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Economic simulation in Julia: A gentle introduction
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EconSimTutorial.ipynb
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"source": [
"using Plots"
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"metadata": {},
"source": [
"# Economic simulation in Julia: A gentle introduction\n",
"\n",
"Max Kapur | [maxkapur.com](https://www.maxkapur.com)\n",
"\n",
"This tutorial explains how to **simulate a tâtonnement process** (aka Walrasian auction) using Julia.\n",
"\n",
"Prerequisites:\n",
"\n",
"1. You have installed the [Julia programming language and the IJulia](https://julialang.github.io/IJulia.jl/stable/manual/installation/) extension, which lets you use Jupyter notebooks like this. \n",
"2. Maybe some multivariable calculus?\n",
"\n",
"## Scenario\n",
"\n",
"This tutorial considers the *price paths of perishable goods.*\n",
"\n",
"Consider a market with `n` traders. Each produces a fixed amount of good `i` in each time period `t`.\n",
"\n",
"Traders try to sell out their entire supply each time period. No salvage value.\n",
"\n",
"The demand for each good is determined entirely by the prices. Traders do not know the shape of their demand curves. All they can do is \"guess and check.\""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Problem data\n",
"\n",
"First we generate a square matrix `A` that determines the shape of the demand curve."
]
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"execution_count": 2,
"metadata": {},
"outputs": [
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"data": {
"text/plain": [
"6×6 Matrix{Float64}:\n",
" -1.31098 0.478557 0.625543 0.123993 0.911671 0.743691\n",
" 0.601984 -1.45778 0.921821 0.65701 0.753835 0.85025\n",
" 0.194673 0.100195 -1.68408 0.986783 0.957128 0.87066\n",
" 0.612065 0.841071 0.219045 -1.60008 0.42792 0.724267\n",
" 0.4138 0.38046 0.716622 0.635423 -1.14904 0.539066\n",
" 0.701006 0.247234 0.973945 0.221853 0.0451124 -1.11766"
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"source": [
"n = 6\n",
"A = rand(n, n)\n",
"for i in 1:n\n",
" A[i, i] -= 2\n",
"end\n",
"A"
]
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"cell_type": "markdown",
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"source": [
"`q` is a vector of supply quantities. `sum(q) = 1`. "
]
},
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"cell_type": "code",
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"data": {
"text/plain": [
"6-element Vector{Float64}:\n",
" 0.3160576868320888\n",
" 0.3148290559299176\n",
" 0.08798283436539898\n",
" 0.1351738091666288\n",
" 0.13604705660521643\n",
" 0.009909557100749379"
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"source": [
"q = rand(n)\n",
"q ./= sum(q)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Each trader is happy when the demand for her product equals her supply.\n",
"- If she receives excess demand, she will raise her price, since she can still sell out at the higher price. \n",
"- If she has excess supply (too little demand), she will lower her price (but not past zero) in order to attract more buyers.\n",
"\n",
"Whenever the change in prices obeys these two rules, we have a tâtonnement process. \"Prices move in the direction of excess demand.\"\n",
"\n",
"The demand is a function of the prices. It takes the prices of the `n` goods as inputs and returns `n` demand quantities as output."
