# some question on Buera and Nicolini (2004)

I am struggling with understanding one result in Buera and Nicolini (2004) published in JME. In page 538, they write $$A_t(h^t) b_{t-1}(h^{t-1}) = Z_t(h^{t-1})$$ where $$A_t(h^t)$$ is a $$N \times J$$ matrix, $$b_{t-1}(h^{t-1})$$ is a $$J\times 1$$ vector, and $$Z_t(h^{t-1})$$ is a $$N\times 1$$ vector. $$N$$ is the number of states and $$J$$ is the number of maturities. Then, in page 539, they say that

"a necessary condition for the Ramsey to be implementable is that the government can issue non-contingent bonds in at least as many maturities as possible realization of the stock, i.e., $$J \ge N$$.''

For an ease of notation, write the above equation as $$A b = Z$$. Suppose that $$J > N$$. Premultiplying a $$J \times N$$ matrix $$D$$, $$D Ab = D Z$$ If $$D$$ is an inverse of $$A$$, rank($$DA$$) = $$J$$. But, $$\text{rank}(DA) \le \min(\text{rank}(D), \text{rank}(A))\le N$$. Contradiction. I think I am missing something important. I would appreciate if someone helps me with this.

• There is something I can't understand. $A$ is a $n\times j$ matrix, if $n\neq j$ $A$ is a non-square matrix, and can't have an inverse, the inverse of a matrix is defined for square matrices only. Commented Mar 26 at 12:11
• @BakerStreet Perhaps it is a generalized inverse? The article is "an", as in "$D$ is an inverse of $A$". In that case rank of $DA$ would be $N \leq J$ though. Commented Mar 26 at 16:41
• @Giskard maybe, but nothing can be said without reading the paper, it sholuld be specified. Commented Mar 26 at 17:01
• Is this $D$ still in the paper though? I thought the papers part stops at the quotation marks at the middle. Commented Mar 26 at 17:10
• @BakerStreet Here is the relevant page, in case you are interested econ.ucla.edu/fjbuera/papers/matFINAL2.pdf#page=12 Commented Mar 26 at 17:13