Chapter 19. Volatility in Aluminum Production: 12 Exercises

1. Check Your Work from Exercise 3 in the Book:

The variable costs of the 16 smelters should be specified to match the cost curve shown in Figure 19.2.
For example, you might set the variable costs as follows:
variable_cost[1] = 35
variable_cost[2] = 40
variable_cost[3] = 42
variable_cost[4] = 45
variable_cost[5] = 46
variable_cost[6] = 48
variable_cost[7] = 49
variable_cost[8] = 50
variable_cost[9] = 52
variable_cost[10] = 54
variable_cost[11] = 55
variable_cost[12] = 56
variable_cost[13] = 58
variable_cost[14] = 60
variable_cost[15] = 65
variable_cost[16] = 70

The remaining equations could be written as:
capacity_of_a_standard_smelter = 1
smelter_operating?[Smelter] = if(producers'_lagged_price>variable_cost[Smelter]) then 1 else 0
total_number_in_operation = ARRAYSUM(smelter_operating?[*] )

The graph below shows the simulated behavior of the new model. The annual demand is constant at 16 mmt/year, so variability in the ingot price may be attributed to the actions of the sixteen smelters.

You may compare the new results with the results shown in Figure 19.5 to learn if the new model shows greater volatility in prices. Figure 19.13 shows that ingot prices follow a limit cycle with approximately the same period as in Figure 19.5. The prices at the peak of each cycle are around 120 cents/pound, somewhat higher than the corresponding values shown in Figure 19.5. The greater volatility in the new model is probably caused by the "lumpiness" associated with a simulated industry comprised of 16 large smelters.

2. Is Lumpiness Really Responsible for Higher Volatility?

Expand the smelter dimension in the previous exercise to run from 1 to 32. Change the equations to allow the variable cost to run from 35 to 70 cents/pound. Set the variable costs at each smelter to correspond to the shape of the industry cost curve in Figure 19.2, and don't forget to cut the size of the standard smelter in half. Run the new model and compare the results with the results in the previous exercise. Does lumpiness contribute to the simulated price volatility

3. Endogenous Cost of Alumina

Expand the model in exercise 3 from chapter 19 as shown below.
These changes will allow the cost of aluminum ore to vary up or down in response to changes in the ingot price.
Recall that the model in chapter 19 sets the cost of alumina at 15 cents per pound of aluminum that will be extracted from the alumina. You may assume that the ratio of 15 cents/pound to a 60 cent/pound price of metallic aluminum is typical. Define a lagged value of the ingot price with a 3 month lag time to represent short delays in the market adjustments in the cost of alumina:
cost_of_alumina_ore[Smelter] = lagged_value_of_ingot_price*metal_to_ore_price_ratio
lagged_value_of_ingot_price = smth3(ingot_price,alumina_market_adjustment_lag,60)
alumina_market_adjustment_lag = 3
metal_to_ore_price_ratio = .25

Run the new model with different values of the adjustment lag time. Do the endogenous changes in alumina costs make the aluminum cycle more stable?

4. Variable Recycling

Expand the model in Figure 19.3 to allow the fraction recycled to vary with the ingot price. Rather than 12.5% in every year, let the fraction vary from a low of 8%/year to a high of 16%/yr depending on the ingot price. Assume there is no delay in the recyclers' reaction to the price. Does this addition make the system more stable?

5. Lag in the Response of Recyclers

Expand the previous model to introduce a 3 month lag in the reaction of the secondary producers to changes in the ingot price. Does the addition of the time lag make the system more stable?

6. End Use Detail

Expand the model in Figure 19.3 with an extra array to distinguish between different uses of aluminum.
Define a new dimension use to take on the values:

auto	aluminum used in the automobile industry
air	aluminum used in airplanes
house	aluminum used in housing construction
bev	aluminum used in beverage container
other	all other uses of aluminum.

Each use should be assigned a different level of demand, and the total demand should match the 16 mmt/yr in the original model. Assign a different price elasticity, a different product lifetime and a different recycling fraction to each use. Set the recycling fractions to ensure that the total recycling is similar to the 2 mmt/yr of secondary production in the original model. Run the new model with constant recycling fractions. Does the detailed treatment of end uses change the volatility of the simulation?

7. New Scrap as well as Old Scrap -- Equilibrium Diagram

The recycling of used products is sometimes called "old scrap." Another source of aluminum is the recycling of "new scrap" which is found on the floors of the mills and fabricators that change aluminum ingots into finished products. In the United States, new scrap can be twice as large as old scrap (Berk 1982, 51). Expand the model in Figure 19.3 to include aluminum fabrication. Assign one stock to inventories held at smelters and a second stock for inventories held by fabricators (as in Figure 2.10). Split the previous inventory with two thirds to the smelters and one third to the fabricators. You may assume that inventory held at the smelters governs the price of ingots.

