# Why does bond-holding in equilibrium is zero in basic NK?

In Sims’s basic-NK model class notes, page 7 says: "In equilibrium, bond-holding is always zero in all periods: B_t = 0". Yet I’m failing to show it without just assuming it, why bond holding is zero at all t?

For example from FOC, as bonds always appear linearly they just disappear letting only the nominal interest “representing” their role, but when one is going to close the model with the household FOC they are present:

c_t+m_t+b_t = \frac{(1+i_{t-1})b_{t-1}+m_{t-1}}{1+\pi_t}+w_tn_t-T_t+\Pi_t

How to make sure that b_t\equiv B_t/P_t=0\forall t?

1. If the government can issue bonds, then for example B^H_t+B^G_t = 0 instead of B^H_t = 0, but overall B_t = 0. Recall that every market is cleaned in the general equilibrium. When you bypass this condition and start from the labor market clearing condition, by Walras Law, you will still work out b_t=B_t=0. Yet, the labor market condition is usually too trivial to show, so you always see the bond condition.

2. You impose the bond market clearing condition after households’ optimality conditions. Despite b_t = B_t = 0 in your case, it won’t impair the households’ budget constraint.

Perhaps the following document is useful, especially on Page 9, Footnote 3.

If my explanation was wrong, please kindly let me know.

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Let me add some intuition: the point here is the representative agent assumption combined with bonds being private bonds. If all agents were borrowing, there would be nobody they could be borrowing from. If the all were lenders, nobody would like to borrow from them. So bonds across all agents need to be in 0 net supply.

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Just to add some intuition here: My debt is your asset and vice versa. If someone borrows, someone else always lends and therefore if you aggregate across your position and my position you would see that they net out to be exactly zero.

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