TLDR: The paradox goes away if you make price endogenous, i.e., it only occurs because your assumption about the value growth over time that is inconsistent with the profit flows.
The paradox stems from the fact that you’ve made inconsistent assumptions: that the value of the company increases linearly over time, and that the company never generates a flow of profits (i.e., the only value comes from the sale). If profits are zero, the equilibrium price is constant at zero, and investors are indifferent between holding the company and selling it at any point in time. More generally, if the company has some potential for profits (which can be modeled as a flow of profits per unit of time, or as a hazard rate of getting an instantaneous lump sum of profits), the equilibrium price will be set so that the marginal investor is indifferent between holding and selling.
I have a tongue-in-check resolution to the Schrodinger cat variant: if his goal is to set a new world record, he should open the box immediately after the old world record. More seriously, to resolve the paradox, you need to be more explicit about his utility function: how does the value he obtains increase with the amount by which he exceeds the old record? Depending on your choice of utility function, you may or may not have a paradox, and it may or may not be equivalent to the St. Petersburg paradox.
TLDR: The paradox goes away if you make price endogenous, i.e., it only occurs because your assumption about the value growth over time that is inconsistent with the profit flows.
The paradox stems from the fact that you’ve made inconsistent assumptions: that the value of the company increases linearly over time, and that the company never generates a flow of profits (i.e., the only value comes from the sale). If profits are zero, the equilibrium price is constant at zero, and investors are indifferent between holding the company and selling it at any point in time. More generally, if the company has some potential for profits (which can be modeled as a flow of profits per unit of time, or as a hazard rate of getting an instantaneous lump sum of profits), the equilibrium price will be set so that the marginal investor is indifferent between holding and selling.
I have a tongue-in-check resolution to the Schrodinger cat variant: if his goal is to set a new world record, he should open the box immediately after the old world record. More seriously, to resolve the paradox, you need to be more explicit about his utility function: how does the value he obtains increase with the amount by which he exceeds the old record? Depending on your choice of utility function, you may or may not have a paradox, and it may or may not be equivalent to the St. Petersburg paradox.