31:T416,y_vs_alpha = { const svg = d3.select(DOM.svg(width, height)); const bar = svg.append("g") .attr("fill", "steelblue") .selectAll("rect") .data(data_y_vs_alpha) .enter().append("rect") .attr("x", (d, i) => x_alphas(alphas[i])) .attr("width", 1) .attr("y", (d, i) => y_y_vs_alpha(d)) .attr("height", (d, i) => height - margin.bottom - y_y_vs_alpha(d)); svg.append("g") .call(xAxis_alphas); svg.append("g") .call(yAxis_y_vs_alpha); svg.append("text") // x-axis label .attr("transform", "translate(" + (width/2) + " ," + height + ")") .style("text-anchor", "middle") .style("font", "12px sans-serif") // .attr("dx", "0.75em") .text("α"); svg.append("text") // y-axis label .attr("transform", "rotate(-90)") .attr("y", 0) .attr("x",0 - (height / 2)) .attr("dy", "0.75em") .style("font", "12px sans-serif") .style("text-anchor", "middle") .text("Output (GDP)"); return svg.node(); }32:T4bf,labor_market_equilibrium = { const svg = d3.select(DOM.svg(width, height)); const ls_line = svg.append("line") .style("stroke", "black") // colour the line .attr("y1", y_ws(ws[0])) // x position of the first end of the line .attr("x1", x_ls(data_ls_vs_w[0])) // y position of the first end of the line .attr("y2", y_ws(ws[50])) // x position of the second end of the line .attr("x2", x_ls(data_ls_vs_w[50])); // y position of the second end of the line const ld_line = svg.append("line") .style("stroke", "black") // colour the line .attr("y1", y_ws(ws[0])) // x position of the first end of the line .attr("x1", x_ls(data_ld_vs_w[0])) // y position of the first end of the line .attr("y2", y_ws(ws[50])) // x position of the second end of the line .attr("x2", x_ls(data_ld_vs_w[50])); // y position of the second end of the line const labor_eq = svg.append("circle") .attr("r", 5) .attr("stroke", "red") .attr("fill", "red") .attr("cx", x_ls(l_star)) .attr("cy", y_ws(w_star)) svg.append("g") .call(xAxis_ws); svg.append("g") .call(yAxis_l_vs_w); return svg.node(); }33:T59e,l_vs_w = { /* const svg = d3.select(DOM.svg(width, height)); const bar1 = svg.append("g") .attr("fill", "steelblue") .selectAll("rect") .data(data_ls_vs_w) .enter().append("rect") .attr("x", (d, i) => x_ls(d)) .attr("width", 1) .attr("y", (d, i) => y_ws(i)) .attr("height", (d, i) => height - margin.bottom - y_ws(i)); const bar2 = svg.append("g") .attr("fill", "darkred") .selectAll("rect") .data(data_ld_vs_w) .enter().append("rect") .attr("x", (d, i) => x_ls(d)) .attr("width", 1) .attr("y", (d, i) => y_ws(i)) .attr("height", (d, i) => height - margin.bottom - y_ws(i)); /* const ld_line = d3.svg.line() .x((d, i) => return x(i);) .y((d, i) => return y(d);) svg.append("g") .call(xAxis_ws); svg.append("g") .call(yAxis_l_vs_w); svg.append("text") // x-axis label .attr("transform", "translate(" + (width/2) + " ," + height + ")") .style("text-anchor", "middle") .style("font", "12px sans-serif") // .attr("dx", "0.75em") .text("Labor (demanded, supplied)"); svg.append("text") // y-axis label .attr("transform", "rotate(-90)") .attr("y", 0) .attr("x",0 - (height / 2)) .attr("dy", "0.75em") .style("font", "12px sans-serif") .style("text-anchor", "middle") .text("Real wage"); return svg.node();*/ }34:[] 2f:{"title":"A visual introduction to the IS-LM-FE model","cells":[{"id":0,"value":"md`# A visual introduction to the IS-LM-FE model`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1156,"value":"md`\\`NOTE: THIS IS AN EXTREMELY EARLY AND TENTATIVE VERSION OF THIS DOCUMENT. IT IS EXPECTED TO CHANGE ENORMOUSLY AS I LEARN MORE MACROECONOMICS AND MORE d3.js!