Questions 1 and
2 are based on the following scenario:
Ruritania’s
currency is the euro (€), and its only coalmining company is a pricetaker in
the euroarea market for coal. The company sells at €90 per tonne to specialist
wholesalers and faces no transportation costs on the coal it produces. Its only
variable costs relate to two distinct labour types, skilled (engineers etc.)
and unskilled. Skilled workers command an hourly wage of €25, while that of
unskilled workers is €8. Fixed costs are ignored. The firm’s production
function is quadratic in both skilled and unskilled labour hours (respectively
denoted S and U):
q_{c}=100
S–2
S^{2}+200U5U
^{2}
+2SU
.
1.) Using the above information:
(a)
Derive
an expression for the firm’s profit solely in terms of S and U.
(b)
By applying the implicit function
theorem appropriately (i.e. by making use of a total differential for profit),
obtain an algebraic expression for the slope of any given isoprofit contour in
S, U space. Depict the firm’s isoprofit map in S, U
space (i.e. with S on the horizontal axis and U on the vertical),
and discuss how the slope expression provides insight into the shape of the
isoprofit contours.
(c)
Obtain firstorder conditions for the
firm’s optimal combination of S and U. Solve these equations for S
and U by means of Cramer’s rule and state the amount of profit
associated with your solutions (which should be stated to 3 d.p.).
(d)
Obtain a secondorder total differential
(i.e. quadratic form in dS and dU), express this in matrix form,
and investigate whether the secondorder conditions for a maximum or a minimum
are satisfied. Provide a brief prose discussion of this procedure.
2.) Now assume that Ruritania’s government
passes legislation which restricts each firm
to hiring no more than 30 hours of
skilled labour.
(a)
Set up the Lagrangian function, and
proceed to obtain and solve by means of Cramer’s rule the firstorder
conditions for the optimal S, U input combination when the firm
is subject to this constraint. Comment briefly on how your solutions for S
and U, and the associated profit, compare with those obtained earlier
for the unconstrainedcase.
2
(b)
Proceed to obtain appropriate second
derivatives, construct the relevant bordered Hessian matrix, and assess whether
the secondorder conditions for a (constrained) maximum are satisfied.
(c)
Reproduce your S, Uspace
diagram from part (b) of question 1, with the constraint (and any relevant
tangency point) depicted clearly.
(d)
By making use of both the constraint and
the slope expression obtained for part (b) of question 1, derive your S,
U solutions for part (a) of question 2 by a method which does not involve
the Lagrangian.
(e)
Discuss the intuitive interpretation of
the Lagrange multiplier’s solution value, obtained earlier for part (a) of
question 2.
3.)
An agent has two items of noisy information regarding the current average state
of productivity in the economy. Denoting (unobserved) average productivity by ?,
the state of productivity at the agent’s own firm is u, while the
government’s statistics department publicly announces that it believes the
value of ? is v. From his/her experiences over many previous
periods, the agent knows that both the departure of its own productivity u,
and the departure of the government’s estimate v, from the true average
value ?, are drawn from normal distributions. These departures are noise
terms, which respectively have zeromean,
finitevariance distributionsw~N(0,s^{2}) 
andh ~ N (0,s ^{2} ) . The realized values of u and v 
w  h 
are therefore given by: 

u=q+w,  v=q+h 
The
agent observes the realized values of u and v before forming an
expectation of ?’s value; the realized values of the noise terms and of ?
are then unknown. There is no correlation between ? and either of the
noise terms (i.e.E(uq)=
E
(vq)=
0
), and the noise terms are uncorrelated with each other,E
(uv)
=0
.
Obtain an
expression for the agent’s mean square forecast error, when the agent’s
forecast
(i.e. expectation,
or estimate) conditional
on the signals
u and v
has the linear
form: