[SOLVED] CS python ECON 8500 Monetary Theory (MAEP w/ Python)

ECON 8500 Monetary Theory (MAEP w/ Python)
Prof. Marcelo Arbex Winter 2021
Topic #1: Fully Funded Social Security

Student Names:

INSTRUCTIONS

1.Deadline: The group leader should submit the final group project (Jupyter notebook) directly to me ([email protected]), CC: all group members by April 19, 2021; 11:59pm.
2.Rename and save this notebook to a location of your choice by using the File -> Rename option from the menu.
3.All your work must be in this notebook. Save your work often (Ctrl + S).
4.For each question, insert cell below (use the icon +, or Insert -> Insert Cell Below option from the menu) to enter your code (see Exercise 1 for an example).
5.Add comments along your code using # or insert another cell for text and additional explanations.

1.Give an overview of the current structure of retirement/elderly funding in Canada (exclude health). How is it organized (provincial, federal)? What is the share of government expenditure on these programs?

1.Consider the Fully Funded Social Security model from Homework 3.
Assume government mandated savings of quantity $b$ from the young each period
This savings accrues to the same individual when they are older, $t_{ss}=(1+r)b$
The program is fully funded in the sense that promised retirement payments are always backed by sufficient savings, b.
The resulting young problem is:
begin{align} underset{c_t,c_{t+1},s}max& ; log(c_t)+beta log(c_{t+1})
otag \ &s.t.
otag \ c_t+s&leq y-b
otag \ c_{t+1} &leq s(1+r) + t_{ss} end{align}
(a) Define an equilibrium
(b) Write a code to solve the model

1.Calibrate the model to match the Canadian economy.
Set y=1
Set $n$ to match the pop growth rate of Canada over last 30 years
Set $r$ to match the average interest rate on government debt (risk free rate) over the last 30 years.
Set $b$ to match the tax burden of entitlements ($b/y$)
Choose $beta$ to deliver the HH savings rate in Canada (you will have to compute the model and adjust $beta$ until it returns the desired savings rate). * Hint: The last part would be best done by automating the search for an appropriate $beta$ rather than manually.*

1.The model above has assumed unrestricted savings $s$. Suppose we now impose a borrowing constraint ($s>-bar{S}$). The problem then becomes:
begin{align} underset{c_t,c_{t+1},s}max& ; log(c_t)+beta log(c_{t+1})
otag \ &s.t.
otag \ c_t+s&leq y-b
otag \ s & geq -bar{S}
otag \ c_{t+1} &leq s(1+r) + t_{ss} end{align}
(a) In this environment, can a Fully Funded system make agents better off?
(b) Write a code to solve the model. Set $bar{S}=0$.
* Hint: you may want to use the minimization routine we used in class to solve for the optimal savings over a bounded interval*

1.Fully funded systems are essentially a form of forced savings. Why could this be a good thing? Empirically, there is quite a bit of evidence that people dont save enough for their future. In a sense, they dont use the right discount factor when thinking about the future. Economists capture this type of issue with something called hyperbolic discounting, which is a more nuanced argument about discounting that we wont get into.
For us, we will imagine simply that agents solving their problem with the objective function $u(c)+beta u(c^prime)$ while their true present value is $u(c)+hat{beta} u(c^prime)$ where $hat{beta}>beta$. In other words, they dont recognize at the time of decision making how much they value tomorrow relative to today. Assume that the government recognizes this is an issue (i.e knows $hat{beta}>beta$). Set $hat{beta}=max{1.05beta,0.99}$ from your calibrated value in part (3).
(a) Define the problem of a government choosing b to maximize the lifetime utility of the young.
* Hint: recall from the optimal taxation application that the government must take the decisions of the agents as given!That is, the agent responses must be optimal given their own problem.*

(b) With $bar{S}=0$, solve for the optimal $b$ that maximizes a utilitarian planners problem where the correct discount factor $hat{beta}$ is applied. Call this $b^*$.

1.Compare the welfare gain of the planner when $b=b^*$ and $b=0$.