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Introduction to Modern Economic Growth
a price of χ(ν, t) at time t to maximize profits. Since machines depreciate after use,
χ(ν, t) can be interpreted as a “rental price” as well.
The demand for machine of type ν is obtained by maximizing net aggregate
profits of the final good sector as given by (13.2) minus the cost of inputs. Since
machines depreciate after use and labor is hired on the spot market for its flow services, the maximization problem on the final good sector can be considered for each
point in time separately, and simply requires the maximization of the instantaneous
profits of a representative final good producer. These instantaneous profits can be
obtained by subtracting the total inputs costs, the user costs of renting machines
and labor costs, from the value of our production. Since machines depreciate fully
after use, the user cost of renting machine ν at time t is χ (ν, t). Therefore, the
maximization problem at time t is:
(13.5)
1
max
[x(ν,t)]lv∈[0,N (t)] ,L 1 − β
"Z
0
N (t)
1−β
x(ν, t)
#
β
dν L −