are to assume that  remains smaller than , which is
the union bargaining power parameter. Coordination is partial, since the
firm is free to adjust the level of employment without consultation with
the union, after the wage is set. The production function is Y  F (N ).
(a) Derive the wage-setting rule for the arbitrator.
(b) Now simplify your wage-setting relationship in three ways: by
focusing on a full-equilibrium outcome in which wages throughout the
economy are the same (w* = w), by specifying that the unemployment
insurance payment (w ) equals fw, and by assuming that Y  N  , so that
profit maximization on the firm’s part implies Y / N  w. The outcome is
the equation that determines the nation’s unemployment rate.
(c) Use your unemployment rate equation to determine the effect of a
change in society’s institutional arrangements that involves more
widespread profit-sharing (a higher parameter ). According to the
model, what effect does this increased embracing of profit sharing have
on the unemployment rate?
(d) Now we use the model given in this question to explore whether the
general level of interest rates affects the natural unemployment rate. In
this case, the definition of A is altered, since profits are defined by the
expression [Y  w(1  r) N ] where r is the interest rate. Firms need to pay
their wages before receiving their sales revenue, and so interest must be
paid on a loan to cover the wage bill. Does a rise in interest rates affect
unemployment?