Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • icon

      Proving the marginal value theorem

      James V. Stone
      17 March, 2022

      The marginal value theorem holds true under three fairly mild conditions:

      1. The fixed cost T is larger than zero.
      2. The reward function E(t) increases with t.
      3. The slope dE(t)/dt of the reward function decreases with t (i.e. E(t) is a diminishing returns function).

      We wish to prove that the instantaneous reward rate r(t) equals the average reward rate R(t) when R(t) is maximal. To achieve this, we need to find the value of r(t) when R(t) is maximal. To find the maximal average reward rate, we make use of the fact that its slope is zero at a maximum. The average reward rate is defined as R(t)=E(t)T+t, and its derivative is dRdt=1T+tdEdt+E(t)d(T+t)−1dt, where (by definition) dEdt=r(t), is the instantaneous reward rate, and where d((T+t)−1)dt=−1(T+t)2. Substituting Equations ??? and ??? into Equation ???, dRdt=r(t)1T+t−E(t)(T+t)2, where E(t)T+t=R(t), is the average reward rate, so that Equation ??? becomes dRdt=r(t)T+t−R(t)T+t. At a maximum, this is equal to zero, r(t)T+t−R(t)T+t=0. Finally, multiplying both sides by (T+t), and re-arranging yields r(t)=R(t). This proves that the average reward rate is maximal when the instantaneous reward rate equals the average reward rate.

      Back to the main article

      • Log in or register to post comments
      University of Cambridge logo

      Plus Magazine is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms