The output values for the notional example presented Section 5.1, page 110, are not consistent with the model of ship output developed in the paper. That model assumed a deterministic budget and stochastic input costs. However, the example output values in section 5.1 are actually generated by assuming a stochastic budget and deterministic input costs, a case which was not discussed in the paper. In other words, the equations are correct, but the output numbers in the example are not generated with those equations.
The correct output values that are generated using the equations presented in the paper are as follows:
- For the case where a = 1.3, ship production is declining 0.5% per year with 19.0 ships produced over the next 20 years
- For the case where a = 1, the production rate declines at 0.75% per year with 18.6 ships produced over the next 20 years
- For the cost reduction case with a = 1.3, the production run is 25.3 ships over the next 20 years, and for a = 1, the production run is 23.2 ships
The difference in behavior between these output values and those originally included in the paper are due to the fact that ship output is a function of the reciprocal of the input costs. When input costs are stochastic, the resulting growth rate of ship production becomes dependent upon the relative values of the budget and cost growth rates, the cost volatility, and the production function exponent. This is evident upon examination of the definition of αY presented in the paper. Cost volatility provides a boost to αY. When the cost volatility is low relative to the budget growth rate shortfall, the behavior is similar to the case presented in the paper. However, when cost volatility is large relative to the growth rate shortfall, it offsets the detrimental effect of increasing returns to scale under declining budgets and rising costs.
For example, if we reduced cost volatility to σC = 0.1 in the base case, then we get the expected increase in production from the change in a. For a = 1.3, cumulative ship output is 15.9 over 20 years versus 16.5 for a = 1.