Heterogeneity is ubiquitous in firm-level and sectoral data. Equilibrium models, however, typically assume a representative firm, as in Andrew B. Abel and Olivier J. Blanchard (1983). The representative firm paradigm leaves no role for the distribution of capital. We model capital reallocation in a general equilibrium model with two sectors. Capital adjustment costs capture illiquidity in our model, similar to Hirofumi Uzawa's (1969) capital installation technology. We follow Fumio Hayashi (1982) in assuming that the production technology is linearly homogeneous, which allows us to focus on the sectoral distribution of capital, separately from the level of total capital. The two sectors may have different levels of productivity, and we show that the distribution of capital between the two sectors is the single state variable governing investment, growth, and valuation in the economy.
We analytically characterize prices and quantities, including investment, growth, the interest rate, and the price of capital (Tobin's q) at both aggregate and sectoral levels, along with the effects of sectoral heterogeneity and reallocation in the economy. Without adjustment costs, capital is immediately reallocated to the more productive sector. With adjustment costs, the central planner optimally trades off growth against the cost of reallocating capital. Hence, reallocation to the high productivity sector is not immediate, and reallocation itself expends resources. When the more productive sector is initially small, investment exceeds output in the high productivity sector, so output from the less productive sector finances growth in the more productive sector. Nonetheless, investment and growth optimally continue in the initially larger, low productivity sector. This occurs because, while the sector is relatively less productive, its output can be reinvested in the other, more productive, sector. This is more efficient than directly uninstalling capital from the less productive sector and reinstalling it in the more productive sector because of adjustment costs. The capital stock in the less productive sector dwindles over time as its growth rate shrinks, and eventually the economy specializes in the more productive technology. As the economy moves toward specialization, the growth rate is nonmonotonic. At first, the aggregate growth rate falls, because more resources are expended on reallocation, but eventually the growth rate rises as the economy specializes in the high productivity sector. The interest rate follows this same nonmonotonic pattern, first falling and then rising along with the aggregate growth rate because the equilibrium interest rate must rise with the growth rate of aggregate consumption to clear the market.