Wealth Inequality and the Exploration of Novel Technologies

A.1 A General Model

This section presents a comprehensive model where I add all the extensions considered in Section 5 to the baseline model of Section 3.

First, I add an outside option to entrepreneurship. I assume that at the beginning of the first period, agents can choose to become either investors or entrepreneurs. Investors deposit their wealth in a bank for a riskless return; conversely, an entrepreneur undertakes a risky investment in the form of starting a firm and can then choose to explore or exploit (as in the baseline model). Second, relatively to the baseline model, I assume that talented agents are not only better at weeding out inferior actions but also at discovery more productive actions. I assume that a proportion λ of agents is talented, which means that they can exert unobservable effort at a fixed and indivisible monetary cost e > 0 to increase both their unconditional probability of success using the novel approach to the level EpN,H, which is assumed to be in (EpN,L,pC), and their conditional probability of success to EpN,H|S,N, assumed to be in (EpN|S,N,1). For the remaining proportion 1 − λ (the untalented agents), the effort cost is prohibitively high. Third, I assume that banks can observe the wealth of a potential entrepreneur and the outcome of her firm, but effort and project selection are unobservable. Fourth, I consider the general equilibrium case, where the rate of return adjust to clear the credit market. Since the analytical results are more notation-intensive than the in baseline model, for ease of reading, I shorten the notation as follows: p _C = p, EpN,H=qH, EpN,L=qL, EpN,H|S,N=qHS, and EpN|S,N=qHL.

The assumption of efficiency is that, absent the need for financing, the net presented value of a project run by a talented explorer is strictly greater than the net present value from exploitation and investment, both of which are strictly greater than the net present value from untalented exploration, i.e. Y _H − e > max{Y _C , RI} > min{Y _C , RI} > Y _L . This implies that the equilibrium risk-free interest rate must be such that

I do not take a stand on whether exploitation is preferred to investing in first best (this will depend on the equilibrium interest rate).

As in the baseline model, only three types of borrowers need to be considered: talented explorers (H), untalented explorers (L), and exploiters (C). I assume that agents choose to become investors if indifferent between becoming investors or entrepreneurs, and exploiters if indifferent between exploitation and exploration. The tie-breaking assumptions rule out mixed strategy equilibria. Thus, for a given wealth level and menu of contract, all talented agents make the same decision as each other, and all untalented agents make the same decision as each other. As a consequence, for any given wealth class, I only need to consider pair-wise combinations of exploration (with or without effort), exploitation, and investing. Equilibrium contracts can be of two types: pooling equilibrium, in which both types receive an identical contract, and separating equilibrium, in which contracts induce self-revelation of their unobservable ability.

Similarly to the baseline model, it cannot exist a zero-profit separating menu of contract that induces the two types to both enter exploration, since this cannot be incentive compatible. Hence, an equilibrium contract must be either a pooling contract or a separating contract that only one type accepts. There are three possible pooling contracts: one pooling effort-exerting explorers with untalented explorers σHL⋆, one pooling effort-exerting explorers with untalented exploiters σHC⋆, and one pooling shirking explorers with untalented explorers σL⋆.^[10] Each of these contracts must lie on the corresponding zero-profit condition, otherwise banks could undercut each other. These three putative contracts are then given by τHL(A,R)=R(I−A)/q̄HL, τHC(A,R)=R(I−A)/q̄HC, and τ _L (A, R) = R(I − A)/q _L , where q̄HL=λqH+(1−λ)qL and q̄HC=λqH+(1−λ)p are the corresponding Bayesian probability of success of a random applicant.