]
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"data": {
"text/plain": [
"demand (generic function with 1 method)"
]
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"metadata": {},
"output_type": "execute_result"
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"source": [
"function demand(p)\n",
" numer = A * p\n",
" return numer ./ sum(numer)\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"By design, the demand function has the following properties for nonnegative `p`:\n",
"- `demand(p)[i]` is *decreasing* in `p[i]` and *increasing* in `p[j]` for `j ≠ i`. Seems reasonable.\n",
"- We assume the economy is *Walrasian.* In this case, that means `sum(demand) = sum(q) = 1`. The sum of the entries `demand(p)` is always one. \"Aggregate supply equals aggregate demand.\"\n",
"- `demand(p)` is *continuous* and *homogeneous* in `p`. "
]
},
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"cell_type": "code",
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"data": {
"text/plain": [
"6-element Vector{Float64}:\n",
" 0.39355798000386577\n",
" 0.2827696415339973\n",
" 0.3063252829557601\n",
" 0.5626567305955585\n",
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"source": [
"currentprices = rand(n)"
]
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"The excess demand at the current prices:"
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"7×4 Matrix{Any}:\n",
" :price :demand :supply :diff\n",
" 0.393558 0.110984 0.316058 -0.205073\n",
" 0.28277 0.305083 0.314829 -0.00974598\n",
" 0.306325 0.214911 0.0879828 0.126928\n",
" 0.562657 -0.0214525 0.135174 -0.156626\n",
" 0.30958 0.207558 0.136047 0.0715108\n",
" 0.219469 0.182916 0.00990956 0.173006"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"[ :price :demand :supply :diff ;\n",
" currentprices demand(currentprices) q demand(currentprices) - q ]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Importantly, the demand for the `i`th good depends on not only `p[i]`, but also on the prices of the other goods. "
]
},
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"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
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" 1185.68,840.055 1189.64,842.617 1193.59,845.174 1197.55,847.725 1201.51,850.269 1205.46,852.809 1209.42,855.342 1213.37,857.87 1217.33,860.391 1221.29,862.908 \n",
" 1225.24,865.418 1229.2,867.923 1233.16,870.422 1237.11,872.916 1241.07,875.404 1245.02,877.886 1248.98,880.363 1252.94,882.834 1256.89,885.3 1260.85,887.76 \n",
" 1264.8,890.215 1268.76,892.664 1272.72,895.108 1276.67,897.546 1280.63,899.979 1284.59,902.406 1288.54,904.828 1292.5,907.245 1296.45,909.657 1300.41,912.063 \n",
" 1304.37,914.463 1308.32,916.859 1312.28,919.249 1316.24,921.634 1320.19,924.013 1324.15,926.388 1328.1,928.757 1332.06,931.121 1336.02,933.48 1339.97,935.834 \n",
" 1343.93,938.182 1347.88,940.525 1351.84,942.864 1355.8,945.197 1359.75,947.525 1363.71,949.848 1367.67,952.166 1371.62,954.479 1375.58,956.787 1379.53,959.09 \n",
" 1383.49,961.388 1387.45,963.681 1391.4,965.969 1395.36,968.252 1399.32,970.531 1403.27,972.804 1407.23,975.072 1411.18,977.336 1415.14,979.595 1419.1,981.849 \n",
" 1423.05,984.098 1427.01,986.342 1430.96,988.582 1434.92,990.816 1438.88,993.046 1442.83,995.272 1446.79,997.492 1450.75,999.708 1454.7,1001.92 1458.66,1004.13 \n",
" 1462.61,1006.33 1466.57,1008.52 1470.53,1010.72 1474.48,1012.9 1478.44,1015.09 1482.39,1017.27 1486.35,1019.44 1490.31,1021.61 1494.26,1023.77 1498.22,1025.93 \n",
" 1502.18,1028.09 1506.13,1030.24 1510.09,1032.39 1514.04,1034.53 1518,1036.67 1521.96,1038.8 1525.91,1040.93 1529.87,1043.06 1533.83,1045.18 1537.78,1047.29 \n",
