Home Economics Generalized Microeconomics

## MODEL A: UNIFORM DISTRIBUTIONS OF THE PROBABILITY OF EXTINCTION W.R.T. PRICEIn sections 8.1 and 8.2 we assume for simplicity uniform distributions of the probability of extinction with respect to price p, i.e. distribution functions in the for m in the interval = 0 for p < = 1 for p > where p0 is the price at which the possibility of competitors entering vanishes entirely and
= 1 for = 0 for where The following two figures show the probability distribution functions for these distributions:
The probability of survival in this model is maximized by the price p* at which the following function reaches a maximum: for the derivative of function η If we set , we get the following for the argument of the maximum Here, not surprisingly, the optimal price is the average of the price levels at which the threat of extinction due to one of the two reasons under consideration materializes with 100% certainty. ## MODEL B: UNIFORM DISTRIBUTIONS OF THE PROBABILITY OF EXTINCTION W.R.T. PROFITABILITYNow let us assume, more realistically, uniform distributions of the probability of extinction with respect to profitability . We will denote
We assume here extinction distribution functions in the form:
h, p),= 0 for = 1 for where
= 1 for = 0 for where The probability of survival in this model is maximized by the profitability Plots 1 and 2 in Figure 46 represent the distribution functions of the uniform distribution of the risk of extinction Analogously to the previous section, if we set Hence, the optimal price It therefore holds that the survival-probability-maximizing price
## MODEL C: NORMAL DISTRIBUTIONS OF THE PROBABILITY OF EXTINCTION W.R.T. PROFITABILITYIn this model we will work with the normal distribution of the probability of extinction due to the two reasons under consideration with respect to profitability . As it is not generally possible to determine the cumulative distribution function of the normal distribution algebraically, we will use estimates obtained by numerical methods for our subsequent conclusions. In this model we are looking for the profitability π* at which the following survival probability function reaches a maximum
The optimal profitability π* is the root of equation
The symbols used have the following meanings:
Plots 1 and 2 in Figure 47 represent the distribution functions of the probability of extinction due to Figures 48 and 49 show the plot of the probability of survival (i.e. the probability of avoidance of both threats of extinction] against profitability π and the magnitude of the optimal profitability π* for the case where the two distributions have the same standard deviation. In this case, the maximum probability of survival occurs for the profitability which corresponds to the average of the means of the two distributions and which does not depend on the other parameters of the model (the common size of the variance and the slope of the demand function]. Figure 48 assumes normal distributions of the risk of extinction Figure 49 assumes normal distributions of the risk of extinction
= 0.1
= 0.2Figures 50 and 51 illustrate the case where the variances of the extinction probability distributions with respect to the chosen profitability differ significantly. In both figures, plot 3 represents the agent's probability of survival and plots 1 and 2 the probabilities of extinction due to one of the reasons under consideration. The optimal profitability is again π
It is clear that the main determinants of the size of the optimal profit are the means of the two distributions and the relationship between their variances. If we fix the means [say at 0 and 2) we can compare the size of the optimal profit as a function of the variances. When the two variances are the same, the optimal profitability will be close to the average of the two means, i.e. to the diagonal of the square in Figure 52. For small positive variances of both distributions ( It can be shown
We have successfully investigated the problem of the firm's optimal business strategy in conditions of increasing returns to scale given high initial fixed costs of entry into the market and zero marginal costs where the firm is threatened by extinction due to both low profitability and ruinous entry of a competitor. The optimum strategy is a compromise (between the two threats) that involves choosing the price at which the rise in price is associated with an equal fall and rise, respectively, in the probabilities of extinction due to the two reasons (low profit and entry of a competitor). It turns out that this problem has a single solution (with both uniform and normal distributions of the probabilities of extinction). This allows us to construct a supply function. Where the sole threat to the firm is low profitability, this function is coincident with the standard supply function of a monopoly producer. The described approach is therefore obviously again a generalization (and not a refutation) of the standard micro- economic approach. Where both extinction threats are active, the location of the optimum depends also on the relationship between the variances of the probability of extinction. If the variances are identical, the optimal profit is at the level of the arithmetic average of the means of the two distributions. If one of the variances increases in relation to the other, the optimal profit level shifts towards the mean of the distribution having the lower variance, although only when the variance exceeds a certain level. From this we can conclude that under certain conditions (a high and non-uniform degree of uncertainty regarding the various threat factors) there is a risk that a relatively minor increase in uncertainty in a system will result in a qualitative change in the behaviour of the system and its sensitivity to changes in its parameters. This represents a potential element of instability in markets of the type under analysis. |

< Prev | CONTENTS | Next > |
---|

Related topics |

Academic library - free online college e textbooks - info{at}ebrary.net - © 2014 - 2019