# **3. Simulations**

We will in this section illustrate properties of our model through four sets of simulations: 1) ex ante expected probability of energy use or retrofit in period 2; 2) ex ante expected energy and retrofit costs in period 2; 3) energy intensity of the infrastructure chosen in period 1; and 4) overall expected energy use, which combines the energy intensity of infrastructure (3) with the probability of subsequent energy use (1). Under our (we think, reasonable) numerical cases the closedown option turns out to play a very small role for the overall results, and will not be emphasized in the following; but must still be kept in mind.<sup>22</sup> All simulations are made under the assumption that energy and retrofit costs are jointly log-normally distributed, and calculations are done using the scientific program Matlab. Most of the simulations for expected energy cost in period 2 (section 3.2) depart directly from the expected cost expressions (7), and (8) for expected retrofit cost in period 2. For probabilities of using energy and retrofitting respectively (section 3.1) we use (3) and (4), and for energy intensity of infrastructure (section 3.3), we use (13). For expected overall energy use section 3.4), we apply the solution value for _H1_ together with the ex ante probability of energy use, (3).

In all cases where nothing is otherwise stated, the (unconditionally) expected energy and retrofit cost (per unit of energy use and retrofit investment respectively) are both kept constant, at levels _E(q(2))_ = 2, and _E(y(2))_ = 3. ( _E(q(2_ )) is the unconditionally expected energy/environmental cost in period 2; it would be the _actual_ expected cost given no retrofit or closedown. A similar interpretation holds for _E(y(2)_ ).) While both energy and retrofit costs are uncertain, ex ante energy costs are thus generally lower _in expectation_ . The distributions of energy and retrofit costs are both assumed to be jointly log-normal, and either independent, or have a positive correlation coefficient of 0.5. In sections 3.3-3.4, simulations are done for all variants a) – c) of the utility function (15).

# **3.1 Ex Ante Probability of Energy Use or Retrofit in Period 2**

This first set of simulations considers implications of various parametric changes in the model, for the expected value of (per-period) energy costs during period 2. It is useful to first study period 2 behavior; this will be a benchmark for subsequent analysis of period 1 investment and overall energy use, in sections 3.3-3.4 below.

Figure 1 describes the probability that the firm in period 2 will continue using energy at the level initially established in period 1, given a parametric change in expected energy cost, for different degrees of uncertainty about both energy and retrofit cost.<sup>23</sup> In all examples used in figure 1, _Ey_ = 3. We see, as should be obvious, that a low (high) _q_ implies a high (low)

> 22 Two other sets of assumptions in the following also potentially be attacked. One is the log-normal distribution assumption. Log-normality is however a rather robust assumption in this context; see e g Schuster (1984). The other is our assumption on the utility function of energy intensity in infrastructure, where we assume a set of functions parameterized by the demand elasticity with respect to price (thus constant for a given specified function; the CRRA class of functions). While this general class of functions can be viewed as robust, criticism can be raised in particular to the Cobb-Douglas and log specifications; see below.

> 23 Note that the probability of closedown is also accounted for in these calculations, but its probability turns out to be too small to matter for the results. (Throughout, the value of continued use is set at 10, implying that this option will be exercised only when min( _q, y_ ) > 10.)

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