" 1541.74,1049.4 1545.69,1051.51 1549.65,1053.61 1553.61,1055.71 1557.56,1057.81 1561.52,1059.9 1565.47,1061.98 1569.43,1064.06 1573.39,1066.14 1577.34,1068.21 \n",
" 1581.3,1070.28 1585.26,1072.34 1589.21,1074.4 1593.17,1076.46 1597.12,1078.51 1601.08,1080.56 1605.04,1082.6 1608.99,1084.64 1612.95,1086.67 1616.91,1088.7 \n",
" 1620.86,1090.73 1624.82,1092.75 1628.77,1094.77 1632.73,1096.79 1636.69,1098.8 1640.64,1100.8 1644.6,1102.8 1648.55,1104.8 1652.51,1106.8 1656.47,1108.79 \n",
" 1660.42,1110.77 1664.38,1112.75 1668.34,1114.73 1672.29,1116.71 1676.25,1118.68 1680.2,1120.64 1684.16,1122.6 1688.12,1124.56 1692.07,1126.52 1696.03,1128.47 \n",
" 1699.99,1130.41 1703.94,1132.36 1707.9,1134.3 1711.85,1136.23 1715.81,1138.16 1719.77,1140.09 1723.72,1142.01 1727.68,1143.93 1731.63,1145.85 1735.59,1147.76 \n",
" 1739.55,1149.67 1743.5,1151.57 1747.46,1153.48 1751.42,1155.37 1755.37,1157.27 1759.33,1159.16 1763.28,1161.04 1767.24,1162.92 1771.2,1164.8 1775.15,1166.68 \n",
" 1779.11,1168.55 1783.07,1170.42 1787.02,1172.28 1790.98,1174.14 1794.93,1176 1798.89,1177.85 1802.85,1179.7 1806.8,1181.55 1810.76,1183.39 1814.71,1185.23 \n",
" 1818.67,1187.06 1822.63,1188.9 1826.58,1190.72 1830.54,1192.55 1834.5,1194.37 1838.45,1196.19 1842.41,1198 1846.36,1199.81 1850.32,1201.62 1854.28,1203.42 \n",
" 1858.23,1205.22 1862.19,1207.02 1866.14,1208.81 1870.1,1210.6 1874.06,1212.39 1878.01,1214.17 1881.97,1215.95 1885.93,1217.73 1889.88,1219.5 1893.84,1221.27 \n",
" 1897.79,1223.04 1901.75,1224.8 1905.71,1226.56 1909.66,1228.32 1913.62,1230.07 1917.58,1231.82 1921.53,1233.57 1925.49,1235.31 1929.44,1237.05 1933.4,1238.79 \n",
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},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"let thistrader = 1\n",
" # All of the prices except this trader's are fixed.\n",
" fixedprices = 1.0*ones(n)\n",
" \n",
" # Range of prices that this trader will consider.\n",
" pricestoplot = 0.0:0.01:5\n",
" \n",
" # Demand when this trader shifts to one of those prices\n",
" # (and others prices are held fixed).\n",
" observeddemand = similar(pricestoplot)\n",
" \n",
" # Fill the demand container.\n",
" for (i, p) in enumerate(pricestoplot)\n",
" # Replaces this trader's prices with the new value.\n",
" fixedprices[thistrader] = pricestoplot[i]\n",
" # Observe demand and extract this trader's demand.\n",
" observeddemand[i] = demand(fixedprices)[thistrader]\n",
" end\n",
" \n",
" # Plot.\n",
" plot(pricestoplot, observeddemand,\n",
" xlabel=\"price\", ylabel=\"demand\",\n",
" label=\"demand curve\")\n",
" \n",
" hline!([q[thistrader]], label=\"quantity\")\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Equilibrium and t&acirc;tonnement processes\n",
"*Equilibrium* means that no trader has an incentive to change her price&mdash;that is, for all traders, `demand(p) ≤ q`, and we allow `demand(p)[i] < q[i]` only if `p[i] = 0`.\n",
"\n",
"Under some technical conditions on the demand function and how aggressively the traders update their prices, we can show that a t&acirc;tonnement process converges to an equilibrium.\n",
"\n",
"To find the equilibrium here, we will use a price update rule called the *simultaneous t&acirc;tonnement process.*\n",
"\n",
"`α` describes how reactive the traders are."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"7×5 Matrix{Any}:\n",
" :currentprice :demand :supply :diff :newprice\n",
" 0.0641437 0.307904 0.316058 -0.00815368 0.062513\n",
" 0.257614 0.132883 0.314829 -0.181946 0.221225\n",
" 0.139949 0.188913 0.0879828 0.100931 0.160135\n",
