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The AK model with temperature-dependent depreciation rate

The standard AK model is the simplest model of endogenous economic growth [1]. In the AK model the capital is the state variable, and its dynamics obey an equation


where is the savings rate, is the technology parameter and is the depreciation rate (all parameters are assumed to be constant). According to Eq. (1), the capital grows exponentially:


where is the initial capital stock and


is the (background) growth rate.

Following some previous theoretical work on Integrated Assessment modelling [2, 3, 5, 7], we now assume that the depreciation rate is climate-dependent, and that it increases as global warming evolves. Taking the temperature as a proxy of the state of the climate system, we assume the linear temperature dependence of the depreciation rate:


where is the initial value of the depreciation rate, is the constant sensitivity of depreciation rate to temperature change, and is the initial value of the temperature. Then Eq. (1) should be rewritten in the form

, (5)

or, equivalently,

, (6)

where Eq. (3) has been taken into account.

For the sake of model tractability, we assume a very simple exogenous climate scenario with linear temperature growth:

, . (7)

Then it can be easily shown that the solution of Eq. (6) takes the form

. (8)

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