Econ 560a Assignment 2

1. Verify that the analytical solution for the value function in (2.22) is correct, i.e. that this V(s) satisfies Bellman's equation. Show your work!

2. Read sections 4.1 and 4.2 on numerical solution methods in my chapter and pay particular attention to the illustration in section 4.2.2 on "Comparison of Methods for the Auto Replacement Method". Study the method for discretizing the continuous transition densities so the problem can be converted to an approximate problem on a finite state space. Download the Gauss code for solving this problem at

   ftp://gemini.econ.yale.edu/pub/johnrust/handbook/discrete_programs

   and try to replicate the approximate solutions given in fig 14.1 by running the algorithms. The successive approximations algorithm can be used to do this. The code for this is in the file succapp.gpr.

3. Compare the speed and accuracy of other methods for solving the DP problem including a modified successive approximation algorithm with termination determined by the McQueen-Porteus error bounds described on page 653 of my chapter (this code is in the file merrbnds.gpr), and the method of policy iteration described on page 654 (code for this is in policyit.gpr), the modified policy iteration algorithm described on page 655 (molicyit.gpr), and the policy iteration with adaptive state aggregation algorithm discussed on page 656 (adapagg.gpr). Do the results of your comparison of the relative cpu times and errors in the various methods yield the same general conclusions that are in table 14.1 of my chapter?

4. Consider a modified version of the auto replacement problem when there is a secondary market for used automobiles. Using the same parameters as the example problem on page 659, modify the DP problem to allow the following function for cost of replacing an auto from a constant cost of

   \overline P - \underline P = crep= 100000

   to:

   case 1: crep=100000*exp(-4+(s^1.3)/20)/(1+exp(-4+(s^1.3)/20))

   case 2: crep=100000-10*(100-x)^2

   Solve the modified DP problem on a discrete state space with 100 states from 1 to 100. Plot the optimal decision rules and value function for this modified problem. Does the decision rule still take the form of a simple optimal stopping rule, or does a more complex decision rule emerge?

   Hint: modify the Gauss code. The cost of replacing an automobile is set in vector crep in the procedure initc.g. Modify this appropriately to allow the above cases of non-constant costs of replacement. For plotting, the following code is an example for plotting the implied costs of replacement functions in the two cases given above. You should be able to modify this code to plot the value function and decision rule.

 /* Gauss code to plot out cost of replacement functions for
    cases of non-constant cost of replacing an automobile for
    homework 2 of Econ 560a */

 c=100000;
 library pgraph;
 x=seqa(1,1,100);
 crep=c*exp(-4+(x^1.3)/20)./(1+exp(-4+(x^1.3)/20));
 crep1=c-10*(100-x)^2;

#IFUNIX

let v = 100 100 640 480 0 0 1 6 15 0 0 2 2;
wxy = WinOpenPQG(v,"XY Plot","XY");
call WinSetActive(wxy);

#ENDIF

_pdate="";
title("Alternative Assumptions about Costs of Replacing an Auto");
xlabel("Auto Mileage (thousands)");
ylabel("Cost ($ x 10)");
xy(x,crep~crep1);

#IFUNIX

call WinSetActive(1);

#ENDIF

5. The modified dynamic programming problem still assumes that if you decide to trade your old existing auto, that you will always purchase a brand new one. However you can also decide to purchase a used car on the market. Auto rental companies such as Hertz or Avis claim that the least cost strategy is to buy a slightly used car, say about one or two years old, and then hold it for about 5 years. Hertz or Avis may have an ulterior motive in giving this advice since they like to sell their rental cars on the used car market and it is in their interest to make sure that consumers would pay as high a price as possible for their used cars. Use dynamic programming to check whether the Hertz or Avis advice does in fact correspond to an optimal trading strategy for cars that minimizes expected discounted trading costs. Modify the Gauss code to allow people the option to either buy a brand new car, or a car in any condition x, where x is an integer running from 0 to 99, representing the mileage on the auto in hundreds of thousands of miles. Assume that the price of cars as a function of x is given by

   P(x)=10*(100-x)^2

   or by

   P(x)=101831.56/(1+exp(-4+(x^1.3)/20))

   and assume there are no transactions costs to trading in the automobile market other than the net cost of selling your current car in condition y for another car in condition x equal to P(x)-P(y). What does the DP solution give you for the optimal trading strategy?

6. What can you say in general about trading automobiles when there are zero transactions costs as opposed to positive transactions costs? Formulate your model with a continuous state space on the nonnegative real line similar to the model outlined in my Handbook of Computational Economics chapter, but now assuming there is a secondary market for buying or selling automobiles with the equilibrium price of autos given by a function P(x) which is decreasing in x. Characterize the nature of optimal trading strategies in the case of zero transactions costs and positive transactions costs.