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Fri 5 Aug, 2022 04:37 am

A representative agent lives for an infinite number of years. He produces output ๐ฆ๐ก using capital ๐๐ก via the following production function:

๐ฆ๐ก=๐(๐๐ก)=๐๐ก.

In each year the representative agent decides the allocation of ๐ฆ๐ก between consumption, ๐๐ก, and investment, ๐๐ก. Assuming that capital depreciates fully at the end of every year, ๐๐กโก๐๐ก+1. The objective of the representative agent is to maximise his lifetime utility which is given by

ฮฃ๐ฝ๐ก(๐0๐๐กโ๐12๐๐ก2๐๐ก=0),

where 0<๐ผ0<๐ผ1, 0<๐ฝ<1 and ๐<โ (i.e. T is finite), subject to the resource constraint and the initial condition for capital, ๐0>0.

(a) Derive the value function for the representative agent.

(b) Show that the marginal utility of consumption is equal to first derivative of the value function with respect to capital that is, ๐โฒ(๐).

(c)Derive the relationship between the value function and the Lagrange multiplier.

(d)Using the result in part (b), derive the Euler equation in k.

(e) Show the general solution of the difference equation in (d).

(f) Show the transversality condition and explain whether the version of the model where the horizon is infinite (i.e. ๐=โ) is economically meaningful. Justify your answer