Radix Stage 1  Adaptive Pbuy with Vesting
In Brief:
Instead of always buying 1 XRD for 1 USDT during Stage 1, new investors will buy 1 XRD at a price of Pmin + 0.3. The purchase price for XRD in a given period is called “Pbuy”. The Pbuy price of Pmin + 0.3 will naturally increase as Pmin increases each period. In addition, the Genesis investors (and Radix team) will receive 104 million XRD instead of 79 million, but will start out with 10 million and the rest will vest over 5 years.
This model was inspired in part by @tesslerc who proposed a different adaptive model.
Summary of Advantages:
The Adaptive Pbuy With Vesting model increases the Pmin value much more quickly as compared to the original model. It also provides better returns for new investors, and less risk (due to the higher Pmin). These improvements will give new investors more incentive to invest in the Radix network. Otherwise, the new model is very similar to the original model.
What Problems Does This Approach Solve?
This Stage 1 proposal solves the following problems in the current economic model:

In the original model, Pmin does not increase very quickly, especially if the amount of investment is not very high in some periods.

The Adaptive Pbuy with Vesting model increases Pmin more quickly even for lower amounts of investment.
 This reduces risk for new investors, encouraging more to invest.
 Increasing Pmin more quickly during Stage 1 will mean reaching 0.9 more quickly in Stage 2.


In the original model, Stage 1 investors are quickly diluted and do not see a significant return relative to their risk.
 The Adaptive Pbuy with Vesting model provides investors with more Radix for the same fiat, as compared to the current model. This results in a better return on investment for new investors.

In the original model, as the end of Stage 1 approaches, investors would have less and less incentive to buy for two reasons: (a) their rate of return is significantly lower and (b) because they could wait for the DEX to open in Stage 2 and potentially buy for a much lower price (and at worst 10% higher).
 The new model provides better rewards for investors who buy in the middle or at the end of Stage 1.

The Adaptive Pbuy With Vesting model will also reach a Pmin of 0.9 more quickly if we assume the same Stage 2 model as in the original economic paper. Reaching a Pmin of 0.9 faster will mean that it requires less total new investments to reach stage 3.
Conclusion:
All of the above advantages provide better incentives to the Stage 1 investors to put in more money, which will increase Pmin more quickly.
This model still provides businesses with a simple way to calculate the cost of buying Radix (since it’s just Pmin plus 0.3).
Details:
There are three main changes proposed by this model. All other aspects of the original paper remain the same. These changes only affect Stage 1:

Adaptive Pbuy. The price at which new investors buy during Stage 1 will increase as Pmin increases. This improves the investors’ return and lowers the risk for new investors which should increase the amounts invested.
 Pmin starts at zero and is equal to nR(t) / M(t) (the amount in the reserves divided by the total Radix in existence). This is the same as in the original paper.
 Pbuy = Pmin + 0.3
 After each period, recalculate Pbuy based on the new Pmin. Pbuy will be capped at a max of 1.1 USDT.
 New investors will all purchase at a price of Pbuy (during Stage 1).
 During Stage 1, all USDT tokens spent will go into the reserves (same as original economic model).

Vesting of the Genesis Investors and Radix Team. Using vesting significantly improves the rate at which Pmin will grow during Stage 1 (see table and spreadsheet below).
 Instead of 78,926,251, the Genesis Investors and Radix Team will get 104,000,000, but won’t get it all at once. The higher total value makes up for not having as much at the start.
 The Genesis Investors and Radix Team start with 10 million XRD, prior to week 1.
 The remaining 94,000,000 will vest over a maximum of 5 years. It could vest faster, if sufficient funds are entering the system (see details below).

The amount of XRD created for redistribution each period will be smaller. Having some redistribution helps investors who hold, including the Genesis Investors, but we no longer need the large redistribution found in the original economic model.
 The amount of Radix created for redistribution will be 0.25 * nR(t’). This will still help existing Radix holders, but will not overinflate the number of Radix.
Comparison
The following table is a comparison of approaches after 6 months of Stage 1 participation. All examples use an initial investment of $10,000 USD. When using the Adaptive method, the Pmin is 72% higher after 6 months (with $10 million in new investment per week), as compared to the original method. And Pmin is over 100% higher after 6 months when there is $1 million in new investment per week.
Pmin  # Radix Owned for Week 1 Investor  Minimum Value at Week 26  Potential Value (at 0.9 Pmin)  

Original Model ($1million/wk)  0.15  15,377  $2,369  $13,839 
Adaptive Model ($1million/wk)  0.33  39,236  $13,038  $35,312 
Original Model ($10million/wk)  0.32  33,147  $10,489  $29,832 
Adaptive Model ($10million/wk)  0.55  46,496  $25,548  $41,846 
Genesis Investors (and Radix Team) Vesting:
 The Genesis Investors (and Radix Team) can see their 104 million vest more quickly if lots of money is coming into the Radix network. In the spreadsheet, the amount that vests each period is either 1/10th of the number of dollars that were spent on new Radix that period, or 361,538 (the amount needed to vest within 5 years max), whichever is higher.
 This gives the Radix Team extra incentive to bring in new investors.
Detailed Spreadsheets:
 The following spreadsheet shows the Stage 1 Adaptive Pbuy With Vesting model. Make a copy of it and customize the parameters to see how they affect the results. Get it here:
 For comparison purposes, I also have created a spreadsheet which reflects the original paper’s formulas (based on the one published by the Radix team) that adds the ability to test it with vesting and also provides the ability to see returns on an investment made in different weeks. Get it here:
NOTES:
 I experimented with various values other than 0.3. Others worked too, but I found that 0.3 was a reasonable compromise between increasing Pmin quickly and giving new investors a good return throughout Stage 1 (and thereby incentive to invest).
 The initial starting amount for Genesis Investors of 104 million was somewhat arbitrary, but based loosely on how much they would make using the original redistribution method.
 The vesting time of 5 years was selected to spread out the impact of the 104 million so that Pmin could increase at a reasonable pace.
 If the initial Genesis allotment was less than the proposed 10 million, Pmin would grow even more quickly during Stage 1, but such a change should probably also require a total allotment of higher than 104 million. The provided spreadsheet allows for such experimentation.
 The original model requires extremely large amounts of investment to reach a Pmin of 0.9 and takes a long time to do so. The Adaptive Pbuy with Vesting model requires less investment and will reach a Pmin of 0.9 faster.