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It is broadly accepted that climate-economic modelling (Integrated Assessment modelling) inevitably implies substantial uncertainties [8]. One of established approaches for taking into account these uncertainties is performing Monte Carlo simulations with Integrated Assessment models (IAMs) [4, 6, 8]. The idea of the method is that instead of one model run a series of model runs is performed with model parameters that are likely to be responsible for critical uncertainties randomly varying from one model run to another, and then the probability distributions of output model variables of interest are constructed on the basis of these model runs.

Given the complexity of state-of-the-art IAMs, many of which need substantial computational resources, the probability distributions can be obtained only numerically. However, in case of simple `toy' models that have analytical solutions it might be occasionally possible to `imitate' this numeric Monte Carlo procedure by exact analytical calculations of probability distributions and of moments of random output variables of interest.

A constructive example of such an `imitation' is provided in the present paper.

The paper is organized as follows.

In Sec. 2 a simple climate-economic model based on the AK model with the endogenous depreciation rate and on the exogenous climate scenario is described and its analytical solution is obtained. In Sec. 3 the uncertainty of climate projections is introduced in the model, and the Monte Carlo procedure is imitated. Sec. 4 provides some numerical examples and discussion. Sec. 5 concludes.

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