Possible Insight

One of the reasons I like my Production Function Space (PFS) hypothesis, is that it clarifies a lot of issues that have puzzled me for a while.  For example, as part of my work on seed-stage startup investing with RSCM, I have struggled with two questions: (1) what’s the difference between a startup  and a small business and (2) why do some large companies have initiatives like venture groups and startup incubators?

To answer these questions, I’m going to have to get a little mathematical.  Don’t worry.  No derivatives or integrals, but I need to introduce some notation to keep the story straight.  First, let’s define production footprint and search area.

A production footprint is the surface that encloses a set of points in PFS.*  If A is a production function for basketballs, B is a production function for baseballs, and C is a production function for golfballs, then pf(ABC) looks like this in two dimensions:

We can also talk about the production footprint of a company like Google.  pf(Google) is the surface that encloses all the production functions that Google currently uses.

A search area is the surface defined by extending the production footprint outward by the search radius.** Imagine that we wanted to see if making basketballs, baseballs, and golfballs would enable a company to make footballs using production function F, we might want to compute sa(r, ABC).  It looks like this in two dimensions:

We can also talk about search areas for a company.  sa(r,Google) is the surface enclosing all the points in PFS within r units of Google’s current production footprint.  If we write just sa(Google), we mean search area using Google’s actual search radius.

With these two basic concepts in place, I can now easily answer questions (1) and (2).  SU stands for startup, SB stands for small business, and LC stands for large company.  So what’s the difference between a startup and a small business?  Well, when they are founded, a startup doesn’t have any production footprint at all and a small business does.  To a first order, pf(SU)=0 and pf(SB)>0.  A startup doesn’t know with any precision how it’s going to make stuff.  A small business does.  Whether it’s a dry cleaner, law office, or liquor distributor, the founders know pretty precisely what they’re going to do and how they’re going to do it.  However, startups have a much larger search radius than small businesses.   r(SU)>r(SB).  Assuming that we can define a search area on a set of production functions the startup could currently implement (but hasn’t yet), I content that also sa(SU)>sa(SB).

This realization was an epiphany for me.  Even though the average person thinks of startups and small businesses as similar, they are actually polar opposites.  They may both have a few employees working in a small office, but one is widely focused on exploring a huge region of PFS while the other is narrowly focused on implementing  production functions within a tiny region of PFS.  I also realized that you need to evaluate two things in a startup: (a) its ability to search PFS and (b) the ability to implement a production function once it locates a promising region of PFS.  But the magnitude of impact for (a) is at least as big as (b) in the very early stages.

Now on to the issue of large companies.  The problem here is search costs.  Remember that, in three-dimensional space, volume increases as the cube of distance.  In PFS, volume increase as an exponent of distance equal to the dimensionality of PFS.  I posit that PFS is high-dimension, so this volume increases very quickly indeed.  Now try to visualize the production footprint and search area of a large company.  A large company has a lot of production functions in play so pf(LC) is large.  But sa(r,LC) increases exponentially from this large volume by a large exponent.

In three dimensions, imagine that pf(LC) is like a hot air balloon.  Extending just 10 feet out from the hot air balloon’s surface encompasses a huge volume of additional air.  But in high-dimension PFS, the effect is… well… exponentially greater.  So a large company has a problem.  On the one hand, increasing its search radius is enormously costly.  On the other hand, we know that Black Swan shifts in the fabric of PFS will occasionally render a huge volume dramatically less profitable, probably killing companies limited to that volume.  So it’s only a matter of time before sa(r,LC) is hit by one of these shifts, for any value of r.

Obviously, there will be some equilibrium value of r where the cost balances the risk, but that also implies there’s an equilibrium value for the expected time-to-live of large companies.  Yikes!  This explains why the average life expectancy of a Fortune 500 company is measured in decades. Another epiphany.

