International Journal of Applied Mathematics and Numerical Research  |  ISSN (Online): 3107-7110  |  Double-Blind Peer Review  |  Open Access  |  CC BY 4.0

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     2026:2/3

International Journal of Applied Mathematics and Numerical Research

ISSN: (Print) | 3107-7110 (Online) | Open Access

Optimal Investment and Consumption under Inflation and Labour Income Uncertainty

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Abstract

This paper develops a finite-horizon stochastic-control model for the joint determination of household consumption, saving and portfolio allocation when both the price level and labour income are uncertain. The formulation begins in nominal units and uses Itô’s formula to obtain internally consistent real returns and real earnings. Consequently, inflation affects the household not only through a reduction in purchasing power, but also through asset inflation covariances and the covariance between nominal wage growth and the price level. The household invests in a real locally riskless account and multiple risky assets, receives stochastic labour income, consumes in real units and values a terminal bequest under constant relative risk aversion preferences. A dynamic programming principle yields a Hamilton Jacobi Bellman equation in financial wealth, real income and time. In a dynamically complete benchmark, labour income is capitalised as risk adjusted human wealth. The problem then reduces to a Merton problem in total wealth, producing closed-form consumption and portfolio rules. The optimal financial portfolio contains a speculative demand and a human capital hedge; its sign and magnitude depend on the correlation of income innovations with traded returns. For incomplete markets, borrowing limits and portfolio constraints, the paper gives a reduced nonlinear HJB equation, boundary conditions and a monotone policy-iteration scheme. Analytic comparative statics show how risk aversion, inflation exposure, income volatility, retirement horizon and market completeness alter saving and risky investment. A transparent illustrative calibration is used to verify the closed-form formulas and to specify reproducible numerical experiments without presenting synthetic calculations as observed household evidence. The framework provides a tractable bridge between financial mathematics, actuarial life-cycle analysis and household policy design.

How to Cite This Article

PN Nangolo (2026). Optimal Investment and Consumption under Inflation and Labour Income Uncertainty . International Journal of Applied Mathematics and Numerical Research (IJAMNR), 2(4), 36-49. DOI: https://doi.org/10.54660/IJAMNR.2026.2.4.36-49

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