Introduction to Modern Economic Growth constraint at time t again takes the form (14.1) C (t) + X (t) + Z (t) ≤ Y (t) , where C (t) is consumption, X (t) is aggregate spending on machines, and Z (t) is total expenditure on R&D at time t We again assume that there is a continuum of machines used in the production of a unique final good Since there will be no expansion of inputs/machine variety, we normalize the measure of inputs to 1, and denote each machine line by ν ∈ [0, 1] The engine of economic growth here will be process innovations that lead to quality improvement Let us first specify how the qualities of different machine lines change over time Let q (ν, t) be the quality of machine line ν at time t We assume the following “quality ladder” for each machine type: (14.2) q (ν, t) = λn(ν,t) q (ν, 0) for all ν and t, where λ > and n (ν, t) denotes the number of innovations on this machine line between time and time t This specification implies that there is a ladder of quality for each machine type, and each innovation takes the machine quality up by one rung in this ladder These rungs are proportionally equi-distant, so that each improvement leads to a proportional increase in quality by an amount λ > Growth will be the result of these quality improvements The production function of the final good is similar to that in the previous chapter, except that now the quality of the machines matters for productivity We write the aggregate production function of the economy as follows: ∙Z ¸ 1−β q(ν, t)x(ν, t | q) dν Lβ , (14.3) Y (t) = 1−β where x(ν, t | q) is the quantity of the machine of type ν of quality q used in the production process An implicit assumption in this production function is that at any point in time only one quality of any machine is used This is without loss of any generality, since in equilibrium only the highest-quality machines of each type will be used This production function already indicates where the Schumpeterian process of creative destruction will come from: when a higher-quality machine is invented it will replace (“destroy”) the previous vintage of machines 611