Q theory of investment

Question 1

Model assumptions

Answer 1

Convex adjustment costs (macro fluctuations)

• Positive adjustment cost (both for investment and disinventment) C(0)=0
  
  • Previous models assumed no cost of adjustment so that the firm’s problem was static
• Total cost is increasing when I>0, decreasing otherwise
• Marginal cost is increasing in the size of the adjustment
• At the firm level pi(K_t) is exogeneous

Question 2

Individual firm’s problem

Answer 2

Firms choose the optimal path of investment, discounted by r (cost of your funds).

• Prices and industriy-wide capital are exogeneous

Question 3

Hamiltonian

Answer 3

Remember the multiplier is the present value of an additional unit of capital at time t

Question 4

Definition of q

Answer 4

• Shadow value of an extra unit of capital at time t (valued at time t)
• q is the market value of the last unit of installed capital / slide 36
• Here, marginal and average value of capital coincide

Question 5

FOC interpretation

Answer 5

1. Value of one marginal unit of capital equal its cost
2. Marginal revenue of an installed unit of capital equal its marginal user cost
3. q is the market value of the last unit of installed capital

Question 6

Integral expression

Answer 6

Value of an additional unit of capital equals the discounted sum of the stream of additional future revenues that this unit will bring to the firm

• At time t, q_t summarizes all the relevant info about the future needed to make an investment

Question 7

Behavior of a firm capital

Answer 7

A firm investment is a monotonically increasing function of the diff between the value of the marginal unit of capital and its purchasing cost

Question 8

Equilibrium

Answer 8

Defined by just two conditions:

• Firms solve their optimization problem
• Aggregate investment is the sum of individual investment over all firms
  
  • K_t=Nk_t
• All firms are equal and as a result their choices are identical

Question 9

Steady state equilibrium

Answer 9

• Price of investment good is constant (Pijoan defines it as 1)
• Aggregate capital is stable (K dot=0)
• Value of installed capital is constant

In the steady state, the revenues from a unit of installed capital (pi(K)) just offset its opportunity cost, the forgone intereset rq

Question 10

Stability and comparative statics

Answer 10

• There is saddle path stability
• There are differences between:
  
  • Permanent unexpected changes
    
    • Initial jump, evolution along the stable arm
    • Adjustment costs make the firm adjust gradually
  • Transitory unexpected changes
    
    • Costs of adjusting capital make firms reverse the changes in order to spread out the adjustment costs

Question 11

Implications of the q theory of investment

Answer 11

• q is a sufficient statistic for investment
  
  • This might not hold in practice due to:
    
    • Cash contrained firms (there is dependence between financial and investment decisions)
    • Monopoly power
    • Capital isn’t homogeneous