The Appreciation Condition For a Single Tax On Wealth

What if, during her 2020 Presidential Campaign, Elizabeth Warren had advocated, instead of adding a new tax on wealth, eliminating all taxes on economic activity (income, capital gains, sales, inheritance, value added, etc.), replacing them with a single tax on net assets in excess of the homestead exemption for bankruptcy protection?

Perhaps the greatest fear of a wealth tax, more accurately called a tax on liquid value of net assets, is that it would expropriate liquidation value.  In fact, it can appreciate the value of net assets.  To calculate the level of appreciation it is helpful to assume a single tax on wealth and then measure the difference in liquidation value.  Under replacement of other taxes by a single tax on wealth, there is appreciation of non-monopoly* assets due to greater demand being placed on them as barriers to entrepreneurs fall by freeing up capital currently locked in place by rent-seeking.  Appreciation is calculated by subtracting the owner's original value under activity (income) taxation, from the prospective buyer's value under asset (wealth) taxation.  That is to say, buyer figures are after the hypothetical change in tax policy.  Owner figures are before the hypothetical change in tax policy:

\[\frac{  \mathit{buyer\_ income}-\mathit{buyer\_ expense}}{\log{\left( \mathit{buyer\_ interest\_ rate}+\mathit{asset\_ tax\_ rate}+1\right) }}-\frac{\left( 1-\mathit{income\_ tax\_ rate}\right)  \left( \mathit{owner\_ income}-\mathit{owner\_ expense}\right) }{\log{\left( \mathit{owner\_ interest\_ rate}+1\right) }}\]

WHERE
income_tax_rate = the aggregate tax rate on economic activities, such as income, capital gains, value added, etc.
asset_tax_rate = the net asset tax rate (on liquid value)
owner_income = the owner's expected gross periodic income from the asset
buyer_income = the buyer's expected gross periodic income from the asset
owner_expense = the owner's expected periodic expenditure on the asset
buyer_expense = the buyer's expected periodic expenditure on the asset
owner_interest_rate = the periodic interest rate paid by the owner in borrowing to purchase the asset
buyer_interest_rate = the periodic interest rate paid by the buyer in borrowing to purchase the asset

The more positive this difference goes, the greater the appreciation of liquid value resulting from the change in tax policy.

It is important to note that the above formula assumes the buyer does not enjoy a standard deduction -- for example a homestead deduction as normally protected under Chapter 7 bankruptcy.  Such a deduction is an ordinary feature of wealth tax proposals and would frequently come into play in a change to single tax on wealth as tenants  purchase their residences from landlords.

*monopoly assets -- or more precisely, the monopoly-valuation of assets -- arise from privatized positive network externalities, the classic exemplar being the phone network n² value scaling law due to its value scaling with the number of potential connections. Modern examples are the network effect monopolies arising from the Internet, such as social media companies that retain ownership of the user data and, thereby, the social connections made via those platforms.

The derivation follows:

\[\tag{profit_stream}\frac{\left( \mathit{income}-\mathit{expense}\right) \, \left( 1-\mathit{income\_ tax\_ rate}\right) }{{{\left( \mathit{interest\_ rate}+\mathit{asset\_ tax\_ rate}+1\right) }^{t}}}\]

\[\tag{net_present_value}\frac{\left( \mathit{income}-\mathit{expense}\right) \, \left( 1-\mathit{income\_ tax\_ rate}\right) }{\log{\left( \mathit{interest\_ rate}+\mathit{asset\_ tax\_ rate}+1\right) }}\]

\[\tag{AT_ NPV}\frac{\mathit{income}-\mathit{expense}}{\log{\left( \mathit{interest\_ rate}+\mathit{asset\_ tax\_ rate}+1\right) }}\]

\[\tag{IT_ NPV}\frac{\left( \mathit{income}-\mathit{expense}\right) \, \left( 1-\mathit{income\_ tax\_ rate}\right) }{\log{\left( \mathit{interest\_ rate}+1\right) }}\]

\[\tag{Buyer_NPV}\frac{  \mathit{buyer\_ income}-\mathit{buyer\_ expense}}{\log{\left( \mathit{buyer\_ interest\_ rate}+\mathit{asset\_ tax\_ rate}+1\right) }}\]

\[\tag{Owner_NPV}\frac{\left( 1-\mathit{income\_ tax\_ rate}\right)  \left( \mathit{owner\_ income}-\mathit{owner\_ expense}\right) }{\log{\left( \mathit{owner\_ interest\_ rate}+1\right) }}\]