\\``","pinned":false,"hidden":false,"mode":"ojs"},{"id":2,"value":"md`The IS-LM-FE model is perhaps the simplest model of the macroeconomy that nevertheless captures many of the most important macroeconomic ebbs and flows that govern the modern economy.\n\nIn this document, we will examine a version of the IS-LM-FE model, based upon the version presented in *Macroeconomics* by Abel, Bernanke, and Croushoure (henceforth \\`ABC\\`).`","pinned":false,"hidden":false,"mode":"ojs"},{"id":790,"value":"md`As the name suggests, the IS-LM-FE model has three components. It looks at the conditions under which the economy reaches __general equilibrium__, a state of simultaneous equilibrium in the three key component markets of the economy: the __labor market__, the __goods market__, and the __asset market__. Let us look at each of these markets separately before putting them all together in the IS-LM-FE model.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":6,"value":"md`## The labor market`","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"md`The labor market is concerned with the supply and demand of labor in the economy. Labor matters because it is both a key input for the *production* of economic output as well as a determinant of the wages earned by people.\n\nWe begin with the production function, which relates economic output *Y* to the __total factor productivity__ *A*, the __capital stock__ *K*, and the __labor input__ *L*:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":18,"value":"tex.block`Y(K, L; A) = A f(K, L),`","pinned":false,"hidden":false,"mode":"ojs"},{"id":21,"value":"md`where *f* can be any function with the required characteristics. One of the most common forms that this function takes is the __Cobb-Douglas function__, which takes an additional parameter α that represents the percentage share of the economy contributed by one of the two inputs (say labor).\n\nIn this case, the economic output *Y* is given by`","pinned":false,"hidden":false,"mode":"ojs"},{"id":26,"value":"tex.block`Y(K, L; A, \\alpha) = A K^{1 - \\alpha} L^{\\alpha}.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":203,"value":"Y_CD = function (K, L) {\n\tif ((A < 0) || (K < 0) || (L < 0) || (alpha < 0) || (alpha > 1)) {\n return NaN;\n } else {\n return A * math.pow(K, 1 - alpha) * math.pow(L, alpha);\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":257,"value":"md`We can now choose some values for the total factor productivity and for α (keeping in mind that labor’s share of the economy is currently around 70%):`","pinned":false,"hidden":false,"mode":"ojs"},{"id":215,"value":"viewof A = {\n const div = html`

