Group overview: Investment It is always costly for the firms to involve in big swings in capital. The cost may involve installment, search for capital and potentially slowed-down production during the capital adjustment, which are in addition to the capital purchase cost. There are several types of models that capture such adjustment cost in capital. The motivations of such models also come from empirical evidence that firms' capital stock responds slowly to shocks in productivity and that Tobin's (average) \(q\) is not always equal to 1.

We are going to discuss some baseline models that are useful to describe this phenomenon in aggregate level, not micro level, because such models assume homogeneity in firms. Among those models, some assume convex adjustment cost, while others assume non-convex adjustment cost (in particular assuming wedge between buy and sell prices of capital, or assuming fixed cost of adjustment, or assuming financing constraint of individual firms). We will be mainly talking about the convex adjustment cost model, and the capital-buy-sell-price-wedge model.

Before starting talking about those two models, we can briefly think about the other two models: - Fixed cost model: it is assumed that every time the firm makes adjustment, it pays a fixed cost determined by its productivity, which has similar result as in the capital-buy-sell-price-wedge model. - Financing constraint model: it is assumed that every individual firm faces different cost of financing, so their costs of buying capital are more than capital price and are heterogeneous.

There are also many other interesting settings in the literature. But it is natural to start thinking about this problem with a convex adjustment cost.

Definition 32.1 (Tobin's average \(q\) and marginal \(q\)) Suppose a firm has a value \(V_0(s^0,k_0(s^{-1}))\) in current period 0, then $$ > \text{Tobin's average }q\equiv\frac{V_0(s^0,k_0(s^{-1}))}{k_0(s^{-1})},\qquad \text{Tobin's marginal }q\equiv\frac{\partial V_0(s^0,k_0(s^{-1}))}{\partial k_0(s^{-1})} > $$

Proposition 32.1 With CRS production technology and CRS adjustment cost, Tobin's average \(q\) and marginal \(q\) are equivalent.

Proof Suppose the firm starts with \(k_0\), and denote its corresponding optimal policy functions by \(h_t^{\star}(s^t;k_0)\) and \(k_{t+1}^{\star}(s^t;k_0)\). Then, if the firms start with \(\hat k_0=\lambda k_0\) for \(\lambda>0\), the policy functions \(\hat h_t(s^t;\lambda k_0)=h_t^{\star}(s^t;k_0)\) and \(\hat k_{t+1}(s^t;\lambda k_0)=\lambda k_{t+1}^{\star}(s^t;k_0)\) are still feasible because of the CRS property of both production function \(f(k,zh)\) and adjustment cost \(C\!\left(\frac{k_{t+1}}{k_t}\right)k_t\). So, the feasibility of \(\hat h_t(s^t;\lambda k_0)\) and \(\hat k_{t+1}(s^t;\lambda k_0)\) implies $$ > V_0(s^0,\lambda k_0)\ge\lambda V_0(s^0,k_0) \tag{32.2} > $$ Reversely, suppose the firm starts with \(\lambda k_0\) for the same \(\lambda>0\), and denote its corresponding optimal policy functions by \(h_t^{\star}(s^t;\lambda k_0)\) and \(k_{t+1}^{\star}(s^t;\lambda k_0)\). Then, if the firms start with \(k_0\), the policy functions \(\frac{1}{\lambda}h_t^{\star}(s^t;\lambda k_0)\) and \(\frac{1}{\lambda}k_{t+1}^{\star}(s^t;\lambda k_0)\) are still feasible, so the feasibility implies $$ > V_0(s^0,k_0)\ge\frac{1}{\lambda}V_0(s^0,\lambda k_0) \tag{32.3} > $$ Multiply (32.3) by \(\lambda\) and combine with (32.2): \(V_0(s^0,\lambda k_0)=\lambda V_0(s^0,k_0)\). Denote Tobin's average \(q\) by \(v_0(s^0)\equiv\frac{V_0(s^0,k_0)}{k_0}\), so \(v_0(s^0)=\frac{k_0 V_0(s^0,1)}{k_0}=V_0(s^0,1)\), and $$ > \frac{\partial V_0(s^0,k_0)}{\partial k_0}=\frac{\partial(k_0 V_0(s^0,1))}{\partial k_0}=\frac{\partial(k_0 v_0(s^0))}{\partial k_0}=v_0(s^0) > $$ which implies \(\dfrac{V_0(s^0,k_0)}{k_0}=v_0(s^0)=\dfrac{\partial V_0(s^0,k_0)}{\partial k_0}\), i.e. Tobin's average \(q\) = Tobin's marginal \(q\). \(\blacksquare\)

Remark 32.1 The monotonicity of \(\gamma_t(s^t)\) and \(v_t(s^t)-f_{k,t}(s^t)\) comes from the convexity in adjustment cost function. If the adjustment cost is not convex, then we generally won't have this result.

References - Lucas and Prescott. "Investment Under Uncertainty." Econometrica (1971). - Hayashi. "Tobin's Marginal \(q\) and Average \(q\): A Neoclassical Interpretation." Econometrica (1982).