Internal venture groups and incubators represent a hack that attempts to circumvent this cold, hard calculation. The problem is that it’s difficult to explore a region of PFS without actually trying to implement a production function in that region.  Sure, paper analysis and small experiments buy you some visibility, but not very much.  Also, in most cases, you don’t get very good information from other firms on their explorations of PFS, unless you observe a massive success or failure, at which point it’s too late to do much about your position.  That’s why search costs are so darn high.  Enter corporate venture groups and startup incubators.

These initiatives require some capital investment by the large company.  But this investment is then multiplied by the monetary capital of other investors as well as the human capital of the entrepreneurs.  With careful management, a large company can get almost as much insight into explorations of PFS by these startups as it would from its own direct efforts.  Moreover, because startups are willing to explore PFS farther from existing production footprints, the large company actually gets better search coverage of PFS.

This framework answers a key question about such initiatives.  To what extent should corporate venture groups and startup incubators restrict the startups they back to those with a “strategic” fit”?  If you believe in my PFS hypothesis, the answer is close to zero or perhaps even less than zero (look for startups in areas outside your company’s area of expertise).  Otherwise, they’re biasing their searches to the region of PFS that’s close to the region they’re already searching.  It doesn’t increase the ability to survive that much.  As far as I know, Google is the only company that adopts this approach.  I think they’re right and now I think I understand why.  Epiphany number three.

* There are some mathematical details here that need to be fleshed out to define this surface.  But I don’t think they add to the discussion and my topology is really rusty.

** More mathematical details omitted.  The only important one is that the extension outward doesn’t have to be by a constant radius.  It can be a function of the point on the pf. In that case, r is a global scaling parameter for the function.

One of the interesting questions in economics is why markets coordinate some forms of production but firms coordinate others.  Or to put it more sharply, if centralized economic planning doesn’t work for countries, why does it work for firms?

In principle, it should be possible to coordinate production solely through market transactions among individuals.  Everyone would be some combination of an independent contractor and a capital owner.  The challenge of course would be establishing the necessarily fluid markets, not to mention all the time everyone would have to spend negotiating contracts for their labor and capital.

Thus the standard approach to explaining firms starts with the Transaction Cost theory (typically attributed to Coase).  Whenever the cost of a market-based mechanism is higher than a firm-based mechanism, a firm will end up coordinating production.  Of course, to anyone who has ever started a new firm or worked any length of time at a large one, this explanation is not terribly satisfying.  How do you know the relative costs in the first place and then explain the apparently wasted resources at large companies?  Newer theories have tried to do better, but none seem to tell the whole story.  See here for an overview of the literature.

Another unsatisfying aspect of mainstream Theories of the Firm is that they don’t explain the observed interactions of firms with the macroeconomy very well.  For example, firms don’t responsively lower existing salaries when the demand for labor goes down or the supply goes up.  Firms also seem to forego internal price-based mechanisms that would allow them to respond more flexibly to macro shifts in cost and demand.  Even more puzzling, there’s no good explanation of why and when some large firms grow dramatically while others die off. I’ve seen some attempts to address particular questions, but they seem like a patchwork rather than a coherent framework.

The always insightful Arnold Kling refers to a tweet from fellow GMU economist Garett Jones as one possible explanation: “Workers mostly build organizational capital, not final output”.  Unfortunately, this is a tweet, not a theory.  I was sort of playing around with the idea to see if I could get a decent theory out of it and I think I may have something.  It ties together microeconomics, entrepreneurship, search theory, options theory, principal-agent theory, and group dynamics.

The basic idea is: firms don’t produce products; firms produce production functions.  It seems obvious in retrospect, but a cursory search of the literature didn’t turn up anything similar.  Someone has probably thought of this before.  But perhaps I got lucky.  So I’ll run with the ball for now.

What Is a Production Function?

In economics, a production function is an abstract model of how an economic actor turns inputs into outputs.  Basically, it represents the formula or a recipe for a product.  Typically, we write Q=f(X), where Q is a vector of output quantities, f is the production function, and X is a vector of input quantities.  While “real” production functions have very specific outputs and inputs, economists often simplify the world by assuming each actor only produces one output, Goods, and there are a standard set of general inputs such as Labor, Capital, and Land.  If we know the cost curves of Labor, Capital, and Land, we can calculate a cost curve for Goods and examine the tradeoffs and synergies among Labor, Capital, and Land.

