Patrick Power PRO
Economics PhD @ Boston University
Functors
a
T a
b
F b
Type Constructor
f
\textrm{fmap } f
- A functor is a function from one category to another the preserves the cateogorical structure
- From a programing perspective, it maps types to types and functions to functions
- We represent this in Haskell via a
- parameterized type constructor
- placing the type in the functor typeclass with the fmap typeclass method
(\to) \ r \quad \textrm{is a parameterized type constructor}
f
a
f :: r \to a
b
f :: r \to b
a
(\to) \ r \ a
b
f
\textrm{fmap } f
(\to) \ r \ b
instance Functor ((->) r) where
fmap f g = (\x -> f (g x))(\to) \ r \ a
a
(\to) \ r \ a
b
f
instance Functor ((->) r) where
fmap f g = (\x -> f (g x))Professor writes down the Model
Student solves optimization problem
Learning how to solve well-posed problems
Typical class
Lecturer writes down question
Student writes down the model
Solver solves the optimization problem
Judgement
Calculus
Statistics
Newspapers/Podcasts
Psychology
Sociology
Finance
Core of Economics
This Class
Write down question
Write down the model
Solver solves the optimization problem
You
Real World Economics
Microeconomic model
Model
\underset{x \in F_p^{-1}(\mathcal{X})}{\textrm{maximize}} \ U_{\alpha}(x)
\underset{x \in F_p^{-1}(\mathcal{X})}{\textrm{maximize}} \ \mathbb{E}[f(x, z^*(x))]
Where to write math
On the Computer
- Easily capture heterogeneity
- Easily model uncertainty
- Solving the model is easy (call a solver!)
- Build the model step-by-step
- Easy to visualize
- We've already covered each aspect! (Nothing new!!)
- Hard to capture heterogeneity without some planning
- Hard to model uncertainty without more statistics
- Manually solve the model (takes time)
- Hard to build the model step-by-step
- Cannot visualize most aspects
If you think something is new, you missed a class
On Paper
xs = jnp.linsapce(0., 5.0, 7)initial value
final value
Number of elements
Vector of linearly spaced values
Vector of Values
def f(x, y):
return 2*x + jnp.sin(y)A Function
Two inputs
x, y \longmapsto 2x + \sin(y)
Output
g = partial(f, 2)A Function
Partial Evaluation
Is a higher order function (takes a function as input and returns a function)
Pass f as input
evaluate at x=2
def f(x, y):
return 2*x + jnp.sin(y)A Function
Two inputs
x, y \longmapsto 2x + \sin(y)
Output
h = partial(f, y=3)A Function
Partial Evaluation
Is a higher order function (takes a function as input and returns a function)
Pass f as input
evaluate at y=3
def f(x):
return 2*x A Function
One Input
Pass in a function
vectorize function
x \longmapsto 2x
z = jax.vmap(f)Vectorized Function
Output
Please write down a model that provides some insight about which types of firms might increase their production in the US in response to the Pandemic
Application # 1
Copy of Applications
By Patrick Power
