A Model of Manufacturers and Buyers of Cars over the Business Cycle Illustrating Competitive Manufacturing

We illustrate competitive manufacturing with an original theoretical model of manufacturers and buyers of cars over a business cycle that have peak and off-peak demand periods. There are two types of plants manufacturing cars, plantK and plantL, each having linear total costs with absolute capacity limits. PlantK operates with low VC and high FC by being capital intensive. PlantK is output-rates rigid since it produces throughout the business cycle and always at capacity. PlantL operates with low FC and high VC by relying on outsourcing major components and parts. PlantL is output-rates flexible since it produces only in the peak-demand periods. We show results under SRMC pricing. Then we examine an alternate arrangement which increases demand irregularity. We show, under conditions of the model, that the added cost to supply irregular demand should be small because of the low FC of plantL. We show, under the conditions of the model, that the added gain in consumer surplus to have irregular demand supplied should be large because consumers will have more available for the peak periods. The main policy implication of this theoretical model—for regularly recurring cycles—is to urge focus, even in the off-peak periods, on adequate capacity for the peak periods.

cycle to the dominant role of fixed costs that are incurred irrespective of output rates: "It is needless to point out that overhead costs play a fundamental part in the behavior of business at every stage of that many-sided phenomenon, the business cycle.The part they play is most paradoxical.For they make regular operation peculiarly desirable and peculiarly profitable, so that business feels a definite loss whenever output falls below normal capacity, yet it is largely due to this very fact of large fixed capital that business breads these calamities for itself, out of the laws of its own being.And the largest businesses, which have the highest percent of constant costs due to invested capital, are, as we have seen, precisely the ones which fluctuate the most, so far as employment is an index.There is something about the commercial-industrial system which bewitches business so that it does just the thing it is trying to avoid, and is held back from doing just the thing it yearns to do-maintain steady operation and avoid idle overhead.And while the contributing causes of this strange auto-hypnosis are many and of varied character, technical, financial, commercial, and psychological; the underlying fact of large capital plays a central part, and the inelasticity of cost, sunk cost, and the shifting and conversion of overhead cost are all facts of major importance." 1 The US manufacturing industries are now some 6 or 7 years in a recession, as the figures in Table 1 show 2 .
In his 1923 book John M. Clark illustrated the calculations for expected average cost, ( ) E AC , for a manu- facturing plant making a car 3 .See

Traditional Manufacturing versus High-Value Manufacturing
In traditional manufacturing the focus is on the production phase of a product.In high-value manufacturing the recommendation is for manufacturers to concern themselves with the entire manufacturing value chain: "A New Definition of High-Value Manufacturing...A successful manufacturing industry goes beyond production, it means thriving research and development (R&D), design, supply management, sales and marketing as well as after sales services... Highly successful manufacturers do not need to rely on production alone and they can accommodate effective outsourcing." 4tsourcing means buying components and parts instead of making them.In high-value manufacturing firms  are increasing product flexibility, meaning which products they make.In traditional manufacturing, as here and in John M. Clark's writings, the industry is composed of manufacturers that produce a particular product, such as a car.In high-value manufacturing firms are part of other industries depending on what products they sell.In traditional manufacturing, outsourcing increases a firm's output-rate flexibility of production of a particular product.

An Original Model of Manufacturing and Buying Cars over the Business Cycle
We illustrate an original model of manufacturing and buying cars over the business cycle.The product is homogeneous in that all cars are assumed identical in looks, driveability and value in the market.We assume fluctuating demand over a business cycle of a number of years, with peak periods, part of the cycle, and off-peak periods, the balance of the cycle.We assume car manufacturers set two prices, one at the peak and one for the off-peak times of the business cycle.We assume no price collusion among car manufacturers.We assume car manufacturers know the consumer-demand schedules for their cars produced.We assume zero expected profits for all car manufacturers in long-run equilibrium.Initially we assume SRMC pricing.

Car Manufacturing over the Business Cycle: The Supply Side
We assume a single homogeneous product, Q, cars.We assume ease of entry of new car manufacturers.We assume a business cycle of two states of demand, 1 1 D w and 2 2 D w , off-peak and peak, each with a likelihood, where the likelihoods add to one.There are two types of car manufacturing plants, plant K and plant L .Car manufacturing plants require durable and specific assets, and have linear short-run total-cost curves with absolute capacity limits.Car manufacturing plants have a per-car variable-operating cost b , per-car capacity costs β (fixed costs per-year per-plant divided by maximum cars production rate per-year per-plant) and per-plant capacity q (maximum cars production per-year per-plant).
We envision investors and managers walking into a car manufacturing plant store that has two shelves: each with a model plant n that costs, say, $1,000,000 to build.On one shelf is a model of plant K and on the other shelf is a model plant L (see Figure 1).Investors or entrepreneurs can order any multiple or fraction of the model plants.No economies of scale exist for plants.Thus the long-run marginal cost (LRMC) and long-run average cost (LRAC) for plants in the car manufacturing plant store are horizontal.These customers of the car manufacturing plant store have to decide plant K and choose a K n or plant L and choose a L n .The assets are durable and specific meaning that the plants will last a long time, say 50 years, and are useful only for making cars.

Key Assumptions
The key assumptions of the model are: q q as in Figure 2. The curves in Figure 2 must cross or else the lower one will dominate.
A2: Demand fluctuates with frequencies, 1 w in off-peak and 2 w in peak and 1 2 1 w w + = .A3: We assume SRMC (short-run marginal-cost) pricing behavior.With linear TC functions and SRMC pricing, plants will operate at either 0% or 100%.
A4: We assume market prices in off-peak times 1 P : and market prices in peak times 2 P : 2 L b P < .Thus plants K operate at capacity at all times, while plants L shutdown in 1 t and operate at capacity in 2 t .Total cars manufactured and sold in the industry in the off-peak period is 1 Q where 1 Total car manufactured and sold in the industry in the peak period is 2 Q where 2 Q n q n q = + .A5: Long-run equilibrium requires zero expected profits for both plant types.

