# Alternatives to Monte-Carlo simulation

Imagine I have a model of economy of a region, which consists of

1. several companies,
2. importers and
3. population.

Let's assume that all local companies in question produce

1. food and agricultural products,
2. acquire all of the inputs from companies outside the region and
3. sell all of the final output to the population.

At every cycle, following things happen:

1. local companies import inputs,
2. local companies produce final output from the inputs,
3. local companies distribute the output to the population,
4. the population consumes the outputs of the companies and produces waste.

Every parameter the model, including

1. the technological coefficients of companies (how much of every type of input is required to produce 1 unit of final output),
2. consumer demand (how much food and agricultural products every person consumes)

are estimates, i. e. they are specified as ranges. For example, every person in the region consumes 0.8-1.1 tons food per year with a 90 % confidence interval.

If I repeat the cycles several times, I get an estimate of

1. how much resources the region imports (R1) and
2. how much food and agricultural products the population of the region consumes (R2).

These 2 numbers are the results of running the model.

But in order to get meaningful results, I need to run this simulation with different parameters because - as stated above - I don't have exact values of most of them. I have only ranges from A to B and the true value lies between A and B with 90 % probability.

The number of parameters is equal to

2 types of companies * 14 (number of inputs) + 1 (consumption of food) + 1 (consumption of agricultural products) = 30.

I can run the simulation N times, and in every run set every parameter to a random number in its interval. Then, I would get N result sets (one result set consists of two values R1 and R2).

From these result sets, I could calculate the minimum and maximum value of R1 and R2.

The problem with this approach is that the number of possible combinations of the 30 parameter values is huge.

Are there any other methods, which allow to determine the ranges of R1 and R2 when their values depend on parameters, which can't be measured precisely (when the values they depend on are from-to estimates) ?

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"the true value lies between A and B with 90 % probability." This is absolutely false. Also: are you new to modeling? This model seems needlessly complicated. –  TBrendle Oct 28 '13 at 8:58
What one usually does is to fix all but one parameter and then one tries to get some information about the systems behaviour by varying this parameter hopefully resulting in a simplification of the model. –  Uwe Stroinski Oct 28 '13 at 9:05
If $R1$ and $R2$ have nice functional forms in terms of the parameters, you may be able to reduce dimensionality of the problem. The distributions of parameters are defined by confidence intervals. Assuming they are normal, linear combinations of normal variables are normal so a linear production function will have a normal distribution. $R1$ and $R2$ seem to be related by the fraction wasted and a nice functional form for fraction wasted may also cut on some work. Extreme values of R1 and R2 require extreme values of input parameters but you only have confidence intervals. –  user96614 Oct 28 '13 at 18:27

Following the advice of @Uwe Stroinski, you could do some comparative statics first to simplify things. For example, if the model were simple enough, you could look at the first-order conditions that describe a solution to the model. Take one of the parameters, $\theta_1$. Given the range of values that the other parameters could take on, does the partial derivative of the equilibrium values $R^*_1$ or $R^*_2$ with respect to $\theta_1$ give the range of values of the other parameters have an unambiguous sign? You would then know that for parameters in the given ranges that the effect of $\theta_1$ is monotonic. If so, when you go about calculating $R_1$ and $R_2$ for many combinations of the parameters, you would only have to consider combinations that include the extreme values of $\theta_1$ and ignore the points in between.