Capacity Constraints

Bertrand-Edgeworth Model

In the last Market Simulation, MS-171 Bertrand Competition, we explored the basic micro-economic model in which all firms set their own price, with each firm assuming the other competitors in the market will not change price. The result of that model predicted that Competitors of Commodity Products will always reduce Price down to an unprofitable Marginal Cost.

In 1897, Francis Edgeworth criticized the Bertrand Competition Model. He showed that Capacity Constraints would not lead Competitors to cut their Prices down to Marginal Cost. These Competitors could, therefore, profitably sell Commodity Products. This gave rise to the Bertrand-Edgeworth Model.

In this Market Simulation, we take the same workflow from MS-171 Bertrand Competition and modify it according to Francis Edgeworth so that each Competitor has a Production Limitation.

This Case Study provides a high-level overview of the workflow without detailed explanation. It assumes you are already somewhat familiar with KNIME and Market Simulation. If not, start by reviewing the Building Blocks and Community Nodes.

Add Capacity Constraint

This Market Simulation workflow makes adding a Capacity Constraint to Spacely Sprockets and Cogswell Cogs very easy.

The list of ‘Competitive Rivals’ defined in the ‘Input Product Array’ by the ‘Table Creator’ node is modified such that a ‘Capacity’ column is added. Each Competitor has been set a limitation to sell a maximum Quantity of 4,000 Products. These Capacity Constraints then flow down through the rest of the workflow and into the Market Simulation nodes which run the Price experiments.

Capacity Column

New Price Trend

As before, the Price Strategy trends for both Competitors can be compared and plotted after the 30 iterations of the Pricing Loop.

Now, as Francis Edgeworth predicted, Price no longer trends towards Marginal Cost of $50, and Profit no longer trends towards zero. Instead, the Price of both Products hovers between $70 and $80, with Profitability hovering between $85,000 and $100,000.

Pricing Loop Results

Chart #1
Price vs Cost

Chart #2
Expected Quantity

Chart #3

Edgeworth Paradox

Francis Edgeworth (1845 - 1926)

Wikipedia: The Edgeworth Paradox describes a situation in which two Competitors cannot reach a state of equilibrium by charging a stable Price (pure strategy).

Unlike the Bertrand Paradox, the situation of both Competitors setting Price equal to Marginal Cost is not an equilibrium, since Capacity Constraints mean that either Competitor can raise its Price to generate Profits. Nor is the situation where one Competitor charges less than the other an equilibrium, since the cheaper Competitor can profitably raise its Price towards the expensive Competitor’s price. Nor is the situation where both Competitors charge the same Profitable Price, since either Competitor can then lower its Price marginally and Profitably capture more of the Market.

See also: the Edgeworth Box.

Chart Interpretation

The table above shows the results from Spacely Sprocket’s Price Experiments when Spacely decides to increase Price. The three Price Experiment results include:

  1. Raise Spacely’s Price
  2. No Price Change
  3. Lower Spacely’s Price

The ‘No Price Change’ produces the worst expected result. In the previous Loop Iteration, Spacely expected to sell out the maximum Capacity of 4,000 Products at a Price of $71.77. But in reality, after Cogswell also changed Price, Spacely sold only 3,553 Products.

Spacely calculates that, if he was to lower Price, he would again expect to sell out the maximum Capacity of 4,000 Products at an increased Profit. But, if he was to increase Price, then Profitability would further increase (up from $77,353 to $84,790).

Unfortunately, Cogswell makes a similar calculation and also changes the direction of his Price Strategy.

These constantly changing Pricing Strategies results in a chaotic Market whereby steady-state Prices are never reached.