\n \n \n\tA\n

\n \n \n\tα\n

`;\n const range = div.querySelector(\"[type=range]\");\n const number = div.querySelector(\"[type=number]\");\n div.value = range.value = number.value = 0.7;\n range.addEventListener(\"input\", () => number.value = div.value = range.valueAsNumber);\n number.addEventListener(\"input\", () => range.value = div.value = number.valueAsNumber);\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":525,"value":"md`To see the effect of different levels of α on output, enter values for *K* and *L* here:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":529,"value":"viewof K_test = {\n const div = html`

\n \n\tK\n

`;\n const number = div.querySelector(\"[type=number]\");\n div.value = number.value = 1;\n number.addEventListener(\"input\", () => div.value = number.valueAsNumber);\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":533,"value":"viewof L_test = {\n const div = html`

\n \n\tL\n

`;\n const number = div.querySelector(\"[type=number]\");\n div.value = number.value = 10;\n number.addEventListener(\"input\", () => div.value = number.valueAsNumber);\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":570,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":45,"value":"md`Given these parameters and input levels, we can then obtain the __marginal product of labor__ *MP_L* by differentiating *Y* with regard to *L*:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":55,"value":"tex.block`MP_L(K, L) := \\frac{\\partial Y}{\\partial L} (K, L; A, \\alpha) = \\left( A \\cdot K^{1- \\alpha} \\cdot \\alpha \\right) L^{\\alpha-1} `","pinned":false,"hidden":false,"mode":"ojs"},{"id":65,"value":"md`If we chose a different production function, say a quadratic production function in which capital stock is assumed to be a constant, we would have a different expression for *Y*:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":68,"value":"tex.block`Y(L; A, f_1, f_2) = A \\left( f_1 L - \\tfrac{1}{2} f_2 L^2 \\right)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":213,"value":"Y_quad = function (L) {\n\tif ((A < 0) || (L < 0)) {\n return NaN;\n } else {\n return A * (f1 * L - 0.5 * f2 * L * L);\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":261,"value":"md`and hence for *MP_L* as well:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":72,"value":"tex.block`\\implies MP_L(L) := \\frac{dY}{dL} = A \\left( f_1 - f_2 L \\right).`","pinned":false,"hidden":false,"mode":"ojs"},{"id":74,"value":"md`This latter expression is the one used in \\`ABC\\` for the derivation of the IS-LM-FE model, and so I will stick with it for now.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":267,"value":"md`Let us now choose some values for the two parameters *f*₁ and *f*₂, so that we can do some computations with this expression later. (Unlike α, these are not standard economic parameters to the best of my knowledge, and don’t have any specific names.)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":229,"value":"f1 = 11","pinned":false,"hidden":false,"mode":"ojs"},{"id":232,"value":"f2 = 5","pinned":false,"hidden":false,"mode":"ojs"},{"id":80,"value":"md`### Labor demand *L^D*`","pinned":false,"hidden":false,"mode":"ojs"},{"id":85,"value":"md`Under our assumption that all firms are identical, they are going to face the same equation above, and ask themselves the same question: How much labor do we want?\n\nIf firms are *Homo economicus* writ large, then the answer should be straightforward: a firm will hire more and more labor, until the additional *benefit* of adding one more unit of labor is exactly canceled out by the additional *cost* of hiring that additional unit of labor. The additional benefit from an extra unit of labor is, of course, nothing other than the marginal productivity of labor; the additional cost is simply the real wage *w*.\n\nThus the labor demand *L^D* is the value of *L* such that`","pinned":false,"hidden":false,"mode":"ojs"},{"id":88,"value":"tex.block`MP_L(L^D) - w = 0 \\implies A \\left( f_1 - f_2 L^D \\right) - w = 0.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":118,"value":"md`Assume that firms don't have the power to alter *A*, *f₁*, or *f₂*. Then firms’ labor demand is a function of the real wage:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":120,"value":"tex.block`L^D(w) = \\frac{f_1}{f_2} - \\frac{1}{A f_2} w.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":489,"value":"labor_demand = function (wage) {\n\tif ((f1 < 0) || (f2 < 0) || (A < 0) || (wage < 0)) {\n return NaN;\n } else {\n return math.eval(ld, {A: A, f1: f1, f2: f2, w: wage});\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":97,"value":"md`This linear equation in two variables, *L^D* and *w*, is the equation of the labor demand curve in (*w*, *L*)-space.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":81,"value":"md`### Labor supply *L^S*`","pinned":false,"hidden":false,"mode":"ojs"},{"id":96,"value":"md`What happens on the labor supply side? We assume a population of identical *Homo economicus* workers, each of whom chooses between labor and leisure when facing the real wage *w* prevailing in the labor market. \n\nIf we chose to do so, we could craft a utility function for the workers, and use that to determine the optimal allocation of labor versus leisure. However, to keep things tractable for now, let’s simply note our intuition that, as the real wage rises, the labor supply *L^S* should increase as well. (Of course, given that some fraction of this real wage is going to go towards taxes, we should strictly look only at the after-tax real wage, *(1-t)w*.)\n\nTo keep things simple, let us suppose that this relationship between the labor supplied and the real wage is a linear relationship: `","pinned":false,"hidden":false,"mode":"ojs"},{"id":100,"value":"tex.block`L^S(w) = l_0 + l_w (1-t) w,`","pinned":false,"hidden":false,"mode":"ojs"},{"id":500,"value":"labor_supply = function (wage) {\n\tif ((l0 < 0) || (lw < 0) || (wage < 0)) {\n return NaN;\n } else {\n return math.eval(ls, {l0: l0, lw: lw, t: t, w: wage});\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":102,"value":"md`where *l₀* represents the base labor supply (the number of workers who will work even for free), *l_w* is the rate at which labor supply increases with an increase in the real wage, and *t* is the effective tax rate faced by the workforce.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":348,"value":"viewof l0 = {\n const div = html`

