This is an interesting question that I would like to answer you.
There are many definitions of the complexity of a system that consider a complex system as characterized by a set of agents that are interacting and interchanging information in an environment with uncertainty. These definitions are not useful for management if they do not let to establish a quantitative measure, because we cannot manage something that we cannot measure. Ontonix has proposed a definition of complexity that follows this scheme based on structure and uncertainty, but this definition lets to measure complexity in a system that is defined using a set of state (or control) variables. In this way, we can measure the complexity of an economy with the same variables that analysts are using to evaluate it.
What is the meaning of structure? Cause-Effect Relationships between two variables. For instance, if the oil price increases, the general price index increases.
What is the meaning of uncertainty? The variability degree existing in this relationship.
Both components applied to all of the agents of the system represent the complexity of a system. Complexity is a property of the system, or, of the economy, and, as an important thing, it can be measured! Then, it can be managed!
Now, I am going to explain you with an actual example, the way that innovation is related to complexity. We can analyze car sales. What happens if the price of the oil increases?
If the transport expenses of the families are the sum of the fuel and the amortization of the current vehicle, the higher fuel price the lower capability to amortize a new vehicle, and to buy a new vehicle will be less feasible. The family economy, related to the car acquisition, of this example has a certain complexity degree that only lets to buy fuel impelled cars. The decision will be easier if oil price is stable, but if oil price has high variability it will be more difficult to dare to buy a vehicle. With stable oil prices, to manage the transport in our family economy is easy. But with very changing oil prices, to manage our family economy becomes more complex.
Now we can innovate. Put electric cars in the market in a country as France with energy mix that has a strong component of nuclear + renewable.
The decision is more complex attending to the buying options, because we need to decide between cars with fuel or electricity engine. However, we have reduced the uncertainty of the expenses because electric cars will have a stable energy price and, on the other hand, is making easy our decision.
Mathematically, innovation has increased the number of relations between the variables (car selling depends on oil and energy prices), increasing the complexity of the economy, but it will decrease the uncertainty of the energy price opposed to the transport expenses of the families when the number of electric cars were sufficient. That is, it will decrease the other complexity component.
And now, what car will you buy? It is reasonable thinking, assuming same performance and price, of all of us would buy electric cars, and fuel cars would disappear, that is, innovation would increase complexity and maximum complexity and, in a second period innovation would decrease complexity.
It can that this scenario is not sole. It could happen that oil prices become stable by the effect of a substitutive product (decreasing complexity in the same way). We think that there is nothing like competence, while others permit it to us.
This point, that can looks like Science-fiction because we are referring to a not very far future, is something that happens every day in the economy in non-innovating investments: When a gas company acquires an electricity company, is thinking about the compensation of its business uncertainty through the vertical integration, although it has not measure the complexity of the business.
Complexity is a thing that nobody has measured and controlled because the advantages of doing it were unknown. Fortunately, nowadays we can find tools that let us to measure and analyze the meaning of complexity in the economy.
I expect that this little reflection can help you to understand the relation among business strategy, innovation and complexity.