I start by considering the putative contract pooling effort-exerting talented and untalented explorers. Let V(i, j, τ) be the expected payoff of an agent of realized talent i = {H, L} (i.e. talent after effort choice), choosing the project j = {N, C}, and under a repayment τ. This contract can be accepted by the untalented agent if the participation constraint, VL,N,τHL>RA, and the incentive compatibility constraint, VL,C,τHL<VL,N,τHL, are both satisfied. Solving these for A leads to

Likewise, consider the putative contract with both effort-exerting talented explorer and untalented exploiters. This contract can be offered to the untalented agent if the participation constraint, VL,C,τHC>RA, and incentive compatibility constraint (versus exploration), VL,C,τHC<VL,N,τHC, are satisfied. Equivalently,

Imagine the bank offering both σHL⋆ and σHC⋆. A low ability agent will prefer the former to the latter if

The wealth levels ϕ _HL , ϕ _HC , ϕ _NC , ϕHCIC, and ϕHLIC naturally divide the (R, A) space into twelve areas, as shown in Panel (a) of Figure A.1. In the areas (1)–(4), the low ability agent prefers investing to both pooling contracts, and thus a pooling contract is not offered in equilibrium. In the areas (5) and (6), the low ability agent prefers σHL⋆ to both investing and σHC⋆. Since σHL⋆ is also incentive compatible, it can be offered. In area (7), L would like to be offered σHC⋆ but this is not incentive compatible; however, σHL⋆ is both incentive compatible and preferred to investing, and thus it can be offered. In areas (8)–(11), L prefers σHC⋆ to both σHL⋆ and investing, and it is incentive compatible. Finally, in area (12), L prefers σHC⋆, which is not incentive compatible; however, σHL⋆ cannot be offered because L would prefer to become an investor (the equilibrium contract is derived below).

Figure A.1:

Contracts offered.

The relevant conditions are summarized in Panel (b) of Figure A.1. In the area labelled “investing”, low ability agents prefer investing to accepting a pooling contract. In the blue area labelled “pooling (HC)”, the pooling contract that can be offered by banks is σHC⋆; in the green area labelled “pooling (HL)”, the equilibrium pooling contract is σHL⋆. In the remaining white area, a pooling contract cannot be offered, because σHC⋆ is not incentive compatible, whereas σHL⋆ does not satisfy the participation constraint.

When untalented agents prefer investing to any pooling contract, talented agents can be offered a separating contract. Given the assumption on efficiency, the only separating contract that I need to consider is the one inducing talented agents to become effort-exerting explorers. The first-best zero-profit exploration contract is τ _H (A, R) = R(I − A)/q _H . An untalented agent with equal wealth A does not accept τ _H if she prefers investing to exploring using this contract, VL,N,τH≤RA, and to exploiting using this same contract, VL,C,τH≤RA. Solving these two conditions for A results in

In Figure A.2, I update Figure A.1(b) with the new threshold ϕ _I (R).^[11] This shows that for certain wealth and risk-free interest rate combinations, a separating exploration contract with effort is incentive compatible, as shown in the area labelled “separating”. I have also shown that there is an area where the only contract that can be offered is a pooling contract. In the area labelled “pooling (HC)”, the equilibrium pooling contract is the one pooling effort-exerting explorers with exploiters; in the area labelled “pooling (HL)”, the equilibrium pooling contract is the exploration one.

Figure A.2:

Separating and pooling contracts.

I still need to determine which contract is offered in the area between ϕ _HL , ϕ _I , and ϕHCIC. The separating exploration contract requiring effort is not incentive compatible, as it is also accepted by low ability agents, but a pooling contract cannot be offered. To graphically show the equilibrium for these wealth class, it will prove useful to consider also possible repayments in case of failure τ^F; note that these be negative, given the limited liability assumption. This is represented in Panel (a) of Figure A.3, where R and A have been chosen so that in the first panel we are in the white area on the left of ϕ _HC , while on the second panel we are in the white area on the right of ϕ _HC from of Figure A.2.^[12] The pooling contract cannot be offered either because low ability agent prefers investing to both pooling contracts (second panel), or because the pooling exploration contract does not satisfy the participation constraint while the exploitation and exploration pooling contract is not incentive compatible (first panel).^[13] In both cases, however, the banks can offer a separating pair of contracts, like yL,yH. Effort-exerting high ability agents strictly prefer y _H ,^[14] while low ability agents are indifferent among y _L , y _H , and investing: thus, they choose to become investors by assumption. This is a Bertrand-Nash equilibrium where banks make positive profits, since no profitable deviations are available.^[15]

Figure A.3:

Contracts offered.