\n \n \n\tl₀\n

\n \n \n\tl_w\n

\n \n \n\tt (effective tax rate)\n

`;\n const range = div.querySelector(\"[type=range]\");\n const number = div.querySelector(\"[type=number]\");\n div.value = range.value = number.value = 0.35;\n range.addEventListener(\"input\", () => number.value = div.value = range.valueAsNumber);\n number.addEventListener(\"input\", () => range.value = div.value = number.valueAsNumber);\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":114,"value":"md`This equation in two variables, *L^S* and *w*, defines the labor supply curve in (*w*, *L*)-space.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":106,"value":"md`### Labor market equilibrium (*L^D* = *L^S*)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":110,"value":"md`The labor market is *in equilibrium* when the labor demand is precisely equal to the labor supply. In graphical terms, this represents the point of intersection in (*w*, *L*)-space of the labor demand curve and the labor supply curve. This point is usually symbolically represented by (_w\\*_, _L\\*_), with _w\\*_ being the __equilibrium real wage__ and _L\\*_ being the __equilibrium labor__ (also known, somewhat misleadingly, as the __full-employment level of labor__).\n\nMathematically,`","pinned":false,"hidden":false,"mode":"ojs"},{"id":112,"value":"tex.block`L^D(w^{\\star}) = L^S(w^{\\star}) = L^{\\star},`","pinned":false,"hidden":false,"mode":"ojs"},{"id":117,"value":"md`so that we can now solve the two equations simultaneously to get`","pinned":false,"hidden":false,"mode":"ojs"},{"id":139,"value":"tex.block`\\Bigg\\{\\,\n\\begin{aligned}\n L^{\\star} &= \\frac{f_1}{f_2} - \\frac{1}{A f_2} w^{\\star} \\\\\n L^{\\star} &= l_0 + l_w (1-t) w^{\\star}\n\\end{aligned}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":148,"value":"tex.block`\\implies\n\\begin{aligned}\n L^{\\star} &= \\dfrac{l_0 + A f_1 l_w (1-t)}{1 + A f_2 l_w (1-t)} \\\\\n w^{\\star} &= \\dfrac{f_1 - f_2 l_0}{1 + A f_2 l_w (1-t)} \\cdot A\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":366,"value":"md`For the parameters we have entered above, the labor market equilibrium is at:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":367,"value":"tex.block`\\begin{aligned}\n L^{\\star} &= ${round(l_star)} \\\\\n w^{\\star} &= ${round(w_star)}.\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":158,"value":"md`Given equilibrium in the labor market, the economy is said to be at the __full-employment level of output__ *Y^FE*. This is given by`","pinned":false,"hidden":false,"mode":"ojs"},{"id":171,"value":"tex.block`Y^{FE}(L^{\\star}; A, f_1, f_2, l_0, l_w, t) = A \\left( f_1 L^{\\star} -\\tfrac{1}{2} f_2 \\left( L^{\\star} \\right) ^{2} \\right).