貝爾曼方程（Bellman Equation）也被稱作動態規劃方程（Dynamic Programming Equation），由理查·貝爾曼（Richard Bellman）發現。
貝爾曼方程是動態規劃（Dynamic Programming）這些數學最佳化方法能夠達到最佳化的必要條件。此方程把「決策問題在特定時間怎麼的值」以「來自初始選擇的報酬比從初始選擇衍生的決策問題的值」的形式表示。藉此這個方式把動態最佳化問題變成簡單的子問題，而這些子問題遵守從貝爾曼所提出來的「最佳化還原理」。
Bellman equation
Each producer also faces a Poisson hazard rate of losing any of its products capturing that products in practice get obsolete. A similar interpretation is presented in Luttmer (2011). He assumes that producer of a di¤erentiated commodity needs a blueprint to produce and these blueprints depreciate in a one-hoss-shay fashion. While each firm takes the value of as given,its equilibrium value, would be determined by the aggregate creation in the economy which will
be discussed below. Firms exit if all of their products are destroyed. There is no re-entering once exit occurs.Assuming a constant interest rate, r; the Bellman equation for a producer of e¢ ciency,';andnumber of products, n, is formulated as follows：
This equation shows that current value of firm is equal to the sum of three terms. The first term on the right hand side shows the current product net of R&D costs. The other two terms show the net future value of the firrm. The second one is the gain in value caused by the innovation of a new variety and the last one is the expected loss associated with a loss of a randomly chosen product.
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