The Optimal Health-Investment Share: A Calibrated Planner’s Problem for Chronic Disease Burden

Technical Summary: The Optimal Health-Investment Share

Problem Statement
This paper addresses a gap in the economic literature regarding the optimal allocation of national income to health. While extensive theoretical work exists on managing acute, time-limited epidemics (e.g., SIR/SIRS models for lockdowns and vaccination), there is no calibrated structural benchmark for the long-run, steady-state health-investment share required to manage chronic, non-eradicable disease burdens (e.g., cardiovascular disease, diabetes). The central question is: what share of GDP should a social planner permanently allocate to health investment to maximize welfare, given that chronic burdens do not follow epidemic curves but rather a constant-incidence accrual process?

Methodology
The author develops a continuous-time optimal-control model featuring a benevolent social planner. The framework integrates the following elements:

• State Dynamics: The disease burden stock, $B(t)$, follows a constant-incidence law of motion. New burden accrues at a constant rate $\pi$ (representing background factors like aging and lifestyle), while existing burden is reduced by health investment $I_h(t)$ via an isoelastic, diminishing-returns technology: $\delta(I_h) = \delta_0 + \delta_1 I_h^\alpha$. This contrasts with density-dependent transmission terms used in epidemic models.
• Economic Mechanism: Output is linearly reduced by the burden stock ($Y = y(1 - \phi B)$). The planner maximizes the discounted stream of log consumption net of a linear disutility of burden.
• Solution Approach: The infinite-horizon problem is solved using Pontryagin's Maximum Principle. The solution is cross-verified against a Hamilton-Jacobi-Bellman (HJB) dynamic-programming formulation to ensure correctness.
• Stability Analysis: The paper derives a closed-form sufficient condition for saddle-path stability using a Jacobian analysis of the linearized system, confirming the existence of a unique stable manifold.
• Calibration: The model is calibrated separately for the four World Bank income groups (Low, Lower-Middle, Upper-Middle, High). The calibration strategy involves "inverting" the steady-state system in closed form. Using observed data for health spending shares (WHO), burden proxies (WHO/World Bank non-communicable disease mortality), and output (World Bank), the author recovers the unobservable structural parameters, specifically the health-system-productivity parameter ($\delta_1$) and the social disutility of burden ($\chi$).

Key Results

1. Structural Determinants of Spending: The calibration reveals that the health-system-productivity parameter ($\delta_1$), which measures how effectively spending converts into burden reduction, rises monotonically with income (from 0.0064 in Low-income to 0.0114 in High-income countries). This single parameter, alongside epidemiological incidence rates, accounts for the entire cross-country gradient in optimal spending and burden levels.
2. Uniqueness and Stability: Unlike models of acute epidemics that may generate multiple equilibria (poverty traps), this model yields a unique, saddle-path-stable steady state for every income group. The cross-country inequality in disease burden is therefore a result of structural differences in technology and epidemiology, not a coordination failure or a failure to coordinate on a better equilibrium.
3. Welfare Counterfactuals: A policy simulation demonstrates that mandating a Low-income country to adopt the High-income spending share (8.5% of GDP) without improving its underlying health-system productivity ($\delta_1$) yields negligible reductions in disease burden (from 24.4% to 24.2%) and results in a net welfare loss (equivalent to a 0.14% permanent reduction in consumption). Conversely, pairing the spending increase with the High-income productivity level yields a substantial welfare gain (12.6% consumption equivalent).
4. Independence from Income Level: The optimal spending share is analytically independent of potential output ($y$). Differences in spending shares across countries are driven by the efficiency of the health system ($\delta_1$) and the incidence rate ($\pi$), not by the inability of poorer countries to "afford" spending.

Significance and Claims
The paper claims to provide the first calibrated, cross-country structural benchmark for the permanent health-investment share required for chronic disease management. Its primary contribution is demonstrating that the observed global gradient in health spending and disease burden is structural rather than coordinational.

The author argues that the literature's focus on acute epidemics has obscured the steady-state reality of chronic disease. By adapting the optimal-control framework to constant-incidence burdens and calibrating it to real-world data, the paper shows that low-income countries are not necessarily "underspending" relative to their own constraints; rather, their optimal spending level is low because their health-system technology is less productive. Consequently, the paper concludes that policy targets focusing solely on increasing spending shares are ineffective and potentially welfare-reducing unless accompanied by investments that improve the underlying productivity of the health system. The findings suggest that development efforts should prioritize building health-system capacity (raising $\delta_1$) rather than simply mandating higher expenditure targets.