Next, assume a 26% fabrication loss factor in fabrication. This means the smelters must deliver 126 pounds to the fabricator for every 100 pounds of finished aluminum products. The extra 26 pounds will fall to the floor as "new scrap." Define a new scrap recycling fraction at 80%, so 80% of the new scrap would be returned to the smelter's inventory. Build the new model with initial conditions set for equilibrium. Document your new model with an equilibrium diagram similar to the diagram in Figure 19.3.

8. New Scrap as well as Old Scrap -- Dynamic Results

Change the initial amount of inventory held at the smelters. This change will allow the model to simulate the production cycle. Run the model with different assumptions on the fraction of new scrap that recycled. Does the recycling of new scrap makes the system more stable?

9. Technological Advance in the Costly Smelters

Expand the model in the 4th exercise of chapter 19 to simulate the investment in advanced technology at the smelters with highly variable operations. Suppose a new technology becomes available in the 48th month of the simulation which reduces the electricity requirement from 7 to 5 kwh per pound. It also reduces the labor costs by 4 cents/pound. Technological possibilities are described by Russell (1981), Jarrett (1987) and INEL (1987). The illustrative benefits in this exercise might be obtained if we were to perfect the "stable cathode/inert anode process". Run the new model with the assumption that smelters 12-16 adopt the new technology when it becomes available. Does their investment allow them to operate these smelters on a more constant basis?

10. Technological Advance in All Smelters

Repeat the analysis from the previous exercise with the assumption that all 16 smelters invest in the new technology when it becomes available. Does the technological advance allow smelters 12-16 to operate on a more constant basis? Do you see any changes in the fraction of the time that smelters 1-8 are in operation?

11. Unstable Cycles Growing Into a Limit Cycle

Figure 19.11 shows an outward growing spiral that encounters limits after only one or two cycles. These cycles grow rather quickly into a limit cycle, so it may be hard to see the transition from growing oscillations to a limit cycle. To gain a clearer picture of this pattern, repeat the simulation with the initial inventory set just slightly higher than the 3.4 mmt equilibrium value shown in Figure 19.3. Document your work with a new scatter graph similar to Figure 19.11?

12. The Cobweb View of Production Cycles

The final exercise deals with the cobweb theory of cyclical behavior. The cobweb model is a graphical approach explained by Samuelson (1964, p. 396) and Meadows (1970, p. 13) and illustrated for the world aluminum industry in the diagram below.

The dark lines show a linear supply curve and a linear demand curve. The intersection point is the market clearing price and quantity. In this case, we expect ingots to sell for 65 cents/pound, and we expect supply and demand to balance at 15 mmt/yr. The slope of the supply curve may be described in terms of the "supply elasticity" which is found by:

relative change in Q: 1/15 = 6.7%
relative change in P: 6/65 = 9.2%
supply elasticity: 6.7%/9.2% = 0.73

The downward slope of the demand curve may be described in a similar manner. In this case, a price increase of around 5.5 cents/pound is expected to lower the demand by 1 mmt/yr:

relative change in Q: 1/15 = 6.7%
relative change in P: 5.5/65 = 8.5%
demand elasticity: 6.7%/8.5% = -0.79

The value of -0.79 turns out to be quite close to the long run price elasticity observed in the construction sector.

Now, to begin the graphical analysis, imagine that producers are unable to deliver the market clearing quantity of 15 mmt/yr. For example, a strike may limit their production to 14 mmt/yr, the point on the lower left of the supply curve. Now follow the upward arrow to see the market clearing price of 69 cents/pound. Now follow the arrow to the right to see the quantity of ingots that would be produced in the following time period. If the price were 69 cents/pound, producers would produce around 15.8 mmt/yr. Follow the downward arrow, and you will learn that the market clearing price in the next time period would be around 62 cents/pound.

The graphical analysis shows the arrows leading to the center of the diagram. This implies that the interplay of supply and demand will lead to converging oscillations. But you can tell that the graphical analysis might also have delivered sustained oscillations or even divergent oscillations because slight change in the slopes could deliver different results. For example, we would expect sustained oscillations if the elasticities are identical. The cobweb would yield divergent oscillations if we imagine a flatter supply curve.

With this background, you should be able to answer the following cobweb questions:

(1) where does the slope of the supply curve appear in the system dynamics model?
(2) where does the slope of the demand curve appear in the system dynamics model?
(3) how is the intersection of the two curves found in the system dynamics model?
(4) does the cobweb model lead us to think that a consumer response to price changes would lead to greater volatility in production cycles (as seems to be the case in Figure 19.10)?