`\n/* \ntex.block` \n\\\\begin{aligned}\n Y^{FE}(L^{\\\\star}; A, f_1, f_2, l_0, l_w) &= A \\\\left( f_1 L^{\\\\star} -\\\\tfrac{1}{2} f_2 \\\\left( L^{\\\\star} \\\\right) ^{2} \\\\right) \\\\\\\\\n &= A \\\\left( f_1 \\\\frac{l_0 + A f_1 l_w}{1 + A f_2 l_w} -\\\\tfrac{1}{2} f_2 \\\\left( \\\\frac{l_0 + A f_1 l_w}{1 + A f_2 l_w} \\\\right) ^{2} \\\\right)\n\\\\end{aligned}` \n*/","pinned":false,"hidden":false,"mode":"ojs"},{"id":431,"value":"md`For the parameters above, the full-employment level of output is equal to`","pinned":false,"hidden":false,"mode":"ojs"},{"id":432,"value":"tex.block`Y^{FE} = ${round(Y_FE)}.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":618,"value":"md`### Graphing the labor market equilibrium`","pinned":false,"hidden":false,"mode":"ojs"},{"id":620,"value":"md`We have already seen that the labor demand curve and the labor supply curve are both drawn in (*w*, *L*)-space, and intersect at the labor market equilibrium of (_w\\*_, _L\\*_). We can depict this visually as well. \n\nNote, however, that the full-employment level of output *Y^FE* does not show up here. Traditionally, that is depicted instead in (*r*, *Y*)-space, where *r* is the __real interest rate__. We have not talked about the real interest rate so far because in this model it does not affect the labor market. The upshot of this is that *Y^FE*, although determined by the goings-on in the labor market, will be a straight vertical line in (*r*, *Y*)-space.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":899,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":769,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"md`## The goods market`","pinned":false,"hidden":false,"mode":"ojs"},{"id":801,"value":"md`The goods market is concerned with two things: individuals’ decisions to allocate their resources between consumption and saving, and firms’ need to borrow resources in order to invest for future growth.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":802,"value":"md`### Savings`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1160,"value":"md`In the IS-LM-FE model’s account, people can choose to either save or consume. According to \\`ABC\\`, the __desired consumption__ *C^D* can be modeled by:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1161,"value":"tex.block`C^D(Y, r; c_0, c_Y, c_r, t_0, t) = c_0 + c_Y Y^{\\text{disp}}(Y; t_0, t) - c_r r,`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1163,"value":"md`where *Y*^disp is __disposable income__ and *r* is the __real interest rate__. \n\nThe three *c*-parameters here are important:\n* *c*₀ captures the effect on desired consumption of all variables other than income and interest rate.\n* *c_Y* is the __marginal propensity to consume__; it is the fraction of disposable income that people are willing to consume (saving the rest).\n* *c_r* captures the sensitivity of desired consumption to changes in the real interest rate; as the rate rises, people will consume less because their money will be worth more in the future.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1242,"value":"viewof c0 = {\n const div = html`

