HPE is an acronym used by Scott Sumner for the Hot Potato Effect. In Sumner's view, base money, best epitomized by hand-to-hand currency, impacts spending on output through this effect.
If the existing quantity of currency is greater than the demand to hold it, then those holding excess currency balances spend them. But those receiving the currency now have excess balances, and they spend them.
Currency is like a hot potato, where holding onto it results in burned hands, and so it is thrown to someone else, who also throws it again.
What brings the process to an end? As people spend the currency, they bid up prices of goods and services. Higher prices reduces the real quantity of currency. Higher prices raise the nominal demand for currency. The HPE continues until prices rise enough so that the real quantity of currency falls to the real demand for currency. An alternative framing is that HPE continues until prices rise enough so that the nominal demand for currency rises to meet the nominal quantity of currency.
While the increased expenditure and demand for assorted goods is essential to the HPE, is should not be taken too literally. Suppose there is as surplus of apples at the current market price. Some of the sellers will be frustrated, unable to completed desired sales. To remove their frustration, they undercut the prices of their competitors. If successful, a firm can then sell all it wants at a slightly lower price. However, this largely shifts the surplus to these competing firms who are even more frustrated, and motivated to not only match the lower price, but the undercut that price as well.
What brings the process to an end? The lower prices reduce quantity supplied by each and every seller, while the lower prices increase quantity demand. This reduces the surplus and when the price is low enough, quantity supplied equals quantity demanded. There is no longer a motivation for any firm to undercut the prices of the others. The equilibrium price and quantity is reached.
Does this mean that an increase in supply or decrease in demand should be expected to result in a surplus and then a gradual price war which only gradually results a new equilibrium price and quantity? No. It is the existence of some such process that motivates those actually setting the prices to adjust them to the equilibrium level. If there were a surplus, sellers could simply lower prices to the new equilibrium level and produce the equilibrium quantity.
In my view, the competitive market process is best characterized by specialist entrepreneurs choosing price and quantity combinations that they believe will avoid the frustration of surpluses or shortages. They often make errors, and they try new price and quantity combinations. The result is more like a direct shift to new equilibrium prices and quantities and nothing like a surplus, then price wars, then quantity adjustments, and so a gradually decreasing surplus until an equilibrium is reached. The story of surpluses and shortages and the incentives to adjust prices describes a counterfactual. What would happen if prices failed to adjust.
Similarly, those selling various goods and services don't have to wait for round after round of hot potato transactions of currency. They could immediately raise prices to a point where the real quantity of currency equals the real demand for currency.
What about production? Sumner advocates looking at the relationship of nominal income to the nominal demand for currency. If real output is constant, then nominal income is proportional to the price level. The increase in the price level needed to adjust the nominal demand for currency to the quantity implies a proportional increase in nominal income.
But what if real output is dependent on expenditure? As long as the real demand for currency is unit elastic with respect to real income, then the nominal demand for currency remains proportional to nominal income. The HTE raises spending on output. The increased spending on output could result in higher prices, higher output, or some combination. Still, the result is higher nominal income which raises the nominal demand for currency to meet the existing quantity.
So, those selling and producing various goods and services don't have to wait for round after round of HTE transactions, they could instead each set prices and produce a quantity of output so that when it all adds up it generates a level of nominal income such that the nominal demand for currency equals the nominal quantity of currency.
I don't think the competitive market process is usefully characterized in this way. I think the primary difference between the quantity of currency and the demand to hold it, and the supply and demand for apples or some other good or service, is a difference in specialization. Currency has no market of its own. Currency has no price of its own. Those who make the adjustments to bring the nominal demand for currency into equilibrium with the quantity are not making adjustments in the quantity of currency. I think these differences are important.