Download Problem F.1

Macroeconomics 2
06.11.2011
Exercises
Christian Groth
Problem set F
I sidste dobbelttime blev jeg spurgt, om jeg havde yderligere opgaver liggende inden
for emnet “Tobin’s q og faste investeringer”. Mit svar var ja og jeg lovede at lægge et
eksempel ud på fagets hjemmeside. Her er det, men bemærk, at udgangspunktet er en
situation med ufuldkommen konkurrence, hvilket ikke rigtigt er med i det endelige pensum
(der er lidt om det i kapitel 2, afsnit 5, men det er ikke pensum).
F.1
Tobin’s q and imperfect competition. Consider a firm supplying its own differen-
tiated good in the amount  per time unit at time  The production function is

 =  1−

0    1
(*)
where  and  is capital and labor input at time .
The nominal wage and the nominal general price level in the economy faced by the
firm are constant over time and exogenous to the firm. So the real wage is an exogenous
positive constant, . The demand,   , for the firm’s output is perceived by the firm as
given by


  1
(1)

where  is the price set in advance by the firm (as a markup on expected marginal
  = −
cost), relative to the general price level in the economy,  is the given large number of
monopolistically competitive firms in the economy,   is the expected overall level of
demand, and  is the (absolute) price elasticity of demand. The interpretation is that the
firm faces a downward sloping demand curve the position of which is given by the general
level of demand, which is exogenous to the firm. We assume that  is kept fixed within
the time horizon relevant for this analysis.
The increase per time unit in the firm’s capital stock is given by
̇ =  −  
  0
0  0 given,
where  is gross investment per time unit at time  and  is the capital depreciation rate.
We assume that  is high enough to always be above actual marginal cost so that it always
pays the firm to satisfy demand. Then cash flow at time  is
 =   −  −  − ( )
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where ( ) is a capital installation cost function satisfying
(0) = 0 (0) = 0 00 ()  0
a) Given  =    find  as a function of  .
b) Given that the real interest rate faced by the firm is a constant   0 set up
the firm’s intertemporal production and investment problem as a standard optimal
control problem, given that the firm wants to maximize its market value. Hint: given
the function found in a) there is only one control variable and one state variable.
c) Derive the first-order conditions and state the necessary transversality condition
(TVC) for a solution. Hint: the TVC has the standard form for an infinite horizon
optimal control problem with discounting.
d) The optimal investment level,   can be written as an implicit function of  Show
this. Construct a phase diagram for the ( ) dynamics, assuming that a steady
state with   0 exists. Let the steady state value of  be denoted  ∗ .
e) For an arbitrary 0  0 indicate in the diagram the movement of the pair (   )
along the optimal path.
f) Express the level of gross investment in steady state,  ∗  as a function of  ∗ 
g) By curve shifting in the phase diagram, sign  ∗  and  ∗   
h) Comment by relating the results in g) to the signs of the partial derivatives of the
investment function in a standard IS-LM model.
i) By curve shifting in the phase diagram, sign  ∗  Comment.
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