" 0.110997 0.229927 0.135174 0.0947533 0.129948\n",
" 0.18076 0.13899 0.136047 0.00294268 0.181348\n",
" 0.246536 0.00138312 0.00990956 -0.00852644 0.24483"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Simultaneous tâtonnement process\n",
"α = 0.2\n",
"function priceupdate(p)\n",
" res = max.(0, p + α * (demand(p) - q))\n",
"\n",
" # Normalize prices to sum to one.\n",
" # By homogeneity, this does not change the demand at the new prices.\n",
" res ./= sum(res)\n",
"\n",
" return res\n",
"end\n",
"\n",
"currentprices = rand(n)\n",
"currentprices ./= sum(currentprices)\n",
"\n",
"[ :currentprice :demand :supply :diff :newprice ;\n",
" currentprices demand(currentprices) q demand(currentprices) - q priceupdate(currentprices) ]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Notice that finding the new price requires only *observing* the demand. The traders do not need to \"know\" `A` or the equation for the demand."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5-element Vector{Vector{Float64}}:\n",
" [0.24999526764969154, 0.05379016810095744, 0.1840174685508432, 0.043633874197122585, 0.113627205339323, 0.3549360161620622]\n",
" [0.20686240711359824, 0.06180587386195269, 0.1885411717150587, 0.06779294835911062, 0.12356735001088376, 0.351430248939396]\n",
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]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function pricepath(p, T)\n",
" pricecontainer = Vector{Float64}[]\n",
"\n",
" for t in 1:T\n",
" push!(pricecontainer, p)\n",
" p = priceupdate(p)\n",
" end\n",
" \n",
" return pricecontainer\n",
"end\n",
"\n",
"\n",
"p0 = rand(n)\n",
"p0 ./= sum(p0)\n",
"T = 50\n",
"\n",
"pricecontainer = pricepath(p0, T)\n",
"pricecontainer[1:5]"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
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]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(hcat(pricecontainer...)', label=reshape([\"p$i\" for i in 1:n], 1, :))"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"┌ Info: Saved animation to \n",
"│ fn = C:\\Users\\Max\\418fc766c478eda3d13b976ff3330ab4\\tmp.gif\n",
"└ @ Plots C:\\Users\\Max\\.julia\\packages\\Plots\\vVVub\\src\\animation.jl:104\n"
]
},
{
"data": {
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/>"
],
"text/plain": [
"Plots.AnimatedGif(\"C:\\\\Users\\\\Max\\\\418fc766c478eda3d13b976ff3330ab4\\\\tmp.gif\")"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ymax = 1.02*maximum(maximum, pricecontainer)\n",
"\n",
"anim = @animate for t in 1:T\n",
" plot(hcat(pricecontainer[1:t]...)', label=reshape([\"p$i\" for i in 1:n], 1, :),\n",
" xlim=(1, T), ylim=(0, ymax))\n",
"end\n",
"\n",
"gif(anim, fps=10)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Equilibrium golf\n",
"Above, we used *simulation* to find the path of the prices over time. \n",
"\n",
"However, what if all we want to know are the equilibrium prices? Then there is no need to store the intermittent values of `p`."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6-element Vector{Float64}:\n",
" 0.03350699141417744\n",
" 0.1082420013987998\n",
" 0.2500169789174714\n",
" 0.1316339330508862\n",
" 0.19567665567476994\n",
" 0.2809234395438952"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function equilibrium(p, T)\n",
" # priceupdate(priceupdate(priceupdate( ... (p))))\n",
" for t in 1:T\n",
" p = priceupdate(p)\n",
" end\n",
" \n",
" return p\n",
"end\n",
"\n",
"p̄ = equilibrium(p0, 100)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We will know `p̄` is an equilibrium if calling `priceupdate()` yields the same value."
