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# financial economics

Fri 5 Aug, 2022 04:37 am
A representative agent lives for an infinite number of years. He produces output 𝑦𝑡 using capital 𝑘𝑡 via the following production function:
𝑦𝑡=𝑓(𝑘𝑡)=𝑘𝑡.
In each year the representative agent decides the allocation of 𝑦𝑡 between consumption, 𝑐𝑡, and investment, 𝑖𝑡. Assuming that capital depreciates fully at the end of every year, 𝑖𝑡≡𝑘𝑡+1. The objective of the representative agent is to maximise his lifetime utility which is given by
Σ𝛽𝑡(𝑎0𝑐𝑡−𝑎12𝑐𝑡2𝑇𝑡=0),
where 0<𝛼0<𝛼1, 0<𝛽<1 and 𝑇<∞ (i.e. T is finite), subject to the resource constraint and the initial condition for capital, 𝑘0>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, 𝑉′(𝑘).
(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. 𝑇=∞) is economically meaningful. Justify your answer

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