673 lines
80 KiB
HTML
673 lines
80 KiB
HTML
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<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"/><title>Unit 5: Factor Markets</title><style>
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/* cspell:disable-file */
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/* webkit printing magic: print all background colors */
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html {
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-webkit-print-color-adjust: exact;
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}
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* {
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box-sizing: border-box;
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-webkit-print-color-adjust: exact;
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}
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html,
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body {
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margin: 0;
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padding: 0;
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}
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@media only screen {
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body {
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margin: 2em auto;
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max-width: 900px;
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color: rgb(55, 53, 47);
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}
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}
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body {
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line-height: 1.5;
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white-space: pre-wrap;
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}
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a,
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a.visited {
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color: inherit;
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text-decoration: underline;
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}
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.pdf-relative-link-path {
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font-size: 80%;
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color: #444;
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}
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h1,
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h2,
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h3 {
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letter-spacing: -0.01em;
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line-height: 1.2;
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font-weight: 600;
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margin-bottom: 0;
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}
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.page-title {
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font-size: 2.5rem;
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font-weight: 700;
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margin-top: 0;
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margin-bottom: 0.75em;
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}
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h1 {
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font-size: 1.875rem;
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margin-top: 1.875rem;
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}
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h2 {
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font-size: 1.5rem;
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margin-top: 1.5rem;
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}
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h3 {
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font-size: 1.25rem;
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margin-top: 1.25rem;
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}
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.source {
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border: 1px solid #ddd;
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border-radius: 3px;
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padding: 1.5em;
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word-break: break-all;
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}
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.callout {
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border-radius: 3px;
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padding: 1rem;
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}
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figure {
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margin: 1.25em 0;
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page-break-inside: avoid;
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}
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figcaption {
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opacity: 0.5;
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font-size: 85%;
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margin-top: 0.5em;
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}
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mark {
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background-color: transparent;
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}
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.indented {
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padding-left: 1.5em;
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}
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hr {
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background: transparent;
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display: block;
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width: 100%;
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height: 1px;
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visibility: visible;
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border: none;
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border-bottom: 1px solid rgba(55, 53, 47, 0.09);
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}
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img {
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max-width: 100%;
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}
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@media only print {
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img {
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max-height: 100vh;
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object-fit: contain;
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}
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}
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@page {
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margin: 1in;
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}
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.collection-content {
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font-size: 0.875rem;
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}
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.column-list {
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display: flex;
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justify-content: space-between;
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}
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.column {
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padding: 0 1em;
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}
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.column:first-child {
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padding-left: 0;
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}
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.column:last-child {
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padding-right: 0;
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}
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.table_of_contents-item {
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display: block;
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font-size: 0.875rem;
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line-height: 1.3;
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padding: 0.125rem;
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}
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.table_of_contents-indent-1 {
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margin-left: 1.5rem;
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}
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.table_of_contents-indent-2 {
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margin-left: 3rem;
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}
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.table_of_contents-indent-3 {
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margin-left: 4.5rem;
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}
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.table_of_contents-link {
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text-decoration: none;
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opacity: 0.7;
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border-bottom: 1px solid rgba(55, 53, 47, 0.18);
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}
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table,
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th,
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td {
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border: 1px solid rgba(55, 53, 47, 0.09);
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border-collapse: collapse;
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}
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table {
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border-left: none;
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border-right: none;
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}
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th,
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td {
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font-weight: normal;
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padding: 0.25em 0.5em;
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line-height: 1.5;
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min-height: 1.5em;
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text-align: left;
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}
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th {
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color: rgba(55, 53, 47, 0.6);
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}
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ol,
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ul {
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margin: 0;
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margin-block-start: 0.6em;
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margin-block-end: 0.6em;
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}
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li > ol:first-child,
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li > ul:first-child {
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margin-block-start: 0.6em;
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}
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ul > li {
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list-style: disc;
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}
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ul.to-do-list {
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text-indent: -1.7em;
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}
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ul.to-do-list > li {
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list-style: none;
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}
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.to-do-children-checked {
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text-decoration: line-through;
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opacity: 0.375;
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}
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ul.toggle > li {
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list-style: none;
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}
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ul {
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padding-inline-start: 1.7em;
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}
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ul > li {
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padding-left: 0.1em;
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}
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ol {
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padding-inline-start: 1.6em;
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}
