Game Theory Foundations: Production-Possibility Frontiers

The next concept relating to efficiency is the idea of production-possibility frontiers. This is the idea that given a limited amount of resources and accounting for systematic diminishing returns, that there is some limit where you cannot create more of one product without sacrificing another. In the simplest case, this is described as a curve on a graph with two axis each reflecting one product (a more complicated model with more products can be represented with more dimensions in a multi-dimensional model, but is beyond the scope of this discussion).

The most popular form of this model includes the Guns or Butter model as well as the efficient frontier in market portfolio theory.

In the Guns or Butter model, a government has the choice of spending its resource on foreign or defense projects (Guns) versus domestic or civilian projects (Butter). With a limited budget, there is some limit where in order to get more of one, they have to sacrifice production of the other. Plotting the points along this curve produces the production-possibility frontier.
Excusing the violent association with the "Guns" component, it is possible to instead substitue the idea of exportable goods and globalization which affect the balance of trade between countries (a topic for another time).

An efficient frontier in market portfolio theory describes a securities portfolio which includes securities in the most efficient weights and correlations such that the portfolio exhibits no un-systematic risk and is composed entirely of non-diversifiable risk (Capital market line - CML). Here the frontier is presented a little differently with the resource simply being combinations of different securities betas versus their expected return to form the efficient frontier (which intersects with the investor's marginal utility curve to determine the efficient market portfolio).

It seems like everywhere we go we naturally run into these frontier limits. How can we describe them in game theory? As it turns out, there are some models which describe behaviour in different stages of game theory.

Positions on the efficient possibility frontiers can not produce more of one benefit without sacrificing another. In game theory, this manifests as a zero-sum game. That is to say for one participant to gain (show on the production possibility frontier as one axis), another has to lose out).

Points contained within the curve (but not on the curve) allow for gains of one benefit or the other without having to make any sacrifices. This is represented in game theory as a non-zero sum game. That is to say, that at the end of the day, it is possible for both parties to benefit positively without the expense of the other.

You'll notice that the theme here is one of efficiency. In an environment where no more benefit can be extracted from the system with out additional resources (maximum efficiency), it becomes a zero-sum model. However, if production is inefficient, there is potential for a non-zero sum model (recouping the inefficientcy as non-zero sum gains - getting more of A without comprimising B).

This is also an indicator of efficiency and competition. In an industry with a maturing life cycle, transactions, deals and strategy will slowly being to have less non-zero sum models.

Game Theory Foundations: Economies of Scale or Diminishing Returns?

Before we start out foray into game theory, I'd like to do a quick review of economic efficiency framed as economies of scale versus diminishing returns for growing enterprises. In the examples provided by economics textbooks, they will often express diminishing returns as using less efficient resources in order to produce the same task (which reflects a decreasing marginal utility / benefit / revenue).

However, in the abstract, we are also told about economies of scale, where adding staff and capacity can result in productivity gains beyond what is expected (Specialization of tasks and the Model T assembly line strategy). In evaluating a scenario, how can we tell which stage we are in?

The best way to understand your efficiencies is by understanding the production bottle necks of your current factors of production. That is to say that you can fight off diminishing returns if you add resources to the weakest link of your system (similar to the idea of critical path of a PERT chart). While adding resources in general will cause you to gain diminishing returns in the one production line, it should create synergies that recoup the drop in efficiency.

Now, if you are sensitive to seemingly meaningless buzzwords like "synergy" which tend to be overused, you'll probably recoil a little as I did upon hearing that word so let's break it down to less abstract terms with an example.

Example: A factory has 200 workers working on 20 machines. The optimal ratio of workers to machines is 12 to 1 for the purposes of scheduling and capacity planning. Although adding the 21st machine will bring some benefit (versus not having the machine at all), it is clearly diminishing versus adding the 3rd machine. In this scenario, adding another 40 workers would help bring the worker / machine level up to its target concentration. Note that beyond that, adding another worker begins to diminish the benefit of workers and the value of adding a machine starts to increase again.

Now assume that you have a limited number of resources to apply to your production. What mix of workers to machines will you have?
This introduces the idea of production-possibility frontiers which we will discuss in the next post.

Game Theory - An "Emerging" Field of Study and Interest

While game theory, the study of strategic interactions, has been a field of study in social psychology for sometime, it has become an increasingly important field which on which management science has recently begun to focus on. Particularly, the logical analysis of decision making processes to optimize success as defined within the model.

