For the production function for /, gross investment as a fraction of the initial home value, we assume that/(/) = I5. This choice of investment function is admittedly arbitrary, but it is roughly plausible, and seems suitable for calibrating the model for illustrative purposes. With this investment function, we would see that in the absence of any risk sharing arrangement the homeowner would, maximizing profits, set / = 0.25. This means that over an 8 year period the homeowner would invest about 3% of the value of the home per year. It also means that if the homeowner were to consider an additional improvement of 10% of the initial value of the home, beyond the improvements optimally made under a conventional mortgage, then the additional improvement would be mapped into approximately a 9% additional improvement in the resale value of the home, substantial, but not quite worth making as an investment. We assume a value for D equal to 0.5. This implies that if the homeowner lives in the home but makes absolutely no investment at all in value-maintaining or value-improving activity of any kind (including not resisting the impulse to remove value from the home and sell at below market price) then the home would lose half its value in eight years. We will not normally observe such sharp declines in value because there are usually some incentives to maintain value. With this value of Д a profit-maximizing homeowner, who maximizes equation (1) in the absence of any risk management contract distortions, will, by investing 25% of the home’s initial value, cause the home value to remain unchanged after eight years at 1.00 times the initial value.

The h(l) curves corresponding to Figures 1-4, computed using expressions (5) through (9), are shown in Figures la-4a. On each figure a 45 degree line is also shown, a line with a slope of one.

Note that for all four figures, the h{J) curves are smooth, even though the g( V) function that was used to derive them had a kink in it in two cases. The smoothness of the h(T) curves reflects the presence of uncertainty: the homeowner does not know which section of the broken straight line will be relevant after the final value of the home is discovered. Thus, one might say that all four curves look basically similar in overall appearance, despite their different derivations. They all show an upward slope and a negative second derivative. But, this similarity obscures important differences to the homeowners in terms of the implied profit maximizing behavior.