What are the terms of the profitable separating menu of contract? From Panel (a) of Figure A.3, the contract offered to the low ability agent (when neither the zero-profit separating contract nor a pooling contract can be offered) is given by the intersection of V(L, N, τ ) = RA and the zero-profit condition from a pooling exploration contract. Solving the corresponding system of equations,

reveals that the contract offered to the low ability agent, y _L , is

The terms of the contract offered to the high ability agents, y _H , are derived by the intersection of V(L, N, τ ) = RA with the vertical axis, τ^F = 0, i.e.

The high ability agent’s expected utility associated with this contract is

The assumption on efficiency ensures that high ability agents prefer this contract to investing. Setting (A.16) equal to YqH+qHqHS+p−qHp−R̂qHI−A and solving for R̂ reveals that the interest rate on this contract is

This completes the set of contracts offered by the banks, as summarised in Panel (b) of Figure A.3.

I now consider the agent’s decision given that banks offer one of the pooling contracts above. An untalented individual with wealth A has three options: she can become an investor, she can become an exploiter, or she can become an explorer. Panel (b) of Figure A.1 above gave us the preferred option for each R and A combination.

Likewise, a talented individual with wealth A has four options: (a) she can become an investor, (b) she can become an exploiter, (c) she can become a shirking explorer, or (d) she can become an explorer who exerts effort. By the assumption on efficiency, effortful exploration is always preferred to exploitation, and pooling with low ability exploiters is always preferred to investing.^[16] A shirking high ability agent is no different from a low ability agent, and thus the analysis is the same as above. In addition to those conditions, effortful exploration is preferred to shirking if

and to investing if

Panel (a) of Figure A.4 summarises all participation and incentive compatibility constraints in the (R, A) space. This allows us to complement the analysis regarding the banks’ problem by adding the decisions of the agents.

Figure A.4:

Occupational choices. INV stands for investing, C for setting up a conventional firm, N for untalented exploration, and NE for effortful exploration.

In the areas (1)–(9), banks offer the zero-profit separating exploration contract to high ability agents, high ability agents accept it and exert effort, whereas low ability agents prefer investing. In the areas (10)–(16), the profitable separating exploration contract is offered, high ability agents accept it and exert effort, whereas low ability agents become investors. In the areas (17)–(18), only the pooling exploration contract can be offered, but this does not satisfy the participation constraint of the high ability agents, who thus decide to become investors. Low ability agents do not explore because they would be identified as low ability, and thus they prefer to invest. In the areas (19)–(21), the pooling exploration contract is offered, but high ability agents do not exert effort. Therefore, they are no different from low ability agents and bank, to break even, must ask for the interest rate consistent with the low ability applicants. Thus, all agents prefer to invest. In the areas (22)–(25), the pooling exploration contract is offered, high ability agents exert effort and low ability agents become explorers. In the areas (26)–(34), the exploitation and exploration pooling contract is offered, high ability agents exert effort and low ability agents become exploiters.

These insights are summarised in Panel (b) of Figure A.4. In the yellow area, the zero-profit separating exploration contract is offered by the banks, high ability agents accept it and exert effort, whereas low ability agents become investors. Similarly, in the red area, low ability agents become investors, and high ability agents become effort-exerting explorers, but banks make positive profits. In the grey area, only the pooling exploration contract can be offered, but high ability agents do not become explorers either because they do not want to (light-grey area), or because they are unwilling to provide effort (dark-grey area): everyone would be treated as a low ability agent by the banks, and thus all agents prefer to become investors given the efficiency assumption. In the green area, the pooling exploration contract is offered, high ability agents provide effort and low ability agents become explorers. Finally, in the blue area, low ability agents become exploiters and high ability agents become effort-exerting explorers, thanks to the pooling exploration and exploitation contract.

