Assignment 3
Microeconomics for MFE, UofT
Due by the end of Wednesday, December 3rd
This problem is roughly based on DeMarzo, Peter, and Yuliy Sannikov “Op-
timal Security Design and Dynamic Capital Structure in a Continuous-Time
Agency Model,” The Journal of Finance (2007).
In this problem, you are required to implement DeMarzo-Sannikov model
and do comparative statics. Consider the following baseline parameters: L = 25,
R = 0, µ = 10, σ = 5, r = .1, γ = .15, λ = 1, K = 30.
1. Find the principal’s value function P (W ). Recall that P (W ) is given by
µ+ γWP ′(W ) + 1
2
λ2σ2P ′′(W ) = rP (W )
on W ∈ [R, W̄ ] with boundary conditions P (R) = L, P ′(W̄ ) = −1, and
γW̄ + rP (W̄ ) = µ. One way to solve this equation is by :
(a) constructing a grid of slopes α = P ′(R), and solving first the differ-
ential equation with initial conditions P (R) = L and P ′(R) = α;
(b) then, for each α, you can find W̄ (α) from γW̄ (α) + rP (W̄ (α)) = µ;
(Hint: this is not necessarily solvable for every value of α.)
(c) finally, you can determine α for which P ′(W (α)) = −1.
2. The model has the following parameters: L, R, γ, µ, λ. Pick any two
parameters p1 and p2. Describe what happens to the firm’s debt D =
1
r
(
µ− γ
λ
W̄
)
and its credit limit CL = (W̄ − R)/λ as you vary each one
of the two parameters. Does the effect always go in the same directions?
Provide intuition for your results.
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