Deriving a transversality condition

General DSGE Modeling
1

Hello everyone,

My question may seem too basic, but now I have wondered about this. It has to deal with transversality conditions.

Reading the Obstfeld and Rogoff textbook Foundations of International Economics on this matter, I understand that they always derive their transversality conditions by iterating forward the budget constraint of the representative agent (Chapter 10 for example). Hence, I was trying to follow the same logic, but it seems that the transversality condition I obtained looks way too complicated.

The representative agent supplies labor and capital to intermediate good producing firms, receives real income from wages w_t, a rental return on capital rr_t, and nominal profits from the ownership of firms. The agent then divides its resources to purchase consumption C_t and investment I_t. The agent trades domestic and foreign nominal risk-less bonds B_{H,t} and B_{F,t}.

Hence, the budget constraint the agent faces (under incomplete asset markets) is given by:

\begin{gather} \begin{split} P_t(C_t+I_t)+\frac{B_{H,t}}{1+i_t}+\frac{S_tB_{F,t}}{(1+i^*_t)\Theta\Big(\frac{S_tB_{F,t}}{P_t}\Big)} \leq B_{H,t-1}+S_tB_{F,t-1}+P_tw_tL_t+P_trr_tK_t\\ +\int_0^1\Pi_t(j)dj \end{split} \end{gather}

and capital follows the law of motion K_{t+1} = (1-\delta)K_t + I_t. The cost function \Theta(\cdot) drives a wedge between the return on foreign currency denominated bonds received by domestic and by foreign residents.

I would like to ask for suggestions of paths I could follow to derive the transversality condition considering this budget constraint or if I should adopt a different strategy to derive such condition. A second question is what role the \Theta(\cdot) function would have in the transversality condition as well.

Thank you,

Alejandro

2

You still would need to iterate forward in time. You will end up with the appropriately discounted future value of all assets being 0. So you will have TVCs fro K, B_H and B_F. The cost function simply increases the discount factor applied. The complication you will arrive at is that the discounting involves and infinite product of interest rates.

3

This is what I do:

  1. Rearrange terms of the budget constraint:

B_{H, t-1}+S_tB_{F,t-1}+P_t(rr_t+1-\delta)K_t = P_tK_{t+1} + \frac{B_{H,t}}{1+i_t} + \frac{S_tB_{F,t}}{(1+i^*_t)\Theta(\cdot)} + P_tC_t - P_tw_tL_t - \Pi_t

  1. Forward one period the latter expression

B_{H,t} + S_{t+1}B_{F.t} + P_{t+1}(rr_{t+1} + 1-\delta)K_{t+1} = P_{t+1}K_{t+2} + \frac{B_{H,t+1}}{1+i_{t+1}} + \frac{S_{t+1}B_{F,t+1}}{(1+i^*_{t+1}) \Theta (\cdot)} + P_{t+1}C_{t+1} - P_{t+1}w_{t+1}L_{t+1} - \Pi_{t+1}

  1. Then, I first tried to isolate B_{H,t}, substitute in the original budget constraint, and just repeat the process T periods to derive the TVC. However, when I tried isolating B_{F,t}, substituting it in the budget constraint, and repeating the process T periods to derive the TVC, I noticed that my answer was different.

This confuses me a little bit, so my question is what is the correct procedure to derive the TVC.

4

With multiple assets, it’s complicated, because there will be a constraint for each asset.