1. Understand how your computer works and interacts with programming languages.
2. Learn some advanced Julia concepts, especially to improve how fast your code runs.

• Computers are very good at performing calculations and executing algorithms.
• Unfortunately, a computer’s hardware only understands things written in machine language.
• A machine language consists of instructions written using only of binary 0’s and 1’s.
• This machine language cannot easily be understood by humans.

• Programming languages act as the translator between you and your computer

1. You write a set of instructions in a programming language, such as Julia.
2. When you execute your code, these instructions get translated to machine language.
3. Your computer executes the task.

• How fast your code runs will be impacted by:

1. How you write your code.
2. How your instructions get translated to machine language.
3. What computer you are using.

• Programming languages generally fall into two groups:

1. Compiled languages (C, Fortran)

• All of the code gets translated before execution.
• Generates very efficient machine code.
• Cannot code interactively or make changes on the fly.

1. Interpretted languages (Stata, R, Python, Matlab)

• Translates each line of code just before its execution.
• This allows for dynamic and interactive coding.
• Will not generate efficient machine code.

• Julia tries to have the best of both worlds with just-in-time compilation.
• When executing a piece of code, such as a function, for the first time Julia compiles it to machine language.
• When executing the next time, Julia machine language is called directly instead of re-compiling.
• This allows for the benefits of interactive coding of an interpretted language.
• Efficient machine code can also be generated if you follow best practices.

https://docs.julialang.org/en/v1/manual/performance-tips/

• Given a capital \(k\), we have resources: \[ y = k^\alpha + (1-\delta)k. \]
• We live forever and we discount future at rate \(\beta\). Utility from consumption is \(\log(c)\).
  
• We need to decide how much to consume \(c\) and how much capital to save \(k'\).

\[ V(k) = \max_{c,k'} \:\:\: \log(c) + \beta V(k'). \] \[ c + k' = k^\alpha + (1-\delta)k. \]

• We know that this is a contraction mapping, so we can iterate over \(V\) until convergence.
• Guess \(V_1\). Solve

\[ V_2(k) = \max_{c,k'} \:\:\: \log(c) + \beta V_1(k'). \] - Then: \[ V_3(k) = \max_{c,k'} \:\:\: \log(c) + \beta V_2(k'). \] - We know that \(V_n \xrightarrow{} V\) as \(n \xrightarrow{} \infty\).