Typically, economic theory defines a firm in terms of its production function. However, anyone who has ever founded a startup or worked in a large company should find this definition puzzling.  When I’ve started companies, I could not even clearly define what our inputs and outputs would be, let alone the formula for turning the former into the latter.  When I’ve interacted with big companies, I typically see a lot of people and groups who have nothing to do with turning inputs into outputs.  For example, CTO, CIO, CFO, Corporate Development, Business Development, Product Management, Brand Management, and Market Research. In fact, when I think about the technology industry, the fraction of people actually involved in transforming of inputs to outputs, even if you count the relevant management hierarchy, seems pretty small.

So what is it that entrepreneurs and most of the of the people at large companies are actually doing?  My hypothesis is that they are exploring alternative production functions: trying to figure out ways to improve existing businesses and explore opportunities for completely new businesses.  Intuitively, this seems reasonable given my experience, but I’d never thought of how to formalize the concept.

Now some production functions are conceptually “near” current ones in that they represent small refinements to the manufacturing process or modest enhancement to existing products.  Other production function are conceptually “far” from current ones in that they introduce radical new manufacturing technologies or generate revolutionary new products.  I bet if you think about the people you’ve worked with, most of them are involved in figuring out how to produce things or what things to produce, rather than actually producing things. So they are producing production functions.

Production Function Space

The cool way to approach this kind of search problem is to posit a high-dimension space of all the alternatives with a structure that allows us to define the concepts of “near” and “far”.  In this case we have production function space.

The challenge is that most points in this space are not economically viable.  Some of them define products that nobody wants (e.g., Apple Newtons).  Some of them define products that we can’t make at our current level of technology (e.g., flying cars).  Some of them define products that we could make but whose demand curve never intersects its cost curve (e.g., diamond coated toothpicks).

Moreover, there are a lot of dependencies among points in production function space.  So you can’t consider just one point in isolation; you have to consider configurations of points.  Some Goods in one production function are Capital in another production function (e.g., Intel processors).  In other cases, demand for some Goods exists only if there is also demand for other Goods (e.g., third party iPhone cases and Apple iPhones).

Most importantly, the goal is not just to find economically viable points.  The goal is to find points that generate a lot of profit.  Given that changes in technology and fashion cause these points to constantly shift relative to each other and the high dimensionality of the relationships among points, we have a rather complex optimization problem.  I think of it as the potfolio optimization problem from finance, combined with the n-body problem from physics, combined with the protein folding problem from biology.

Given this level of complexity, we would expect that search strategies almost always use heuristics and trial-and-error rather than purely analytic optimization.

The Implications

In subsequent posts, I plan to analyze this hypothesis from several different angles.  I also hope to come up with some predictions that someone could test.  But I thought it would be useful to throw out some gross speculation right now to show how this insight crystallized my thinking and pique your interest in exploring further:

– The value of a firm is the present value of the profit stream from actual current production functions plus the option value of potential future production functions.

– Large firms tend to explore neighborhoods of production function space relatively near the surfaces defined by their current products.

– Startups tend to explore neighborhoods of production function space relatively distant from the surfaces defined by everyone else’s current products.

– Firms that have released successful new products are more valuable because this success implies greater skill in searching production functions space.

– Firms that have released successful new products are also more valuable because they then have a larger beachhead from which to explore greater regions of production function space in the future.  Luck counts.

– Large firms acquire startups in part to increase their ability to search more distant regions of production function space.

– Due to network effects among colleagues and the uniqueness of each firm’s endowments, a given firm’s ability to search production function space is proportional to the number of employees it has and their length of service.

– It is harder to observe the true contribution of a particular employee to searching production function space than it is to observe the true contributions of an employee to a specific production function.  Therefore, the principal-agent problem is worse than we think.

– Because the ability to search production function space increases with the number of people involved, “empire building” is a rational strategy for an employee to increase both his apparent and his actual value to the firm.