Entrepreneurship and Occupational Choice Model

[alanlujan91]

Summary

Entrepreneurs are roughly 7-10% of the U.S. population but hold about 40% of top-1% wealth (Cagetti and De Nardi 2006). HARK has no model of entrepreneurship or occupational choice: all existing agent types receive exogenous labor income, so the agent never starts a business or invests in a productive project with idiosyncratic risk and collateral constraints. A HARK EntrepreneurConsumerType would address questions about wealth inequality, credit constraints, small business policy, the "missing middle" in developing economies, and the wealth concentration effects of entrepreneurial risk.

Connection to Financial Services Practice

The private-equity plugin models the core economics of acquiring and operating businesses with leverage. The returns-analysis skill (private-equity/skills/returns-analysis/SKILL.md) computes IRR and MOIC with returns attribution (EBITDA growth + multiple expansion + debt paydown). The lbo-model skill (financial-analysis/skills/lbo-model/SKILL.md) models sources and uses, debt tranches, and cash sweep mechanics. The deal-screening skill (private-equity/skills/deal-screening/SKILL.md) evaluates investment opportunities against criteria including revenue, margins, growth, and valuation multiples. The unit-economics skill (private-equity/skills/unit-economics/SKILL.md) computes LTV/CAC, ARR cohorts, and SaaS metrics. A HARK entrepreneurship model would answer why agents become entrepreneurs, how much capital they invest, and how collateral constraints distort business formation, providing the structural counterpart to these practitioner valuations.

Mathematical Model

Environment

Time is discrete and infinite (or lifecycle, $t = 0, \ldots, T$). Each period, the agent chooses an occupation: worker or entrepreneur. Workers receive stochastic labor income. Entrepreneurs operate a production technology with idiosyncratic productivity shocks, invest capital subject to a collateral constraint, and bear the risk of business failure.

State Variables

The individual state is $(a_t, z_t^w, z_t^e, \iota_t)$, where we denote by $a_t \geq 0$ wealth (which serves as both saving and potential business collateral), by $z_t^w$ worker productivity (driving wages), by $z_t^e$ entrepreneurial productivity (driving business returns), and by $\iota_t \in {W, E}$ current occupation.

The productivity processes are independent AR(1) in logs:

$$\log z_{t+1}^w = \varrho_w \log z_t^w + \epsilon_{t+1}^w, \quad \epsilon_{t+1}^w \sim \mathcal{N}(0, \sigma_w^2)$$

$$\log z_{t+1}^e = \varrho_e \log z_t^e + \epsilon_{t+1}^e, \quad \epsilon_{t+1}^e \sim \mathcal{N}(0, \sigma_e^2)$$

The persistence $\varrho_e$ and variance $\sigma_e^2$ of entrepreneurial productivity are key parameters governing the risk-return tradeoff of entrepreneurship.

Preferences

$$\mathbb{E}_0 \sum_{t=0}^{\infty} \beta^t u(c_t), \quad u(c) = \frac{c^{1-\rho}}{1-\rho}$$

For the lifecycle variant ($t = 0, \ldots, T$), the terminal condition is $V_{T+1}(a, z^w, z^e) = \Phi(a)$ where $\Phi(\cdot)$ is a warm-glow bequest function. For the infinite-horizon variant, we seek a stationary fixed point.

Technology

An entrepreneur with productivity $z^e$ who invests capital $k$ and hires labor $n$ produces output:

$$y = z^e, k^\nu, n^\omega$$

where the capital share is $\nu$, the labor share is $\omega$, and we assume decreasing returns to scale at the firm level: $\nu + \omega < 1$ (otherwise the firm would want infinite scale). The entrepreneur hires labor at the competitive wage $w$ and retains the profit. (We use $w$ for the competitive wage; this should not be confused with total wealth $w_t$ used in some portfolio choice models.)

$$\pi(z^e, k) = \max_{n \geq 0} ; z^e, k^\nu, n^\omega - wn = (1 - \omega), (z^e)^{1/(1-\omega)}, \left(\frac{\omega}{w}\right)^{\omega/(1-\omega)}, k^{\nu/(1-\omega)}$$

We can write this more compactly as $\pi(z^e, k) = \tilde{z}^e, k^{\tilde{\nu}}$, where $\tilde{z}^e = (1-\omega)(z^e)^{1/(1-\omega)}(\omega/w)^{\omega/(1-\omega)}$ and $\tilde{\nu} = \nu/(1-\omega) < 1$ captures the effective diminishing returns to capital after optimizing out labor.

Collateral Constraint

The entrepreneur can invest capital $k$ up to a multiple $\lambda$ of their own wealth:

$$k \leq \lambda, a_t$$

Capital is rented from a competitive capital market at rate $r + \delta$; the entrepreneur holds no physical capital on their balance sheet. Wealth $a$ serves as liquid financial savings and as collateral to secure the capital rental agreement. The constraint $k \leq \lambda a$ limits the scale of the enterprise: the entrepreneur can operate capital up to $\lambda$ times their own collateral. When $\lambda = 1$, only self-financed investment is possible; when $\lambda > 1$, the entrepreneur effectively borrows $(k - a)$ at the risk-free rate, secured by their wealth.

The collateral constraint is the central friction: productive entrepreneurs with low wealth are constrained below their optimal scale, while wealthy but less productive agents may over-invest relative to the frictionless optimum.

Recursive Formulation

Occupational choice:

$$V(a, z^w, z^e) = \max\bigl{V^W(a, z^w, z^e),; V^E(a, z^w, z^e)\bigr}$$

Worker value:

$$V^W(a, z^w, z^e) = \max_{c,, a'} ; u(c) + \beta, \mathbb{E}\bigl[V(a', z^{w\prime}, z^{e\prime})\bigr]$$

subject to

$$c + a' = (1+r),a + w,z^w$$

$$a' \geq 0$$

The worker enters the period with wealth $a$, earns interest $ra$ and wage $w z^w$, then chooses consumption $c$ and next-period wealth $a'$. The worker can switch to entrepreneurship next period at no cost (or with an entry cost, see below).

Entrepreneur value:

$$V^E(a, z^w, z^e) = \max_{c,, k,, a'} ; u(c) + \beta, \mathbb{E}\bigl[V(a', z^{w\prime}, z^{e\prime})\bigr]$$

subject to

$$c + a' = (1+r),a + \pi(z^e, k) - (r + \delta),k$$

$$k \leq \lambda, a$$

$$a' \geq 0$$

The entrepreneur enters the period with wealth $a$, earns interest $ra$, earns profit $\pi(z^e, k)$, and pays the rental cost of capital $(r + \delta)k$ (where $\delta$ is the depreciation rate).

Entry and Exit Costs

To prevent unrealistic occupational churning, we can add switching costs:

• Entry cost $\phi^E > 0$: An agent switching from worker to entrepreneur pays a one-time cost (capturing business setup, licensing, opportunity cost of learning).
• Exit cost $\phi^W \geq 0$: An entrepreneur reverting to worker may face costs (business dissolution, severance).

With switching costs, the value function tracks current occupation:

$$V(a, z^w, z^e, W) = \max\bigl{V^W(a, z^w, z^e),; V^E(a - \phi^E, z^w, z^e)\bigr}$$ $$V(a, z^w, z^e, E) = \max\bigl{V^W(a - \phi^W, z^w, z^e),; V^E(a, z^w, z^e)\bigr}$$

where the entry cost $\phi^E$ is denominated in wealth units (representing business setup costs, licensing fees, and opportunity costs), reducing the agent's available wealth upon entering entrepreneurship.

General Equilibrium

In general equilibrium, the wage $w$ and interest rate $r$ clear labor and capital markets. The aggregate capital demand by entrepreneurs is $K^d = \int k^*(a, z^e), d\mu(a, z^w, z^e, E)$ where $\mu$ is the stationary distribution over individual states. The aggregate labor supply by workers is $L^s = \int z^w, d\mu(a, z^w, z^e, W)$. With a Cobb-Douglas aggregate production function, equilibrium pins down $w$ and $r$. Note that in this model the "firms" are the entrepreneurs, not an aggregate representative firm.

Key Equilibrium Properties

• Wealth inequality: The model generates fat-tailed wealth distributions because entrepreneurs bear concentrated idiosyncratic risk and accumulate wealth rapidly when productive. This is a key mechanism in Cagetti and De Nardi (2006) for matching the observed right tail of the U.S. wealth distribution.
• Misallocation: Collateral constraints ($\lambda < \infty$) cause capital misallocation: high-$z^e$ agents with low wealth under-invest, while low-$z^e$ agents with high wealth over-invest. The aggregate TFP loss from misallocation is a key object of interest.
• Credit policy: Relaxing the collateral constraint (increasing $\lambda$) improves allocation but may also reduce precautionary saving, with ambiguous welfare effects.

Key References

• Cagetti, M. and De Nardi, M. (2006). "Entrepreneurship, Frictions, and Wealth." Journal of Political Economy, 114(5), pp. 835-870.
• Quadrini, V. (2000). "Entrepreneurship, Saving, and Social Mobility." Review of Economic Dynamics, 3(1), pp. 1-40.
• Buera, F. J. and Shin, Y. (2013). "Financial Frictions and the Persistence of History: A Quantitative Exploration." Journal of Political Economy, 121(2), pp. 221-272.
• Buera, F. J., Kaboski, J. P., and Shin, Y. (2011). "Finance and Development: A Tale of Two Sectors." American Economic Review, 101(5), pp. 1964-2002.
• Moll, B. (2014). "Productivity Losses from Financial Frictions: Can Self-Financing Undo Capital Misallocation?" American Economic Review, 104(10), pp. 3186-3221.
• Hurst, E. and Lusardi, A. (2004). "Liquidity Constraints, Household Wealth, and Entrepreneurship." Journal of Political Economy, 112(2), pp. 319-347.

Implementation Notes

• Occupational choice is discrete (compare $V^W$ and $V^E$ at each state), handled via HARK's upper envelope / DC-EGM approach or the DBlock Control variable mechanism.
• Capital choice $k$ is a continuous optimization nested within value function iteration. Given $\pi(z^e, k) = \tilde{z}^e k^{\tilde{\nu}}$, the unconstrained optimum has a closed form, and the constrained solution is $k^* = \min(k^{\text{unconstrained}}, \lambda a)$.
• With 3 continuous state variables $(a, z^w, z^e)$ plus discrete occupation, the problem is manageable with HARK's existing grid methods (4D with occupation treated as a Markov state).
• For general equilibrium, HARK's Market class can close the model by iterating on $(w, r)$ until factor markets clear, similar to CobbDouglasEconomy.
• Natural calibration targets: fraction of entrepreneurs (~7-10% in the U.S.), their wealth share (~40% of top 1%), and the return distribution of small businesses.
• Connects to AggShockConsumerType for business-cycle analysis (recessions tighten collateral constraints, reducing business formation).
• Because the collateral constraint takes the borrowing rate as exogenous, the strategic default model (Issue Strategic Default and Bankruptcy Model #1733) provides a natural complement by endogenizing credit pricing. Parental wealth transfers (Issue Inter-Vivos Transfers and Dynastic Bequest Model #1737) can relax the collateral constraint (Hurst and Lusardi 2004).