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# Optimization in the One-Sector Model: Application of the Pontryagin Maximum Principle

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# Optimization in the One-Sector Model: Application of the Pontryagin Maximum Principle

DOI link for Optimization in the One-Sector Model: Application of the Pontryagin Maximum Principle

Optimization in the One-Sector Model: Application of the Pontryagin Maximum Principle book

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## ABSTRACT

First, we will eliminate K from (2) by using the definition of k. From the definition, it follows immediately that

k = e--rtl(- 'Yk. Substitute from 1.(9) into (2) and then substitute the resulting expression in the formula for k:

(7) c(t) is proportional to consumption per effective worker. If the marginal felicity of consumption has constant elasticity u, then

(8) Let us introduce a new "price" for consumption per effective worker, say p. We wish to define p so that

p = (e-r 1jP)"q. (10) We now need a differential equation for p to replace (5) for q. Take natural logarithms in (10):

log p = u('Yt-log P) +log q. Differentiate with respect to time:

pjp = p + UT - j'(k) = W - f'(k) 1 (11) from 1.(11).