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In Wickens' Macroeconomics book, in page 552, the author states the following:

«The stochastic problem can be solved using the method of Lagrange multipliers, but there is a problem with this solution.»

However, if I'm not mistaken, it's very usual to see other books using lagrange multipliers to solve this type of problems. Are they wrong? Am I wrong? Is there a way to go around this apparent limitation?

Any help would be appreciated.

Here's a the image:

enter image description here

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Following a comment exchange below another answer, the critical detail of a link offered by the OP as an example of by-passing/ignoring Wicken's comment/argument, and one different from Wickens' formulation, is that in the link's eq. $(4)$,

$$\lambda_t = \beta E_t[\lambda_{t+1}(1+r_t)]$$

the multiplier $\lambda_{t+1}$ appears together with the interest rate of period $t$, which is assumed part of the information set at $t$ and so not a random variable in the specific equation. So the authors can proceed in the 2nd line after eq. $(6)$ to write the marginal rate of substitution as they do, without the need to assume uncorrelatedness between the multiplier/marginal utility of consumption, and the interest rate.

This goes back to how they formulate the income resource constraint (page 2 middle), where it is essentially assumed that the household has, at the beginning of period $t$, available assets or debt $(1+r_{t-1})B_t$. The authors explicitly discuss this "timing convention" immediately after eq. $(1)$

"Note a timing convention -$r_{t-1}$ is the interest you have to pay today on existing debt. $r_t$ is what you will have to pay tomorrow, but you choose how much debt to take into tomorrow today. Hence, we assume that households observe $r_t$ in time $t$. Hence we can treat $r_t$ as known from the perspective of time $t$."

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  • $\begingroup$ Alecos, do you have an opinion on this book by Michael Wickens? $\endgroup$ – An old man in the sea. Oct 2 '16 at 20:45
  • $\begingroup$ I haven't checked it thoroughly. But my first impression when I browsed its pages (I have the first edition), is that it is good for the basic tools (so graduate beginners level). Sometimes such books are very useful because the keep the core very visible. $\endgroup$ – Alecos Papadopoulos Oct 2 '16 at 20:56

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