In order to Tax non-monetary property and wealth the government sets an arbitrary price (they are not even personalized assesments) and often people transact below that price. I have bought and sold below that price. I want my government to declare the highest price it would buy at if I so decided.
Government: X is worth Y
Me: Sold (or not if I so wish)
How can we force (make an incentive compatible mechanism for) the government to reveil its indifference price towards all goods/property to be Taxed?
I want to solve an optimization problem to confirm the incentive compatibility. I am hoping for a stable equilibrium where not only an increase/decrease in the price (any deviation from the indifference price) puts the government in a worse position but also if the government were to declare any other price it would always improve its position be moving closer (the utility function should be unimodal with the mode equal to the government's indifference price and it should have no other local maximum) There should be uncertanty (the government doesn't know the Tax payers' reservation price otherwise the best choice would be to declare their reservation price over your own indifference price).
How I went about trying to solve the problem.
The government has a WTP Y (because either it values the asset at Y or because it can sell at Y if no agent has a higher WTP than the owner then obviously Y=Z) The owner has a WTS Z
The government want to maximize their wealth.
So everytime anything lower than Y (X<Y) will not be declared because The government would lose on taxes (it can increase the tax revenues until Y without worrying about having to purchase)
On the other hand when it declares something higher than Z the owner will sell making the government earn the tax rate but lose MR of 1.
Let's say the tax rate in not constant but proportional to some tax rate for the highest valued asset
$\dfrac{x^2}{2w} + Y - X$ is the function of the of the government
We can see that when x < w (always true since w is the highest valued asset) The government actually loses more than it earns for declaring anything higher
I don't know how to account for uncertainty on Z