David Harvey is not just one of the world's foremost social and economic geographers, but is also one of the world's foremost Marx interpreters. "A Companion to Marx's Capital" is the book form of a series of lectures on Capital, Volume 1, that he has annually held with his college students and which has famously been made available publicly in video format (he is currently fundraising for volume 2). Because of this, the book is not just only about Volume 1, but it is also written to be as accessible to a general public as possible. Moreover, it seeks only to explain, not to defend. Sometimes, this does lead to trouble - Harvey does not entirely seem to grasp that to explain the way a certain figure thought about a topic also means you have to show what arguments he himself would have used to defend his perspective, and when Harvey tries to substitute his own arguments for those of Marx, they are often not the more convincing for it. The book is somewhat weak on making the entirety seem convincing for that reason, but that is something easily solved by referring to his excellent other work, "The Limits to Capital" (
Limits to Capital).
That said, the book is a systematic, clear and engaging explanation of the work, built on a chapter-by-chapter approach. Harvey recommends, especially for the difficult and abstract first chapters, to have a copy of Marx's "Capital", Vol. 1, with you while reading it - the Penguin edition is generally recommended (
Capital: Critique of Political Economy v. 1 (Classics S.)). This is justified also because Marx himself, as Harvey shows, builds up his argument from chapter to chapter, both in terms of introducing ever new and more complicated concepts building on the old, and in terms of showing bit by bit what the contradictions in capitalism are and how capitalism unfolds as a result. Marx's approach is thoroughly steeped in a dynamic analysis which sees movement as the result of a clash of contradictions, in the tradition of Hegel in particular. Harvey does a deft job of explaining what this is and how it works out in the course of Marx's book.
There are of course points where one can have disagreements with Harvey's explanations, and I think at a few points this is warranted. He fails entirely to point out the actual analytical benefits of a value theory as opposed to just a price theory in his discussion of the chapter on money. Because the 'labor theory of value' is an absolutely essential and inalienable part of Marxist analysis, this is a serious problem. He does not explain the relation between industrial and financial capital very well in the chapter on capital and labor power (which he does do in his other major work). Finally, he does not give Marx's statements on the relation between 'historical and moral factors' as well as productivity to value and its flows the full attention it deserves, although admittedly that would reach fairly far for what is to be a basic introduction.
Nonetheless, overall the book is an excellent companion to the work of Marx, if one actually uses it in that way. Although I am very familiar with Marx's books, I have found that placing the two side by side and tracing the arguments as Harvey presents them through the chapters indeed allows for clear and easy insight into the difficult and often poorly written material to an extent that has helped me newly understand it too. This is no mean feat, and it will make the task of actually getting down and reading Capital, often seen as an impossible burden, all the lighter and easier to do. For this, the book is much recommended and a great contribution to popularizing Marx & Engels' enduring insights into society. For the deeper theoretical work, there are many others available.