Optimal Consumption, Portfolio, and Retirement Under Implementation Delay

We develop a continuous-time model of optimal consumption, portfolio allocation, and early retirement that, to our knowledge, is the first to incorporate an implementation delay —a fixed lag <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula> between the retirement decision and the actual cessation of labor and income. Using a dual-martingale approach, we obtain closed-form solutions and quantify how <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula> affects optimal behavior. For example, when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula> increases from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.5</mn></mrow></semantics></math></inline-formula> to 2 years (baseline parameters: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mn>0.04</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>=</mo><mn>0.02</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>=</mo><mn>0.08</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>=</mo><mn>0.2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mi>B</mi></msub><mo>=</mo><mn>0.3</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>), optimal pre-retirement consumption rises by approximately 7%, the risky asset share falls by about 5 percentage points, the expected retirement time increases by over 1 year, and the retirement wealth threshold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>x</mi><mi>R</mi></msub></semantics></math></inline-formula> grows by roughly 10%. These results provide policy-relevant insights for retirement systems where procedural lags can distort incentives and reduce welfare.