Guidelines
1. You should include your Matlab code in the Appendix. The code should be commented (do
not overdo this).
2. Figures should be suitably labeled and titled. You can have them either in the main body or
in the appendix.
3. Follow standard guidelines for referencing, there should be a bibliography section listing all
the references used.
4. Mathematical equations should be properly formatted. Microsoft Word supports inserting
math symbols and equations. If you are familiar with Latex or Scientific Word, you can use
these instead.
5. Some questions have word limits, you should strictly adhere to these limits.
Read the group coursework briefing for more information.
1 Continued Overleaf
There are three questions. You must answer all three.
1. Consider the problem of a value maximizing firm called Progress Panda whose profit function
at time t is given by
Π(Kt, At) = e
AtKθt
where e is the natural exponent, At denotes the productivity level, Kt denotes capital and θ is a
parameter representing the elasticity of output with respect to capital. Assume that there is no
depreciation of capital, so the law of motion of capital is given as
Kt+1 = Kt + It
where It is investments in capital at time t. The price of a unit of capital good is p and investment
is subject to a smooth convex installation cost given by
C(It, Kt) =
γ
2
I2t ,
where γ is a constant. Time is discrete and runs to infinity, t = 0, 1 . . .∞. Future values are
discounted with the factor β. The productivity level for the firm is stochastic and follows an
AR(1) process given by:
At+1 = ρAt + ϵt+1, where ϵt+1
iid∼ N(0, σ2)
Based on the above information, answer the following five questions:
1.1 Derive the static profit function.
[5%]
1.2 Write down the Bellman equation and derive the optimal investment decision condition.
[5%]
1.3 Define marginal Q and provide an economic interpretation.
[5%]
1.4 How is average Q related to marginal Q for this firm?
[5%]
3 Continued Overleaf
1.5 Solve the Bellman equation using dynamic programming. You can calibrate model param-
eters as follows
• β = 0.95
• θ = 0.7
• γ = 0.2
• p = 1.2
• Discretize capital grid with 401 uniformly spaced points in the interval [30, 80].
• Discretize the productivity process using the Tauchen (1986) procedure so that
– A takes 3 states – {Low, Medium, High} = {AL, AM , AH}
– ρ = 0.9
– σ = 0.1
– m = 3 (± 3 st dev shocks)
• A tolerance level of 1e-7 can be used to check for convergence of the value function
iteration.
Plot value function and investment policy functions for all three productivity states. Inter-
pret these graphs. Explain how optimal investment responds to changes in the
i Adjustment Cost Parameter γ; and the
ii Persistence of the Productivity Shock.
[20%]
Hint: To save time, start with a small number of capital grid points like 101 and a bigger tolerance
level. Check if the value functions are converging and the policy functions look like you expect them
to, then move on to a bigger grid and a smaller tolerance level. Note that this might take a while
to converge.
4 Continued Overleaf
2. Consider the problem of another firm called Flying Nemo who faces the same conditions as
Progress Panda with one difference. Instead of a convex adjustment cost of capital, Flying
Nemo faces non-convex adjustment costs. These costs come in two different forms.
1 The first cost is an opportunity cost of investment. The firm’s profitability falls to a λ < 1
proportion of actual profits. That is, effective profits are λπ(Kt, At) in case the firm decides
to invest in capital.
2 The second cost is a fixed cost and is given by FKt where F is the cost and Kt denotes
capital stock at time t.
These are the ONLY two costs facing Flying Nemo.1 Both these costs are independent of the
level of investment. These costs essentially follow the ones in Cooper and Haltiwanger (2006).2
The firm can choose whether to invest (I > 0) or not (I = 0), and the value function is then the
maximum of two options
V (K,A) = max{V a(K,A), V i(K,A)} ∀(K,A)
where V a(K,A) is the value function corresponding to active investment and V i(K,A) is the
value function corresponding to inactivity (no investment).
Everything else remains the same as in question 1, so profits are given by
Π(Kt, At) = e
AtKθt
Capital does not depreciate every period
Kt+1 = Kt + It
and the price of an additional unit of capital good is p. Firm’s productivity follows an AR(1)
process
At+1 = ρAt + ϵt+1, where ϵt+1
iid∼ N(0, σ2)
Finally, time is discrete and the horizon extends to infinite periods. Future values are discounted
with the factor β.
1There are no convex adjustment costs facing Flying Nemo.
2See section 3.3 in their paper.
5 Continued Overleaf
Based on this information, answer the following two questions:
2.1 State the dividends for the two options (active and inactive) and write down the Bellman
equation that characterizes this firm’s problem.
[5%]
2.2 Solve the Bellman equation numerically using dynamic programming. The parameter values
are the same as in question 1.5. The two additional parameters can be calibrated as
• λ = 0.5
• F = 0.25
Plot and discuss value functions and optimal investment policy functions for all three
productivity states. Compare and contrast the results of Flying Nemo with Progress
Panda.
[15%]
Hint: To solve 2.2, the standard value function iteration can be applied with a small modification.
Inside the iteration loop, you will have to calculate two different value functions, one for active
investment option and the other for inactivity option. These value functions should be calculated
in the standard way. The updated value function in each iteration then is the maximum of these
two at each state of the model. As in question 1, start with a small grid and proceed to a finer
grid only when you have convergence. This will save you a lot of time.
6 Continued Overleaf
3. Based on the investment-uncertainty literature answer the following questions:
3.1 Describe four different measures of uncertainty. What are the strengths and weaknesses of
each measure (max. 600 words). [10%]
3.2 Theoretically, how can uncertainty impact investment decisions of firms? Does the impact
of uncertainty on investment depends on the type of costs facing the firm? (max. 600
words) [10%]
(a) Summarize the literature on uncertainty and firm investment. Is there conclusive evidence
on how uncertainty impacts firm level investment? Explain. (max. 1200 words) [20%]
7 End of the Paper