{
"cells": [
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
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],
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]
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{
"cell_type": "markdown",
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"source": [
"This note contains a bunch of calculations associated with the reading on directed search with incomplete information. Since $\\lambda$ is s key word in sagemath, I'll switch the notation so that the probability with which each worker is a good type to be $\\gamma$. Otherwise the notation is the same as in the reading in directed search with complete information. \n",
"\n",
"\n",
"We can start with the question about when we can find an equilibrium were good workers apply to the high wage firm for sure. Recall that the condition is that the good worker prefers to take a chance at the high wage firm if the other worker is also a good worker, or to get the low wage for sure (notice that the presumption is that the other worker will apply for sure at the high wage firm). The condition for this is\n",
"$$\n",
"\\gamma \\frac{w_1}{2} + (1-\\gamma)w_1 \\ge w_2\n",
"$$\n",
"\n",
"This depends in a sort of complicated way on $w_1$, $w_2$ and $\\gamma$. The next calculation finds for all values of $w_1$ and $w_2$, the highest value that $\\gamma$ can have before the good worker decides it is better just to apply to the low wage."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
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{
"data": {
"text/html": [
""
]
},
"execution_count": 1,
"metadata": {
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"output_type": "execute_result"
}
],
"source": [
"var('w_1,w_2,gamma,pi')\n",
"indifference = gamma*w_1/2+(1-gamma)*w_1 == w_2\n",
"i_cond = solve(indifference,gamma)\n",
"show(i_cond[0])"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
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"source": [
"Now, we can plot this for all values of $w_1 > \\frac{1}{2}$ assuming that $w_2 = \\frac{1}{2}$. The region we will be focussing on in what follows will be the region below the curve (in other words the region where it isn't so likely that other worker is a good worker).\n",
"\n",
"We can express this in a slightly different way by figuring out how high $w_1$ has to be to support the equilibrium in which good workers apply there for sure. This is just a slight variant of what we just did."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
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"outputs": [
{
"data": {
"text/html": [
""
]
},
"execution_count": 7,
"metadata": {
},
"output_type": "execute_result"
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],
"source": [
"w_cond = solve(indifference,w_1)\n",
"show(w_cond[0])"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"In words, this says that to support the equilibrium where the good workers apply to the high wage firm for sure, the wage $w_1$ must be higher than $\\frac{2w_2}{2-\\gamma}$."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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"
},
"execution_count": 3,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"plot(i_cond[0].rhs().substitute(w_2=1/2),(w_1,1/2,1))"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Once we are in this region, we can work out what the bad workers will do. Assuming that $pi$ is the probability the bad workers apply to the high wage firm, we want them to be indifferent between the high and low wage firms. This happens when\n",
"$$\n",
"(1-\\gamma)(\\pi\\frac{w_1}{2}+(1-\\pi)w_1) = \\pi w_2+(1-\\pi) \\frac{w_2}{2}\n",
"$$\n",
"This kind of calculation is getting routine:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
]
},
"execution_count": 5,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"w_indiff = (1-gamma)*(pi*w_1/2+(1-pi)*w_1) == pi*w_2 +(1-pi)*w_2/2\n",
"prob = solve(w_indiff,pi)\n",
"show(prob[0])"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Notice in this formula what happens when $\\gamma = 0$. It seems intuitively that if there is no chance the other worker is a good worker, then things reduce to the game we discussed before, and we should get the same formula as we did in the directed search game with complete information.\n",
"\n",
"One interesting question is whether good workers are more likely to get jobs than bad workers are. This depends in a pretty complex way on $w_1$, $w_2$ and $\\gamma$.\n",
"\n",
"The probability a good worker gets a job is pretty straightforward, just\n",
"$$\n",
"\\frac{\\gamma}{2}+(1-\\gamma)\n",
"$$\n",
"The probability the bad worker gets a job is\n",
"$$\n",
"\\gamma(1-\\pi)+(1-\\gamma)(\\pi(\\frac{\\pi}{2}+(1-\\pi))+(1-\\pi)(\\pi+\\frac{1-\\pi}{2}))\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
]
},
"execution_count": 9,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"prob_good = gamma/2+(1-gamma)\n",
"show(prob_good)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
]
},
"execution_count": 10,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"prob_bad = gamma*(1-pi)+(1-gamma)(pi*(pi/2+(1-pi))+(1-pi)*(pi+(1-pi)/2))\n",
"show(prob_bad)\n"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
]
},
"execution_count": 12,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"form = prob_bad.substitute(pi=prob[0].rhs())\n",
"show(form)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"This is way complicated, so we'll do some substitutions. First, as before we can let $w_2 = \\frac{1}{2}$. Then we can take $\\gamma$ to be equal to $\\frac{1}{5}$. Then if you refer back to the previous plot, you can see that there are plenty of values for $w_1$ which will support an equilibrium of the kind we want."
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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"
},
"execution_count": 19,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"plot(form.substitute(gamma=1/2,w_2=1/2),(w_1,1/2,1))+plot(prob_good.substitute(gamma=1/2),(w_1,1/2,1))"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"0.6583592135001262"
]
},
"execution_count": 21,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"find_root(form.substitute(gamma=1/2,w_2=1/2) == prob_good.substitute(gamma=1/2), .5,1 )"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"What this computation is saying is that whether bad workers are more or less likely to get a job depends on the difference between the low and high wage. The computation set $w_2 = \\frac{1}{2}$ and $\\gamma = \\frac{1}{2}$. The numerical result says that if the high wage $w_1$ is lower than .65, then the bad workers will actually be more likely to find a job than the good workers will. Conversely, if $w_1 > .65$ then bad workers will be less likely to get a job. Notice they are less likely to get a job the higher is $w_1$."
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/html": [
""
]
},
"execution_count": 37,
"metadata": {
},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
""
]
},
"execution_count": 37,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"var('y_1')\n",
"F = function ('F')\n",
"t = function('t')\n",
"b = function('b')\n",
"g = (y_1-w_1)*(1-(F(t(w_1,w_2))*(1-b(w_1,w_2)))^2)\n",
"show(latex(g))\n",
"show(latex(g.differentiate(w_1)))\n"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath (stable)",
"language": "sagemath",
"metadata": {
"cocalc": {
"description": "Open-source mathematical software system",
"priority": 10,
"url": "https://www.sagemath.org/"
}
},
"name": "sagemath"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 2
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython2",
"version": "2.7.15"
}
},
"nbformat": 4,
"nbformat_minor": 0
}