• Most economic models involve an agent or set of agents trying to maximize something.
  • Individuals maximize their utility.
  • Firms maximize their profits.
  • Governments maximize welfare.
• These maximization problems are subject to some constraints.
  • Budget constraints.
  • Incentive compatibility constraints.
• Whenever you build a model, it is important to know who your agents are and what constraints they face.

• Static: Choices today do not affect future pay-offs.
  • What type of cereal brand to buy.
  • Labor-leisure choice.
• Dynamic: Choices today do affect future pay-offs.
  • Addiction.
  • Human capital accumulation such as college.
  • Investment choice of a firm.
• Static models are simpler. Start simple.
• Dynamic programming will help solve dynamic models.

• State and control variables can be either:
• Discrete: Finite number of choices:
  • What industry to work in?
  • Whether to go to college?
  • Whether to adopt new technology?
• Continuous: Infinite number of choices.
  • How much to save or invest?
  • What price to set?
  • How many hours to work?
• Not clear which one is simpler, but start simple.

• Infinite Time: Agents possibly live forever.
  • Typically used for firms and government.
  • Ok when agents don’t look far in advance.
  • Solve with value function iteration or something else.
• Finite Time: Agents’ decisions end in some terminal period.
  • Necessary to study how things evolve over the life-cycle.
  • Retirement, human capital accumulation, etc.
  • Solve via backward induction from final period.

• What is optimal human capital accumulation over the life-cycle?
• Time invested in human capital accumulation:
  • Increase our wages.
  • But decreases time spent working.
    
• Model begins at age 20. Workers retire at age 70.
• Agent has a learning ability \(a\) and initial human capital \(h_0\).
• Each period decide how much time to spend learning and working.

• We could start at age 20 and consider all possible HC paths.
  • Takes forever.
  • Even if HC only takes two values, there are \(2^{50} > 10^{15}\) possible paths.
• Better way: Use dynamic programming.
  • Start at age 70. You won’t invest in human capital.
  • Then we can solve at age 69.
  • Then go to age 68, …, finally age 20.

• Agent with ability \(a\) at age \(t\) with human capital \(h\) chooses investment \(I\) in human capital to maximize: \[ V(a, h, t) = \max_I u(c) + \beta V(a, h', t+1) \]
• subject to: \[ c = h (1 - I) \]

\[ h' = (a(hI)^\kappa + h)\epsilon \] \[ \log\epsilon \sim N(\mu,\sigma^2) \] - Terminal condition: \(V(-,-,71) = 0\).

\[ V(a, h, 70) = \max_I u(c) + \beta V(a, h', 71) \] - subject to: \[ c = h (1 - I) \] - Plug in terminal condition: \[ V(a, h, t) = \max_I u(c) \]

• We can solve this! \(I = 0\).

• Now we know \(V(a,h,70)\) for all values of \(a\) and \(h\) on a grid.
• We can interpolate \(h\) and solve: \[ V(a, h, 69) = \max_I u(c) + \beta V(a, h', 70) \]
• Once we solve this, we have \(V(a,h,69)\). Can solve at age 68.
• Then solve at age 67. … Then solve at age 20.
• Let’s code this up.

In this class, we have learned how to:

• Do things on a computer that you can’t do with pencil and paper.

  • Approximating and optimizing functions.

• Do these things quickly.

  • Writing good code in Julia, dynamic programming, parallelization.

• Things we learned feature heavily in solving economic models.

• You will work through many such models in Econ 899 and hopefully your own research.

• Lots of additional resources I have found useful are on my website:
  • https://kevinghunt.github.io/ComputationCamp/index.html#resources
• Good luck with your research and field papers.
• Thank you for your participation!