While the CFA has introduced the idea in the Economics portion of their level I curriculum, I thought that it would be valuable to look beyond what is presented there (including the "optional" sections) and how using game theory can predict strategic level behaviours.

First, the basics, we are familiar with the Hawk-Dove model (a game of chicken) which I used in my investment blog to describe the benefits and pitfalls awaiting investors hoping to gain first movers advantage in the market. I've also used the Prisoner's Dilemma to describe OPEC and Oil.

However, what is what oversimplifies these two examples of game theory is the assumption that they occur only once. That is to say that these models only look at a particular snapshot in time. Whereas we know that the game continues to be played and is modeled by a series of events rather than one discrete iteration. Also, the models used to describe these scenarios are simple 2x2 matrices. As we start to remove some of the assumptions, we will also begin to look at more flexible (and therefore complex) models.

In the next couple of posts and into next week, I will focus on a series of posts aiming to look at more involved cases of game theory and how they can be used to model strategic business behaviour.

Demand Elasticity - Who pays?

In high school, through university and even in the CFA, economics is an important field of study. One of my favourite topics is the concept of elasticity.

Elasticity is literally defined as the percentage change in quantity over the percentage change in price and has several flavours (negative elasticity, positive elasticity for substitutes, negative cross elasticity for complements etc)

Elasticity = %ΔPrice / %ΔQuantity
%ΔPrice = ΔPrice / Pavg = P1 - P2 / Pavg
%ΔQuantity = ΔQuantity / Qavg = Q2 - Q1 / Qavg

Mathematically, note that as you move up and down the curve, the elasticity changes because percentage is affected by the absolute value of the average price. If the average price falls, the elasticity increases because the change becomes larger relative to the average to which it is compared for percentage purposes.

Another way of looking at elasticity is flexibility or bargaining power. If you review Michael Porter's five forces, you can see that elasticity for suppliers or customers increases their bargaining power. That is to say, the more competition and choices available means more options.

Let's look at a good which exhibits perfect inelasticity. This means that regardless of the price, consumers will always consume a constant amount (they set the quantity demanded). This manifests in a horizontal demand curve as shown below:

Note that the determining factor of the price is the supply curve. If there are more suppliers, the supply curve shifts right and the price drops. If there are less suppliers, the supply curve shifts left and the price rises. This is similar to what happens with oil and the "prisoner's dilemma" in OPEC's oligopoly.

Next, look at a perfectly elastic curve. This means that given the slightest change in price, the consumers will dramatically change their spending habits (that is to say, that consumers set the price). This manifests as a horizontal demand curve (at the price they set).
The only power suppliers have here is to set the quantity sold (they are price takers). If there are more suppliers, the supply curve shifts right and there is more quantity sold. Visa versa, if there are less suppliers the supply curve shifts left and there is less quantity sold.

This is important when determining how price changes will affect measures like total revenue, quantity consumed etc. This also applies regardless of whether you are talking about goods sold, wages paid, taxes paid etc.

[Example] The government is thinking of applying a tax on a good which exhibits perfectly elastic demand. Who bears the cost?

1. The supplier
2. The customer
3. The supplier and the customer share the tax burden

[Solution] One way to look at this is that if the good exhibits perfectly elastic demand, then the customers have all the bargaining power. This means that if any supplier were to simply "pass along the tax" and make the consumer pay, the consumer would just go to a different supplier. This means the supplier is forced to take on all the tax. The solution is 1. Notice this also means it eats into the producer surplus.

If the good were perfectly demand inelastic, the suppliers have all the bargaining power then the customer would bear all the tax and the solution would be 2. This would eat into the consumer surplus.

If the good were neither perfectly demand elastic or inelastic, the supplier and customer would split the difference in fractions based on who had more relative bargaining power. Both producer and consumer surplus would diminish. The solution would be 3.

Cash Flow and Operating Cycle

I've written about cash flow with queuing theory as the lifeblood of business in my blog, as well as activity (operations) ratios in my financial profitability analysis series on my investment blog, but I wanted to review an interested concept in the CFA level I regarding the cash conversion cycle.

First let's do a review of the tools and topic. Firstly, what affects operations from a cash flow perspective? Using the direct method, the operating items which affect cash flow is change in working capital and the three items that affect that is Accounts Payable (AP), Accounts Receivable (AR) and Inventory (Inv). Now let's look at a standard process for how changes in each affect the operating cycle.

Order of Operations (like the BEDMAS of elementary arithmetic):

1. Purchase supplies from vendor on credit (AP up, Inv up)
   
2. Process supplies into goods for sale
3. Sell products on credit (Inv down, AR up)
4. Pay back supplier (AP down, cash down)
5. Receive payment from customers (AR down, cash up)

Notice that you don't actually receive any cash until step 5, but you have to pay it back in step 4. This means that you have a negative cash flow until you complete the cycle.

Recall that in the indirect method (calculating CFO from NI):

• If more inventory is made than sold, some "cash value" is retained in Inventory (Inv up, cash down)
• Alternately, if more inventory is sold than made, then you are liquidating your inventory (Inv down, cash up)
• An increase in AP means that you owe your supplier more money. This means that instead of paying with cash, you paid with credit so your cash flow goes up
• A decrease in AP therefore means you paid back your debts
• An increase in AR means that your customers paid you with credit so your cash flow goes down
• A decrease in AR therefore means you were paid back (collected on sales on account)

This next little diagram illustrates the relationship between Operating Cycle, DOH, DSO, Days Payables and Cash Conversion Cycle:
Operating cycle is simply the time it takes from when you purchase supplies to when you collect the cash and is composed of two components, Days Inventory on Hand (DOH) and Days Sales Outstanding (DSO).

• Days Inventory on Hand includes the manufacturing process, as well as storage. In accounting terms, this means works-in-progress (WIP), finished goods, sales cycle.
• Days Sales Outstanding is the time between sales on credit and the collection of cash.
• Cash conversion cycle is the time between when you pay your vendor to when you yourself collect cash. It is the difference between operating cycle and Days Payables.

In looking at which company is more likely to have cash flow problems, cateris paribus, it would be the company with the largest cash conversion cycle. That is to say, it has a low Days Payables (bills due sooner - cash out), but a long Operating Cycle (takes really long to produce and sell goods as well as collect on credit - cash in). So the larger the cash conversion cycle, the worse the operational and implicit structural liquidity.

Meet the Dean - Roger Martin and Integrative Thinking

I figure I'll take a short CFA study break to write about an encounter I had at the Meet the Dean session at Rotman early last week. I thought it was an invitational event for those of us who were accepted, but it turns out there were some people who were still applying, waiting for acceptances or deciding.

Dean Martin spoke about how Rotman is different from other MBA programs and for once, I was actually impressed with the Rotman presentation. This may seem kind of odd, coming from someone who has already "sampled" the proverbial kool-aid so to speak by accepting my offer letter to start in September, but the truth of the matter is that I was more sold on Rotman by my colleagues and friends currently enrolled (or graduated) than I was from the Faculty administration. I'm quite embarrassed to say that the admin simply made it seem like just an MBA program whereas my friends were raving about their experiences.

The reason I bring up this point is that Dean Roger Martin brought it up and addressed it as well. Now from ANY MBA program, you would expect some pomp and circumstance regarding why their program is so fantastic. One of the major issues facing MBA programs today is their incremental value add. For instance, there are some top schools for which recruiting companies have stated they would rather hire students who were accepted, rather than students that had graduated. The reason? Top schools who accept good candidates are simply validating their position as top performers, whereas the marginal benefit of attending a top school doesn't necessarily justify the exorbitant increase in salary.

Dean Martin reframed this postulate as top schools resting on their laurels and not affecting the changes required by society in light of the financial crisis in the markets. He put up a rather simple diagram of a three dimensional box with the dimensions described as depth, breadth and flexibility. He called the current state of MBA education, shallow, narrow and static where it should be deep, broad and dynamic. I can't remember who he was quoting off hand, but he mentioned: "There aren't marketing or finance problems. Only business problems" (reflecting the interdisciplinary relationships).

He used the example of the Blacks-Scholes models for derivatives valuation and that stated limitations in the model made it inappropriate for use in many circumstances. However, this model is widely used in ALL derivatives valuations and therefore leaves models with large vulnerabilities in their assumptions.

The punch line?

Integrative thinking is a framework which systematically creates people who ask the right questions to make the right decisions.