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Introduction to Modern Economic Growth greater investments Therefore, in this model Tobin’s q, or alternatively the costate variable q (t), will play the role of signaling when investment demand is high The q-theory of investment is one of the workhorse models of macroeconomics and finance, since proxies for Tobin’s q can be constructed using stock market prices and book values of firms When stock market prices are greater than book values, this corresponds to periods in which the firm in question has a high Tobin’s q– meaning that the value of installed capital is greater than its replacement cost, which appears on the books Nevertheless, whether this is a good approach in practice is intensely debated, in part because Tobin’s q does not contain all the relevant information when there are irreversibilities or fixed costs of investment, and also perhaps more importantly, what is relevant is the “marginal q,” which corresponds to the marginal increase in value (as suggested by equation (7.65)), whereas we can typically only measure “average q” The discrepancy between these two concepts can be large 7.8 Taking Stock This chapter has reviewed the basic tools of dynamic optimization in continuous time By its nature, this has been a technical (and unfortunately somewhat dry) chapter The material covered here may have been less familiar than the discrete time optimization methods presented in the previous chapter Part of the difficulty arises from the fact that optimization here is with respect to functions, even when the horizon is finite (rather than with respect to vectors or infinite sequences as in the discrete time case) This introduces a range of complications and some technical difficulties, which are not of great interest in the context of economic applications As a result, this chapter has provided an overview of the main results, with an emphasis on those that are most useful in economic applications, together with some of the proofs These proofs are included to provide the readers with a sense of where the results come from and to enable them to develop a better feel for their intuition While the basic ideas of optimal control may be a little less familiar than those of discrete time dynamic programming, these methods are used in much of growth 359