Job Search IV: Modeling Career Choice¶

Contents¶

• Job Search IV: Modeling Career Choice
  • Overview
  • Model
  • Exercises
  • Solutions

Overview¶

Next we study a computational problem concerning career and job choices

The model is originally due to Derek Neal [Nea99]

This exposition draws on the presentation in [LS18], section 6.5

Model features¶

• Career and job within career both chosen to maximize expected discounted wage flow
• Infinite horizon dynamic programming with two state variables

Setup¶

Model¶

In what follows we distinguish between a career and a job, where

• a career is understood to be a general field encompassing many possible jobs, and
• a job is understood to be a position with a particular firm

For workers, wages can be decomposed into the contribution of job and career

• $ w_t = \theta_t + \epsilon_t $, where

  • $ \theta_t $ is contribution of career at time $ t $
  • $ \epsilon_t $ is contribution of job at time $ t $

At the start of time $ t $, a worker has the following options

• retain a current (career, job) pair $ (\theta_t, \epsilon_t) $ — referred to hereafter as “stay put”
• retain a current career $ \theta_t $ but redraw a job $ \epsilon_t $ — referred to hereafter as “new job”
• redraw both a career $ \theta_t $ and a job $ \epsilon_t $ — referred to hereafter as “new life”

Draws of $ \theta $ and $ \epsilon $ are independent of each other and past values, with

• $ \theta_t \sim F $
• $ \epsilon_t \sim G $

Notice that the worker does not have the option to retain a job but redraw a career — starting a new career always requires starting a new job

A young worker aims to maximize the expected sum of discounted wages

$$ \mathbb{E} \sum_{t=0}^{\infty} \beta^t w_t \tag{1} $$

subject to the choice restrictions specified above

Let $ V(\theta, \epsilon) $ denote the value function, which is the maximum of (1) over all feasible (career, job) policies, given the initial state $ (\theta, \epsilon) $

The value function obeys

$$ V(\theta, \epsilon) = \max\{I, II, III\}, $$

where

$$ \begin{aligned} & I = \theta + \epsilon + \beta V(\theta, \epsilon) \\ & II = \theta + \int \epsilon' G(d \epsilon') + \beta \int V(\theta, \epsilon') G(d \epsilon') \nonumber \\ & III = \int \theta' F(d \theta') + \int \epsilon' G(d \epsilon') + \beta \int \int V(\theta', \epsilon') G(d \epsilon') F(d \theta') \nonumber \end{aligned} \tag{2} $$

Evidently $ I $, $ II $ and $ III $ correspond to “stay put”, “new job” and “new life”, respectively

Parameterization¶

As in [LS18], section 6.5, we will focus on a discrete version of the model, parameterized as follows:

• both $ \theta $ and $ \epsilon $ take values in the set linspace(0, B, N) — an even grid of $ N $ points between $ 0 $ and $ B $ inclusive
• $ N = 50 $
• $ B = 5 $
• $ \beta = 0.95 $

The distributions $ F $ and $ G $ are discrete distributions generating draws from the grid points linspace(0, B, N)

A very useful family of discrete distributions is the Beta-binomial family, with probability mass function

$$ p(k \,|\, n, a, b) = {n \choose k} \frac{B(k + a, n - k + b)}{B(a, b)}, \qquad k = 0, \ldots, n $$

Interpretation:

• draw $ q $ from a β distribution with shape parameters $ (a, b) $
• run $ n $ independent binary trials, each with success probability $ q $
• $ p(k \,|\, n, a, b) $ is the probability of $ k $ successes in these $ n $ trials

Nice properties:

• very flexible class of distributions, including uniform, symmetric unimodal, etc.
• only three parameters

Here’s a figure showing the effect of different shape parameters when $ n=50 $

Implementation:

The code for solving the DP problem described above is found below:

The code defines

• a named tuple CareerWorkerProblem that

  • encapsulates all the details of a particular parameterization
  • implements the Bellman operator $ T $

In this model, $ T $ is defined by $ Tv(\theta, \epsilon) = \max\{I, II, III\} $, where $ I $, $ II $ and $ III $ are as given in (2), replacing $ V $ with $ v $

The default probability distributions in CareerWorkerProblem correspond to discrete uniform distributions (see the Beta-binomial figure)

In fact all our default settings correspond to the version studied in [LS18], section 6.5.

Hence we can reproduce figures 6.5.1 and 6.5.2 shown there, which exhibit the value function and optimal policy respectively

Here’s the value function

The optimal policy can be represented as follows (see Exercise 3 for code)

Interpretation:

• If both job and career are poor or mediocre, the worker will experiment with new job and new career
• If career is sufficiently good, the worker will hold it and experiment with new jobs until a sufficiently good one is found
• If both job and career are good, the worker will stay put

Notice that the worker will always hold on to a sufficiently good career, but not necessarily hold on to even the best paying job

The reason is that high lifetime wages require both variables to be large, and the worker cannot change careers without changing jobs

• Sometimes a good job must be sacrificed in order to change to a better career

Exercises¶

Exercise 1¶

Using the default parameterization in the CareerWorkerProblem, generate and plot typical sample paths for $ \theta $ and $ \epsilon $ when the worker follows the optimal policy

In particular, modulo randomness, reproduce the following figure (where the horizontal axis represents time)

Hint: To generate the draws from the distributions $ F $ and $ G $, use the type DiscreteRV

Exercise 2¶

Let’s now consider how long it takes for the worker to settle down to a permanent job, given a starting point of $ (\theta, \epsilon) = (0, 0) $

In other words, we want to study the distribution of the random variable

$$ T^* := \text{the first point in time from which the worker's job no longer changes} $$

Evidently, the worker’s job becomes permanent if and only if $ (\theta_t, \epsilon_t) $ enters the “stay put” region of $ (\theta, \epsilon) $ space

Letting $ S $ denote this region, $ T^* $ can be expressed as the first passage time to $ S $ under the optimal policy:

$$ T^* := \inf\{t \geq 0 \,|\, (\theta_t, \epsilon_t) \in S\} $$

Collect 25,000 draws of this random variable and compute the median (which should be about 7)

Repeat the exercise with $ \beta=0.99 $ and interpret the change

Exercise 3¶

As best you can, reproduce the figure showing the optimal policy

Hint: The get_greedy() method returns a representation of the optimal policy where values 1, 2 and 3 correspond to “stay put”, “new job” and “new life” respectively. Use this and the plots functions (e.g., contour, contour!) to produce the different shadings.

Now set G_a = G_b = 100 and generate a new figure with these parameters. Interpret.

Solutions¶

Exercise 1¶

Exercise 2¶

The median for the original parameterization can be computed as follows

To compute the median with $ \beta=0.99 $ instead of the default value $ \beta=0.95 $, replace wp=CareerWorkerProblem() with wp=CareerWorkerProblem(β=0.99)

The medians are subject to randomness, but should be about 7 and 14 respectively. Not surprisingly, more patient workers will wait longer to settle down to their final job

Exercise 3¶

Here’s the code to reproduce the original figure

Now, we need only swap out for the new parameters

You will see that the region for which the worker will stay put has grown because the distribution for $ \epsilon $ has become more concentrated around the mean, making high-paying jobs less realistic