Objective of Proposition 1
We prove in the following proposition the conditions of indifference for investors to choose between plant K and plant L in LR equilibrium.

Proposition I
Proposition 1 Under Assumptions A1 through A5 with both plants used in long-run equilibrium, then it must be true: (that is, the left-side inequality is violated) then only plant L will be used.If (that is, the right-side inequality is violated) then only plant K will be used.Proof: Investors in plant K have zero expected economic profits per Assumption A5: ( ) This gives us:

G. Aranoff
Investors in plant L have zero expected economic profits per Assumption A5: This gives us: Equations ( 3) and (5) can be combined: For plants L to shut-down in the off-peak period per Assumption must be 1 then, strictly speaking, plants L are indifferent to operating and some may be operating.Using Equation (6), this requires: which is the asserted left-side inequality condition: or By Assumption A4, or yields the right-side inequality condition assertion.

Left-Side and Right-Side Inequality Conditions
The left-side condition in (1) is that If one more car is supplied in both peak and off-peak times, the total cost over the cycle of a 1 car capacity plant operated over the cycle is bq q b in both time periods will exactly cover costs of one extra car operating in both periods.We suggest calling this condition that plant K be more static efficient, in the sense of Clark's use of the term static in that there are no business cycles [2] 5 .
The right-side condition in (1) is that The right-hand condition is that where production is used only in high-demand times, plant 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 plantl L be more output-rates flexible efficient6 .> .Thus, a measure of added cost of supplying irregular demand in the model would be the expected manufactured cars to meet peak demand × the difference in SRAC between the two plants, or:

Definition of the Model and Its Terms and Assumptions
There are two groups in our hypothetical society: Suppliers (manufacturers of cars) and consumers (households who buy cars).Consumers buy cars in a free market on a daily basis from various manufacturers where each manufacturer posts its prices.Consumers pay the lowest price per-car in the local market.The intersection of this price with the consumer-demand schedules (off-peak and peak) determine the quantity of cars the consumers order.
Consumers have a fixed budget for car purchase expenditures.They are price sensitive in buying cars, in the sense that consumers will buy more cars at a lower market price and less cars at a higher market price.Consumers pay market price times quantities purchased, TR P Q = × (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 buy more cars 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 w .Let 1 D be consumer demand for cars 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 car consumers are willing to buy 35 cars.Point H might be that at a market price of $33 per car consumers are willing to buy 37 cars.
Let 2 D be consumer demand for cars on the peak period.Using hypothetical numbers to illustrate, point D could be that, at a market price of $51.9 per car consumers are willing to buy 42 cars.Point J could be that, at a market price of $36 per car consumers are willing to buy 54 cars per day.
The demand curve 1 D , off-peak period demand, occurs with frequency, 1 w , 6/7.The demand curve 2 D .Peak period demand, occurs with frequency, 2 w , 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 ( ) A E TR , expected total reve- nue, and ( ) E Q , expected quantities, as follows: ( ) $33 37 6 7 $51.9 42 1 7 $1,358 ( ) 37 6 7 42 1 7 37.7 Using the illustrated numbers for points K and J, the market equilibrium points for pricing rule B, fixed prices, we can calculate ( ) B E TR , expected total revenue, and ( ) B E Q , expected quantities, as follows: ( ) $36 35 6 7 $36 54 1 7 $1,358 ( ) 35 6 7 54 1 7 37.7

Objective of Proposition II
We prove in the following proposition that consumer surplus is necessarily larger in an arrangement where consumers get more cars for the peak period at the cost of less cars for the off-peak periods whereby consumers pay the same amount and buy the same number of cars 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 cars according to two alternative pricing schemes: a fixed price, P , at all times, versus 1 P for off-peak periods and 2 P 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 quantities purchases are the same.

Proposition II
Pricing Rule Equilibrium Points Frequencies A : varying prices ( ) , D A P ( ) ( ) By definition of ( ) and ( ) By definition of ( ) and By Assumption (11) We can state: ( )

E CS E CS −
rises as price elasticity rises, since the areas of ( )( ) ( )( )

Figure 1 .
Figure 1.SR total-cost curves of plant K and plant L .

Figure 2 .
Figure 2. Plant L added cost of supplying irregular demand: need one more car over the cycle only to meet peak demand.costs of one extra car over the cycle manufactured only in high-demand.

2 ABCDw
If demand for cars were static with no irregularities, then firms would choose only plant K and LRMC K K b β = + .Demand for cars is irregular in the model, fluctuating between 1 1 D w and 2 2 D w .The added cost of supplying irregular demand in the model is borne entirely by plant L where min min SRMC SRMC L K

Figure 2
which shows the added cost of supplying irregular demand for a single plant L (rectangle 2 ABCDw ).

Table 2 .
The ( ) $1301 per car would give zero economic profits only if %CU rates actual equalled expected.For lower %CU, as in Table1, a price of $1301 would give losses to car producers with losses rising as %CU falls.Clark argued for efforts to keep %CU high as the key to efficiency and economic wellbeing.

Table 2 .
Annual budgets at various operating rates.
This is important because it shows consumer-surplus comparisons for perfectly inelastic, zero price elasticity, must be greater than zero, providing that price elasticities of the demand curves are not zero.At zero price elasticity 1 and therefore areas DJE and KGH each equals zero.