Attempt to design neoclassical model with Schumpeterian entrepreneurship to explain Kuznet's curve

 Assumptions – 

1. Economy consists only of entrepreneur-agents.

2. Time is divided into periods; every agent builds one entrepreneurial project per period. 

3. Entrepreneurial profit is the only source of income; and (since all saved profits from one period are re-invested in the next one) all inequality is therefore the difference between the absolute amounts

4. Entrepreneurial projects can either be “genuinely innovative” OR they can compete with others’ projects (I call them “competition”; in order to take a share of their profit. Agents choose between these two options according to whichever is more profitable. 

5. All players are starting with same initial endowments. Let’s assume this endowment is in form of capital. 

6. Any form of capital can be converted into other form without any sunk costs. I am using this fact when referring to a machine named “Machina” which has capability to produce any product but under some constraints.

7. Each player produces only one type of product and if another player enters the market, then both of them produce homogeneous products.

8. This is a complete and symmetric information game that means each player has all information about all other agents and demand of all other products.

9. We are assuming linear demand and marginal cost functions for simplicity of calculation.

10. Free entry and free exit from a market are assumed. 

 

Now let’s start this game. Initially, each player may or may not move to other combination of capital/endowments. But he/she saves a little amount from this to build “Machina” our machine.

“Machina” produces certain quantity of goods in each time period and the cost of producing depends on two factors – 1. Scale factor (s) 2. Technology factor (t)

Total Cost(q)=M(s,t) ….. M stands for “Machina”

These factors are solely assumed to get closer to the reality. Scale factor represents scale of the industry, amount of human and physical capital firm has. Technological factor represents technology available to firm.

So in the first period, each player invests some amount of capital according to his/her savings, hence each “Machina” will be different in scale and technology factor (depends on capital invested in it). Notice here, the scale and technology may or may not be optimal for the corresponding product. (according to long term average cost curve)

Now once, this “Machina” is formed, she starts producing in each time interval. Quantity produced by her will be optimal at given scale and technology.

That means,

Marginal Revenue (MR) = Marginal Cost (MC) 

Now let’s understand this with real life examples. Here, I am explaining the reason behind taking scale and technological factors as parameters.

For a cotton industry, scale factor includes area of land available, labor, amount of raw material. Technology represents whether they are using manual machines or automated machines (assuming automated machines produce more quantity in lesser cost)

For an artist, let’s say Drummer, scale factor means number of parts in his drum set, whether he has 2 or 3 cymbals.  And technological advancement means if he has electronic drum set then with 2 cymbals he will be able to produce sounds of 10 different cymbals. His production quantity will be number of solos produced.  For an employee, scale factor means total amount of time given and technology means his/her skills. And quantity will be his/her number of delivered targets per period.

Now assuming all the markets are imperfect, each agent will produce certain amount of profit each period. Each player will start investing this profit into modifying “Machina” so that he will start moving towards optimal scale (s*). This statement is equivalent firm is expanding/contracting to achieve optimal production (moving from short term to long term, you can refer the graph). And once he/she reaches optimal s*, he/she starts improving technology to reduce cost further at given scale.

In between all these production activities, some agents will find other opportunities which can provide them more benefits than the current ones. Now as this is complete information game, every agent knows marginal cost curve of every other agent’s production and he/she knows the demand curve for that particular product.

Now when an agent makes a transition, scale and technology factor of “Machina” remains same but it may be or may not be optimal for new product market. But due to complete information game, player knows his/her marginal cost curve after transition.

Now let’s do a little math. Assuming entry of another player in Monopoly (for simplification)

Before transition,

Agent A was having monopoly in product P.

Linear demand function= P(q) = a-bq, 

Hence, MR curve = a-2bq

MC curve for “Machina” developed by agent A= c+dq

And using MR=MC, q and later P can be found using these equations.

(Remember a, b, c, d, e, f are known due complete information) 

Now after transition,

Agent A produces q1 products and agent B produces q2 products.

Now, demand function = P(q) = a-b(q₁+q₂)

MC curve for agent A = c+dq₁.

MC curve for agent B = e+fq₂

For agent A, Revenue curve =P×q₁=aq₁-bq₁²-bq₁q₂

Hence, MR curve = a-2bq₁-bq₂

Similarly, for agent B, MR curve = a-2bq₂-bq₁

As, both are profit maximization players, both of them will seek for MR=MC

So we get two equations and two variables. Hence we can find q₁, q₂ and P.

 

Conclusion:

In this way, market share of both of above firms will not be equal, it would be strictly dependent on MC curve which ultimately depends on s and t. As the periods passes, every agent tries to improve s and t or shifts into other market.

Also, it is not always true that less successful players will shift to most successful market but it depends on parameters s and t of their “Machinas” i.e. Cotton produces will not shift directly to IT industry looking at profit as there is big gap of technology which makes it costly to move.

Now let’s talk about agent that has made transition. He/she will first try to achieve optimal quantity of scale and then tries to improve technology (in new market). As his/her profit increases, it is easier for him to move towards optimal s*. Also as bigger market player has already achieved technological advancements, the rate of technological improvement is greater for newly entered player than previous market player. (As we all know finding completely new algorithm is far more difficult than learning to apply existing algorithm)

Hence, in the start, we will find large inequalities in profits as bigger firms grow bigger and growth of smaller ones will be slower. But as more of less successful entrepreneurs starts moving into new markets, growth rate of inequality reduces until it becomes zero when rate of profits for transited entrepreneurs will be greater than well-established ones in same market (point of zero slope). And then it starts reducing.