Figure A.5 shows the possible general equilibria that could ensue. Up to the risk-free interest rate R₁ (defined by ϕHL=ϕHCIC), there are only three wealth classes: a lower-class where everyone become an investor, a (lower) middle-class where high ability agents exert effort and cross-subsidise low ability explorers, and an upper-class where low ability agents become exploiters who cross-subsidise the high ability effort-exerting explorers. Since ϕHLH is strictly increasing in R, the number of entrepreneurs decreases as the risk-free interest rate increases in [R, R₁].

Figure A.5:

General equilibrium.

Between R₁ and R₂ (defined by ϕI=ϕHCIC), we have four wealth classes, since (part of) the upper middle-class shows up, where low ability agents become investors and banks make positive profits on the effortful exploration contract accepted by the high ability agents. The number of entrepreneurs is strictly decreasing in the interest rate both because ϕHLH is strictly increasing in R and because the richest low ability agents (that were previously lower middle-class and are now middle middle-class) switch from entrepreneurship to investing.

Consider a rental rate R₂ and R₃ (defined by ϕHC=ϕHCIC). The number of entrepreneurs is still strictly decreasing in R since ϕHLH is strictly increasing in R (and thus agents move from the lower middle-class to the lower-class where they become investors), ϕ _HL is strictly decreasing in R (more and more agents move from the lower middle-class to the middle middle-class where low ability agents become investors), and ϕHCIC is strictly increasing in R (the upper-class shrinks in favour of the newly formed upper middle-class, and thus the poorest low ability agents of the upper-class move from non-innovative entrepreneurship to investing). For rates between R₃ and R₄ (defined by ϕHL=ϕHLH), the case is similar to the previous one, with the difference that ϕ _HC is steeper than ϕHCIC and thus the number of entrepreneurs decreases even faster as R increases.

For rates between R₄ and R₅ (defined by ϕ _HC = I), the lower middle-class disappears. Since ϕ _HL is strictly decreasing in r, the richest members of the lower-class start entering the (upper) middle-class, and thus the number of entrepreneurs increases. However, since ϕ _HC is strictly increasing in r, the poorest members of the upper-class enter the upper middle-class, pushing the number of entrepreneurs down. Which of these two effects dominates depends on the wealth distribution.

Finally, for rates above R₅, the upper-class disappears. The number of entrepreneurs is strictly increasing in R as the lower-class shrinks.

To summarise, the number of entrepreneurs is strictly decreasing in R up to R₄, it is potentially non-monotonic between R₄ and R₅, and strictly increasing above R₅. Thus, depending on the wealth distribution (and parameters), we could have multiple equilibria. If we restrict our attention to rates below R₄, the equilibrium, if it exists, is unique.

Numerical simulations for this extended model are available on request.

A.2 Wealth Transition Equations

Agents in the first generation are endowed with warm-glow Cobb–Douglas utility functions, which imply that they optimally bequeath a fraction δ of their end-of-life wealth. In turn, this will result in the distribution of endowment G(A) of the second generation. Moreover, agents are risk-neutral, which means that they want to maximize end-of-life wealth. As a consequence, the model for the first generation is qualitatively the same as the one in the baseline model, which means that the static equilibrium is the same.

Let i ∈ [0, 1] indicate a lineage of agents; moreover, I add the subscript f to indicate the corresponding variable or parameter for the first-period generation, when needed. Given the occupational choices of the first generation and the assumption that types are intergenerationally uncorrelated, the wealth transition equations for any lineage i are thus given by

where σ9f and σ10f are the profitable separating exploration and negative-profit exploitation contracts offered to agents with wealth between ALf,AHHf and represented in Panel (b) of Figure 2.