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p̄ ≈ priceupdate(p̄)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can make our equilibrium calculator even shorter using *function composition.*"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"15"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Compare\n",
"4 + 5 + 6"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"15"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"+(4, 5, 6)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.06825583369837018"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sin(exp(sqrt(5)))"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.06825583369837018"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(sin ∘ exp ∘ sqrt)(5)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.06825583369837018"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"∘(sin, exp, sqrt)(5)"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"tags": []
},
"outputs": [
{
"data": {
"text/plain": [
"8-element Vector{typeof(priceupdate)}:\n",
" priceupdate (generic function with 1 method)\n",
" priceupdate (generic function with 1 method)\n",
" priceupdate (generic function with 1 method)\n",
" priceupdate (generic function with 1 method)\n",
" priceupdate (generic function with 1 method)\n",
" priceupdate (generic function with 1 method)\n",
" priceupdate (generic function with 1 method)\n",
" priceupdate (generic function with 1 method)"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fill(priceupdate, 8)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6-element Vector{Float64}:\n",
" 0.03350699141417744\n",
" 0.1082420013987998\n",
" 0.2500169789174714\n",
" 0.1316339330508862\n",
" 0.19567665567476994\n",
" 0.2809234395438952"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"∘(fill(priceupdate, 100)...)(p0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Huge market\n",
"\n",
"When `n` increases, the cost of computing the equilibrium increases quickly, because\n",
"- `demand(p)` takes longer to compute, \n",
"- `priceupdate(p)` takes longer to compute, and\n",
"- more t&acirc;tonnement iterations are required to reach a fixed point.\n",
"\n",
"Nonetheless, we can still do pretty big markets. "
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [],
"source": [
"n = 800\n",
"\n",
"A = rand(n, n)\n",
"for i in 1:n\n",
" # We need to subtract a bigger number from the diagonal\n",
" # to ensure that A is well conditioned.\n",
" \n",
" # Without this, tâtonnement reaches a fixed point, but \n",
" # it has some demand(p)[j] > q[j].\n",
" A[i, i] -= 200\n",
"end\n",
"q = rand(n)\n",
"q ./= sum(q)\n",
"\n",
"p0 = rand(n)\n",
"p0 ./= sum(p0);"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 0.747121 seconds (14.00 k allocations: 37.567 MiB, 5.21% gc time)\n"
]
}
],
"source": [
"@time p̄ = equilibrium(p0, 1000);"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"isapprox(p̄, priceupdate(p̄), atol=1e-8)"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [],
"source": [
"demand(p̄) - q;"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
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"histogram(p̄, xlabel=\"price\", legend=false)"
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"## Further reading\n",
"\n",
"[Walras's Law](https://en.wikipedia.org/wiki/Walras%27s_law) generalizes our `sum(demand) = sum(q) = 1` assumption and argues for its realism.\n",
"\n",
"The market considered here is a special case of an [exchange economy](https://en.wikipedia.org/wiki/Exchange_economy) with one consumer whose behavior is encoded by the `demand(p)` function.\n",
"\n",
"Formal characterization of [competitive equilibrium](https://en.wikipedia.org/wiki/Competitive_equilibrium).\n",
"\n",
"#### Academic references:\n",
"\n",
"Codenotti, Bruno, and Kasturi Varadarajan. 2007. \"Computation of Market Equilibria by Convex Programming.\" In *Algorithmic Game Theory,* edited by Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V. Vazirani, 135&ndash;58. Cambridge University Press. \n",
"\n",
"Fisher, Franklin. 2013. \"The Stability of General Equilibrium: What Do We Know and Why Is It Important?\" In *General Equilibrium Analysis A Century after Walras,* edited by Pascal Bridel, 34&ndash;45. Routledge.\n",
"\n",
"Uzawa, H. 1960. \"Walras' T&acirc;tonnement in the Theory of Exchange.\" *The Review of Economic Studies* 27(3): 182&ndash;94. "
]
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