8107 ECON - MACRO 3 TIPS FOR USING MATLAB
CTIRAD SLAVIK
1. Introductory tips Tip 1: Always have a benchmark case with analytical solution to verify your code or do 2 different procedures. Example: In dynamic programming use log utility and Cobb-Douglas production function with full depreciation. Tip 1: Use a lot of comments so that other peope know what you are doing. Tip 3: How to use help. Besides standard help, you can type help 'command' and you will get description of the command plus related commands. Example: type help * and you will get the description of most algebraic operations. Tip 4: A code: clear all; clears all variables in register tic; starts measuring time define variables the code itself toc; stops measuring time and displays time
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C. Slavik
ECON 8107 - Recitation 3 2. Programming in Matlab
2.1. Objects in Matlab. 1 Every object has its name that you define it with, of course you cannot use the same name for 2 objects of the same type (e.g. vectors) neither for 2 objects of different type (e.g. a vector and a matrix). (1) Scalars - defines simply as beta = 0.96; The semicolon guarantees that the number will not be printed out when the code is running. (2) Vectors - e.g. a = [1,1,2,5,6]; or a = [1 1 2 5 6]; (3) Matrices - similarly, e.g. A = [1 1 2 5 6, 3 4 6 7 7, 1 5 4 2 1] Useful trick: A = zeros(n, m) or A = ones(i, j, k, l). (4) Grid - basic object, in some vague sense it defines values that the particular variable can take. In fact it is just a vector. Example: the grid for capital holdings: (0, 0.1, 0.2, ..., 10) will be defined as: K = (0 : 0.1 : 10) or K = linspace(0, 10, 101) You may want to define a grid that is finer close to zero (i.e. because you believe there is more non-linearity there). K = (0 : 0.1 : or K = logspace(0, 1, 101) - 1 Note: the dot denotes element by element operations. (5) Functions of one variable are just vectors. E.g. define Y = K. (1-alpha). Functions of more variables are appropriately sized matrices. Example1 in Matlab. Note: piecewise linear approximations. 10). 2
1We will not discuss symbolic toolbox that is also available for Matlab.
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C. Slavik
ECON 8107 - Recitation 3
2.2. Basic procedures in Matlab. (1) Finding maxima and minima of a vector, a matrix or a function on the grid. See Example 1 above, type help max. (2) How to iterate, do loops and plots. Throughout below. (3) Finding x : f (x) = 0. Note: this can be used also to find f (x) = 0, i.e. finding a maximum of a function that is not restricted to lie on the grid. Consider the following problem taken from Growth PS3: (l)3 s.t. c,l 3 c (1 + r)a + w z l We will assume r = 0.01, w = 1, z denotes the agent's labor productivity shock and a his asset holding. What kind of problem is this? max log(c) - wz = l2 (1 + r)a + wzl 0 = wzl3 + (1 + r)al2 - wz =: f (l) How to find the l [0, 1] that solves this equation for all possible (a, z)? (a) Matlab has a function fsolve. Example2 and Example3 and function fun in Matlab. (b) Brute force method. The same example as before. We know f (0) < 0, f (1) > 0, the function is continuous, thus the x that we are looking for lies between 0 and 1. Procedure: let a1 = 0, b1 = 1, compute f ( a1 +b1 ). If f ( a1 ) +b1 > 0 2 2 then let a2 = a1 and b2 = a1 +b1 . Compute f ( a2 +b2 ). Iterate. 2 2 Example4 in Matlab. Compare the outcomes and the time it takes. Discuss problems. (c) Gauss-Newton method for the same example as before. Procedure: let x1 be given. Set x2 = x1 - if close enough to 0. Iterate.
f (x1 ) f (x1 ) .
FOC:
Check f (x2 ). Stop
f Alternative: let x1 be given. Set x2 = x1 - f (x1 )) , (0, 1). Iterate (x1 as above.
Compare the outcomes and the time it takes. Discuss problems. 3
C. Slavik 3. Further examples
ECON 8107 - Recitation 3
3.1. Value function iteration. Recall one of the past homeworks. Consider the following sequential problem:
v (k0 ) :=
{ct ,kt }t=0
max
t log ct s.t.
t=0
kt+1 + ct kt + (1 - )k kt , ct 0, k0 given Assume = 0.6, = 0.3, = 20, = 0.5. (1) Let capital take values for the discrete grid (0.05, 0.1, ..., 9.95, 10). Make the original guess that k : v(k) = 0, perform the value function iterations until:
max |vi+1 (k) - vi (k)| < 10-5
k
How many iterations does it take? Report the value function and the policy function for k that you obtain. (2) Apply Howard's improved algorithm. Compare the speed, number of iterations and the results. Example6 in Matlab. Discussion: (1) How to do it if is a random variable following a first order Markov process. (2) Problems associated with these procedures - i.e. forcing the policy to lie on the grid, time ... 3.2. Negishi algorithm. Consider the following problem, similar to the one in the homework. There are 2 agents in the economy. Each has endowment 1 in every period. Preferences of agent i are:
t
t=0
c1-i it 1 - i
Solve for the ADE using Negishi algorithm, assume 1 = 4, 2 = 2. Procedure: (1) Solve the SPP with weight on agent 1 being and on agent 2 being 1. Express c1 as a function of . (2) Compute the prices using the FOC of agent 1 and the allocations calculated above. (3) Compare the value of agent 1'...