\n \n \n\tc₀\n

\n \n \n\tc_Y (marginal propensity to consume)\n

\n \n \n\tc_r\n

\n \n \n\tt₀ (lump-sum tax)\n

\n \n \n\tG (government spending)\n

\n \n \n\ti₀\n

\n \n \n\ti_r\n

`;\n const range = div.querySelector(\"[type=range]\");\n const number = div.querySelector(\"[type=number]\");\n div.value = range.value = number.value = 1;\n range.addEventListener(\"input\", () => number.value = div.value = range.valueAsNumber);\n number.addEventListener(\"input\", () => range.value = div.value = number.valueAsNumber);\n return div;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1400,"value":"investment = function (real_interest_rate) {\n\treturn math.eval(ID, {i0: i0, ir: ir, r: real_interest_rate});\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1439,"value":"md`This is an equation in two variables, *I^D* and *r*, and hence defines a curve known as the *I*-curve in (*r*, *S* ≡ *I*)-space. \n\nWe could also use this expression to draw a surface in (*r*, *Y*)-space representing the desired level of investment for every combination of income and real interest rate.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":806,"value":"md`### Goods market equilibrium (*S^D* = *I^D*)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1445,"value":"md`The goods market is said to be in equilibrium when the desired level of saving equals the desired level of investment. Everything that is saved is thus invested into increasing future capital. This equilibrium is effected by the banks, who adjust the real interest rate to equate desired saving and desired investment.\n\nSince we have algebraic expressions for both desired saving and desired investment, we can write them out as follows:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1447,"value":"tex.block`\\begin{aligned}\nS^D &= I^D \\\\\n\\implies \\left[ [1 - c_Y (1-t)] Y_0 + (c_Y t_0 - c_0 - G) \\right] + c_r r &= i_0 -i_r r \\\\\n\\implies (1-c_Y)(1-t) Y + (i_r + c_r) r &= i_0 + c_0 + (1-c_Y)t_0 \\\\\n\\implies \\frac{(1-c_Y)(1-t)}{i_0 + c_0 + (1-c_Y)t_0} Y + \\frac{i_r + c_r}{i_0 + c_0 + (1-c_Y)t_0} r &= 1.\n\\end{aligned}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1466,"value":"md`This equation in the two variables *Y* and *r* defines a curve in (*r*, *Y*)-space known as the IS-curve: it is the locus of points that represent goods market equilibrium.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":810,"value":"md`### Graphing the goods market equilibrium`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1454,"value":"md`Graphically, the goods market equilibrium occurs—for a fixed value of income of *Y*₀—at the point of intersection of the *S*-curve for *Y*₀ and the *I*-curve in (*r*, *S* ≡ *I*)-space. This point is usually denoted (_r\\*_, _S\\*_ ≡ _I\\*_).`","pinned":false,"hidden":false,"mode":"ojs"},{"id":9,"value":"md`## The asset market`","pinned":false,"hidden":false,"mode":"ojs"},{"id":812,"value":"md`The asset market is concerned with money (cash) and other financial assets. For simplicity, we may assume that there is only one asset alternative to cash.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":813,"value":"md`### Money demand (*M^D*)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":815,"value":"md`### Money supply (*M^S*)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":818,"value":"md`### Asset market equilibrium (*M^D* = *M^S*)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"md`## General equilibrium`","pinned":false,"hidden":false,"mode":"ojs"},{"id":30,"value":"md`## Under the hood`","pinned":false,"hidden":false,"mode":"ojs"},{"id":32,"value":"d3 = require('d3')","pinned":false,"hidden":false,"mode":"ojs"},{"id":35,"value":"math = require(\"https://cdnjs.cloudflare.com/ajax/libs/mathjs/3.20.2/math.min.js\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":824,"value":"md`### Expressions for evaluation`","pinned":false,"hidden":false,"mode":"ojs"},{"id":269,"value":"yq = \"A * (f_1 * L - 0.5 * f_2 * L^2)\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":271,"value":"math.eval(yq, {f_1: f1, L: 10, f_2: f2, A:A})","pinned":false,"hidden":false,"mode":"ojs"},{"id":285,"value":"math.eval(math.derivative(yq, \"L\").toString(), {f_1: f1, L: 10, f_2: f2, A:A})","pinned":false,"hidden":false,"mode":"ojs"},{"id":424,"value":"y_cd = \"A * K^(1 - alpha) * L^alpha\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":48,"value":"tex.block`${math.derivative(y_cd, \"K\").toTex()}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":491,"value":"ld = \"f1 / f2 - (1 / (A * f2)) * w\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":503,"value":"ls = \"l0 + lw * (1 - t) *w\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":1252,"value":"CD = \"c0 + cY * YD - cr * r\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":1203,"value":"T = \"t0 + t * Y\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":1313,"value":"SD = \"(1 - cY) * YD + cr * r - c0\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":1398,"value":"ID = \"i0 - ir * r\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":426,"value":"math.eval(y_cd, {A: A, K: 10, L: 11, alpha: alpha})","pinned":false,"hidden":false,"mode":"ojs"},{"id":187,"value":"l_star = {\n\tif (f1 <= f2 * l0) {\n \treturn NaN;\n } else {\n \treturn (l0 + A * f1 * lw * (1-t)) / (1 + A * f2 * lw * (1-t));\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":199,"value":"w_star = {\n\tif (f1 <= f2 * l0) {\n \treturn NaN;\n } else {\n \treturn A * (f1 - f2 * l0) / (1 + A * f2 * lw * (1-t));\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":345,"value":"Y_FE = {\n\treturn Y_quad(l_star);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":439,"value":"round = function(datum) {\n\treturn Math.round((datum + 0.0000001) * 10000) / 10000;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":297,"value":"tex.block`${math.parse(yq).toTex()}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":740,"value":"md`### Charting variables`","pinned":false,"hidden":false,"mode":"ojs"},{"id":544,"value":"margin = ({top: 20, right: 20, bottom: 30, left: 40})","pinned":false,"hidden":false,"mode":"ojs"},{"id":537,"value":"height = 500","pinned":false,"hidden":false,"mode":"ojs"},{"id":741,"value":"md`#### *Y* versus *α*`","pinned":false,"hidden":false,"mode":"ojs"},{"id":515,"value":"alphas = [...Array(101).keys()].map(x => x/100)","pinned":false,"hidden":false,"mode":"ojs"},{"id":520,"value":"data_y_vs_alpha = alphas.map(x => math.eval(y_cd, {A: A, K: K_test, L: L_test, alpha: x}))","pinned":false,"hidden":false,"mode":"ojs"},{"id":542,"value":"x_alphas = d3.scaleLinear()\n .domain([0, 1]).nice()\n .range([margin.left, width - margin.right])","pinned":false,"hidden":false,"mode":"ojs"},{"id":552,"value":"y_y_vs_alpha = d3.scaleLinear()\n .domain([0, 20]).nice()\n .range([height - margin.bottom, margin.top])","pinned":false,"hidden":false,"mode":"ojs"},{"id":564,"value":"xAxis_alphas = g => g\n .attr(\"transform\", `translate(0,${height - margin.bottom})`)\n .call(d3.axisBottom(x_alphas).tickSizeOuter(0))\n .call(g => g.append(\"text\")\n .attr(\"x\", width - margin.right)\n .attr(\"y\", -4)\n .attr(\"fill\", \"#000\")\n .attr(\"font-weight\", \"bold\")\n .attr(\"text-anchor\", \"end\")\n .text(alphas.x))","pinned":false,"hidden":false,"mode":"ojs"},{"id":568,"value":"yAxis_y_vs_alpha = g => g\n .attr(\"transform\", `translate(${margin.left},0)`)\n .call(d3.axisLeft(y_y_vs_alpha))\n .call(g => g.select(\".domain\").remove())","pinned":false,"hidden":false,"mode":"ojs"},{"id":750,"value":"md`#### Labor market equilibrium`","pinned":false,"hidden":false,"mode":"ojs"},{"id":751,"value":"ws = [...Array(51).keys()].map(x => x/10)","pinned":false,"hidden":false,"mode":"ojs"},{"id":838,"value":"data_ld_vs_w = ws.map(w => labor_demand(w))","pinned":false,"hidden":false,"mode":"ojs"},{"id":843,"value":"data_ls_vs_w = ws.map(w => labor_supply(w))","pinned":false,"hidden":false,"mode":"ojs"},{"id":910,"value":"ls_line = ({p1: ({y: ws[0], x: data_ls_vs_w[0]}), p2: ({y: ws[50], x: data_ls_vs_w[50]})})","pinned":false,"hidden":false,"mode":"ojs"},{"id":863,"value":"x_ls = d3.scaleLinear()\n .domain([0, 10]).nice()\n .range([margin.left, width - margin.right])","pinned":false,"hidden":false,"mode":"ojs"},{"id":913,"value":"x_ls(ls_line.p2.x)","pinned":false,"hidden":false,"mode":"ojs"},{"id":755,"value":"y_ws = d3.scaleLinear()\n .domain([0, 10]).nice()\n .range([height - margin.bottom, margin.top])","pinned":false,"hidden":false,"mode":"ojs"},{"id":765,"value":"xAxis_ws = g => g\n .attr(\"transform\", `translate(0,${height - margin.bottom})`)\n .call(d3.axisBottom(x_ls))\n .call(g => g.select(\".domain\").remove())","pinned":false,"hidden":false,"mode":"ojs"},{"id":767,"value":"yAxis_l_vs_w = g => g\n .attr(\"transform\", `translate(${margin.left},0)`)\n .call(d3.axisLeft(y_ws))\n .call(g => g.select(\".domain\").remove())","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W34","stdlib":"1","titling":"auto"}