Problem 1: Heterogeneous Firms in MATLAB
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Problem 1: Heterogeneous Firms in MATLAB [100 marks]
In this problem, you will analyse a special version of the model by Hopenhayn and Rogerson (1993) presented in class using MATLAB .
Consider the Hopenhayn-Rogerson model presented in Lecture 7 (applying the same notation defined then), but suppose from now on that there are no labour adjustment costs, i.e.,
T = 0.
Assume that firms operate the production function
f (n, s) = snθ ,
where n is the level of employment, θ e (0,1), and s is an idiosyncratic productivity shock which follows a discrete-state Markov process.
1. Write down the Bellman equation that defines the value function of an incumbent firm. Denote this value function by W (s; p), where s is the value of the productivity shock in the current period, and p is the price of the consumption good in terms of labour (which is the numéraire). Note that in the absence of labour adjustment costs there is only one state variable (s).
2. Solve for the optimal policy function for employment n\ (s; p).
3. Next, you will numerically analyse the optimal decisions of firms in this model. Follow these steps:
(a) Download the files parameters.mat and parameters_Readme.rtf from Blackboard. The first file contains the parameters to solve the problem in MATLAB . The second file provides some information about the content of parameters .mat (read it care- fully!).
(b) Load the file parameters.mat into your MATLAB code, and find the strictly positive output price p * > 0 that satisfies the free entry condition:
We(p *) = p * ce, (1)
where We (p) is the value function of new entrants in terms of the output price p, and ce is the entry cost. (Important: avoid solutions with p * = 0!)
(c) Given p * , solve for the productivity cutoff s * such that incumbents exit in the fol-
lowing period if and only if s s s * .
Report the values of p * and s * .
4. Plot the value function W(s; p *) against s.
5. Make a plot representing the free entry condition in (1). To this end, plot the the value function We (p) over a grid of p, and (on the same axes) plot the function p . ce .
6. Now recompute the equilibrium by changing the value of the entry cost to ce = 50. How does p * and s * change? Explain the intuition behind your findings (you may use graphs to support your answer).
Hints
• In (2.b) you need to compute W (s; p), which can be done by applying value function iter- ation. It will also be helpful to write a function file that returns the solution to the value function for any given price p (i.e., to write a function file that takes p as a variable and returns W (s; p) after applying value function iteration).
• When you solve for p * in (2.b), you may use a built in function, such as fsolve. To avoid solutions with p * = 0, use a large value as a guess for p * (say 100).
2022-10-10