The key assumptions of the model are:

A1:, , and as in Figure 2. The curves in Figure 2 must cross or else the lower one will dominate.

A2: Demand fluctuates with frequencies, in off-peak and in peak and.

A3: We assume SRMC (short-run marginal-cost) pricing behavior. With linear TC functions and SRMC pricing, hotels will operate at either 0% or 100%.

A4: We assume market prices in off-peak times: and market prices in peak times:. Thus hotel operates at capacity at all times, while hotel_L shuts down in and operates at capacity in. Total rooms rented in the industry in the off-peak period is where. Total rooms rented in the industry in the peak period is where.

A5: Long-run equilibrium requires zero expected profits for both hotels.

We prove in the following proposition the conditions of indifference for investors to choose between hotel_k and hotel_L in LR equilibrium.

Proposition 1 Under assumptions A1 through A5 with both hotels used in long-run equilibrium, then it must be true:

If (that is, the left-side inequality is violated) then only hotel will be used. If (that is, the right-side inequality is violated) then only hotel_K will be used.

Proof: Applying the zero profit condition to hotel_K:

This gives us:

Applying the zero profit condition to hotel_L:

This gives us:

Equations (3) and (5) can be combined:

For hotels_L to shut-down in the off peak period requires, assumption A4. If then, strictly speaking, hotels are indifferent to operating and some may be operating. Using Equation (6), this requires:

Since, We can write:

which is the asserted left-side inequality condition:

By assumption A4, , hotels_K to earn a positive contribution margin or all hotels, even hotels_K, would choose to shut-down in. Further, because if, then positive expected profits to the owners of hotels_K would emerge. Thus

yields the right-side inequality condition assertion.

The left-side condition in (1) is that. If one more room is needed in both peak and off-peak times, the total cost over the cycle of a 1 room capacity hotel over the cycle is since. A price of will exactly cover costs of one extra room operating in both periods. We suggest calling this condition that hotel be more static efficient, in the sense of Clark’s use of the term static in that there are no business cycles [1] [2] ¹.

The right-side condition in (1) is that

.

Assume we need one more room over the cycle only to meet peak demand. A price of will exactly cover costs of one extra room over the cycle operating only in high-demand.

The right-hand condition is that where production is used only in high-demand times, hotel_L is superior. The right-hand condition requires that SAC_L be flatter shaped than SAC_K. We define output flexibility as the relative flatness of the SAC curve. We suggest calling this condition that hotel_L be more output-flexible efficient².

If demand for hotel rooms were static with no irregularities, then firms would choose only hotel_K and . Demand for hotel rooms is irregular in the model, fluctuating between and. The added cost of supplying irregular demand in the model is borne entirely by hotel_L where .

Thus, a measure of added cost of supplying irregular demand in the model would be the expected rooms to meet peak demand × the difference in SRAC between the two hotels, or:. See Figure 2 which shows the added cost of supplying irregular demand for a single hotel_L (rectangle ABCDw₂).

Rectangle ABCDw₂ shows, in the model of the paper, the added cost to have output-flexible hotel_L, available only to provide for the excess peak over off-peak demand³.

There are two groups in our hypothetical society: Suppliers (owners-managers of hotels) and consumers (households who rent hotel rooms). Consumers rent rooms in a free market on a daily basis from various hotels where each hotel posts its prices. Consumers pay the lowest price per-room per-day in the local market. The intersection of this price with the consumer demand schedules (off-peak and peak) determine the quantity of rooms the consumers order.

Consumers have a fixed budget for room rentals expenditures. They are price sensitive in renting rooms, in the sense that consumers will rent more rooms at a lower market price and less rooms at a higher market price. Consumers pay market price times quantities purchased, (total revenue to suppliers equals market price times quantities).

The demand curve shows the maximum quantities consumers would be willing to purchase at various prices. The assumption is that the demand curve is downward sloping, meaning that consumers would be willing to rent more rooms daily if prices were lower, all else being the same. The area under the demand curve up to the point of quantities of market purchases shows the value to the consumer.

Figure 3 shows a geometric demonstration with varying pricing (alternative A) versus fixed pricing (alternative B) with fluctuating D functions, off-peak period and peak period each with its associated. Let D₁ be consumer demand for rooms during off-peak periods, the great majority of the year, say 6/7th of the year.

Using hypothetical numbers to make the economic concepts clearer, point K could be that, at a market price of $36 per room per day consumers are willing to rent 35 rooms per day. Point H might be that at a market price

of $33 per room per day consumers are willing to rent 37 rooms per day.

Let D₂ be consumer demand for daily room rentals on the peak period. Using hypothetical numbers to illustrate, point D could be that, at a market price of $51.9 per room per day consumers are willing to rent 42 rooms per day. Point J could be that, at a market price of $36 per room per day consumers are willing to rent 54 rooms per day.

The demand curve, off-peak period demand, occurs with frequency, , 6/7. The demand curve. Peak period demand, occurs with frequency, , 1/7.

We define consumer surplus as the area under the demand curve and above the price line. We define expected values, E, as the sum of each outcome times its expected value. Using the illustrated numbers for points H and D, the market equilibrium points for pricing rule A, varying prices, we can calculate, expected total revenue, and, expected quantities, as follows:

Using the illustrated numbers for points K and J, the market equilibrium points for pricing rule B, fixed prices, we can calculate, expected total revenue, and, expected quantities, as follows:

We prove in the following proposition that consumer surplus is necessarily larger in an arrangement where consumers get more rooms for the peak period at the cost of less rooms for the off-peak periods whereby consumers pay the same amount and rent the same number of rooms over the year. We show graphically this increase in consumer surplus. This becomes a maximum willingness for consumers to pay suppliers for that arrangement.

We assume that suppliers are willing to offer rooms daily according to two alternative pricing schemes: a fixed price, , at all times, versus for off-peak periods and for the peak period. We have two basic assumptions in the model: according to both pricing schemes total payments over the week are the same and total food purchases are the same.

Proposition 2 A comparison of alternative pricing schemes, A: varying prices, versus B: fixed prices, under conditions of shifting downward-sloping demand curves shows and rises as demand elasticity rises assuming

and