Rating: 
Summary: Excellent book; the next "classic" of economics
Review: SLP is now a standard text in serious, modern economics. It seriously and thoroughly explains the tools that will be useful for economists in decades to come. No doubt it will "aquire the stature of Samuelson's Foundations or Hicks's Value and Capital". It's nosense that those who can't understand it say "oh, no, too technical". Please! The same was said of Marshall's Principles!
Rating: 
Summary: almost unreadable
Review: This book may be a decent reference material, however it is really hard to read and the tendency to leave everything out as excercises for the reader does not help either.
Rating: 
Summary: Much ado about nothing.
Review: This book was obviously written by non-mathematicians. An exercise in rigorous hand-waving. Rigor without mortis. Anybody who wants to become a mathematical economist would be better off by taking real math courses first. On the other hand, the Economics content is rather prosaic as is tradition in the Minnesota-Chicago school. There are a great many math and probability books out there that are more interesting, not to mention those that are more concerned about empirically relevant Economics. If you think Real Business Cycle models are mathematically rigorous and empirically relevant, well, this piece should become your Bible.
Rating:
Summary: tough, terse, but worth the fight
Review: This is a really challenging book, worth fighting through. Some of the exercises are impossible without outside help, so for self-study the solutions manual by Irigoyen, et al is *essential*.
Rating: 
Summary: Unecessary Sand in Readers' Eyes
Review: This is a really nice book by many standards, particularly in Chapters 4 and 9 which form the heart of the book. But certain aspects of the surrounding formal math chapters are ridiculous: a nice amount of material is in there, but they seem to prove what is trivial and leave what students would really like to see as exercises. This is a pedagogic no-no. They also make a big deal about what should be corollaries and give but a passing mention to fundamentally important theorems - this is a logical no-no.Structurally, notation is also not consistent across chapters. Some chapters are misleadingly titled, e.g. "Markov Processes" is really only about integration of product measures. Another no-no. FINALLY, the authors also seem to have a HIGHLY selective and suspiciously idiosyncratic way of making references. Here there is a deep, deep ethical no-no. Otherwise, it is really the only thing on the market and, in achieving what it sets out to achieve, it does quite well. Perhaps some more micro examples would make it better.
Rating:
Summary: You better read something else
Review: This is among the worst books I've seen in economics. I suggest you read something else in maths such as dynamic programming, stochastic dynamic programming, measure theory to get the neccessary tools instead of reading this book. By studying these serious books in maths, you get the foundations. The arguments do not flow. To much exercises are left for the readers. The explanations are bad and too terse. The references are really bad and creat a lot of diffculties for you to keep track. This book teachs you many thing you can find in Sargent's three books: Dynamic macroeconomics, recursive macroeocnomics and macro theory. But I found those books are much,much more entertaining and systematic. Forget about this book. I first studied this just to get pass with my course requirement however, after that I switched to more serious mathematic books and read Sargent's in complement. That helps.
Rating:
Summary: not a "textbook" on macro...
Review: This is basically a handbook of mathematical methods necessary to study dynamic economic modells. As such it does a very good job, since as the authors note, just presenting the applications without developing the necessary mathematics would require the reader to keep quite a few math books alongside to keep going at an acceptable pace. It should be clear that the applications developed in this book are idiosyncratic (Carnegie-Mellon/Chicago/Minnesota school).
The book certainly doesn't work too well in a first year graduate Macro course, for that it is simply too terse. That a lot of material is left to the exercises certainly has to do with the fact that the book is already quite thick. In general this is going to be a textbook for a course and won't be used for self-study. Therefore i don't think that missing solutions to exercises are too much of a problem.
I think the book is useful for someone with an "ok" math backround, who has not yet had the chance to study dynamic programming, measure theory and lesbesque integration etc. and who wants to go beyond the typical first year macro stuff. If you have a strong math backround this book is rather unnecessary, save maybe the chapters on applications. I give the book 5 stars as a mathematical compendium, as a macro textbook (which the authors to not claim to have produced) it should get less.
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