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ol > li {
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padding-left: 0.2em;
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}
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.mono ol {
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padding-inline-start: 2em;
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}
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.mono ol > li {
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text-indent: -0.4em;
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}
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.toggle {
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padding-inline-start: 0em;
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list-style-type: none;
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}
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/* Indent toggle children */
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.toggle > li > details {
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padding-left: 1.7em;
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}
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.toggle > li > details > summary {
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margin-left: -1.1em;
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}
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.selected-value {
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display: inline-block;
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padding: 0 0.5em;
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background: rgba(206, 205, 202, 0.5);
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border-radius: 3px;
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margin-right: 0.5em;
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margin-top: 0.3em;
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margin-bottom: 0.3em;
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white-space: nowrap;
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}
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.collection-title {
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display: inline-block;
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margin-right: 1em;
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}
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.simple-table {
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margin-top: 1em;
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font-size: 0.875rem;
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empty-cells: show;
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}
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.simple-table td {
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height: 29px;
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min-width: 120px;
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}
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.simple-table th {
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height: 29px;
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min-width: 120px;
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}
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.simple-table-header-color {
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background: rgb(247, 246, 243);
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color: black;
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}
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.simple-table-header {
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font-weight: 500;
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}
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time {
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opacity: 0.5;
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}
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.icon {
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display: inline-block;
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max-width: 1.2em;
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max-height: 1.2em;
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text-decoration: none;
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vertical-align: text-bottom;
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margin-right: 0.5em;
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}
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img.icon {
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border-radius: 3px;
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}
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.user-icon {
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width: 1.5em;
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height: 1.5em;
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border-radius: 100%;
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margin-right: 0.5rem;
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}
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.user-icon-inner {
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font-size: 0.8em;
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}
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.text-icon {
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border: 1px solid #000;
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text-align: center;
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}
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.page-cover-image {
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display: block;
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object-fit: cover;
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width: 100%;
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max-height: 30vh;
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}
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.page-header-icon {
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font-size: 3rem;
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margin-bottom: 1rem;
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}
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.page-header-icon-with-cover {
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margin-top: -0.72em;
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margin-left: 0.07em;
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}
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.page-header-icon img {
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border-radius: 3px;
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}
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.link-to-page {
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margin: 1em 0;
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padding: 0;
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border: none;
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font-weight: 500;
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}
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p > .user {
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opacity: 0.5;
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}
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td > .user,
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td > time {
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white-space: nowrap;
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}
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input[type="checkbox"] {
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transform: scale(1.5);
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margin-right: 0.6em;
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vertical-align: middle;
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}
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p {
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margin-top: 0.5em;
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margin-bottom: 0.5em;
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}
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.image {
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border: none;
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margin: 1.5em 0;
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padding: 0;
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border-radius: 0;
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text-align: center;
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}
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.code,
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code {
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background: rgba(135, 131, 120, 0.15);
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border-radius: 3px;
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padding: 0.2em 0.4em;
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border-radius: 3px;
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font-size: 85%;
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tab-size: 2;
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}
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code {
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color: #eb5757;
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}
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.code {
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padding: 1.5em 1em;
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}
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.code-wrap {
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white-space: pre-wrap;
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word-break: break-all;
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}
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.code > code {
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background: none;
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padding: 0;
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|||
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font-size: 100%;
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color: inherit;
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}
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blockquote {
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font-size: 1.25em;
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margin: 1em 0;
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padding-left: 1em;
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border-left: 3px solid rgb(55, 53, 47);
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}
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.bookmark {
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text-decoration: none;
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max-height: 8em;
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padding: 0;
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display: flex;
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width: 100%;
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align-items: stretch;
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}
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.bookmark-title {
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font-size: 0.85em;
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overflow: hidden;
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text-overflow: ellipsis;
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height: 1.75em;
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white-space: nowrap;
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}
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.bookmark-text {
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display: flex;
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flex-direction: column;
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}
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.bookmark-info {
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flex: 4 1 180px;
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padding: 12px 14px 14px;
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display: flex;
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flex-direction: column;
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justify-content: space-between;
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}
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.bookmark-image {
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width: 33%;
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flex: 1 1 180px;
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display: block;
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position: relative;
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|||
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object-fit: cover;
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border-radius: 1px;
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}
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.bookmark-description {
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color: rgba(55, 53, 47, 0.6);
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|||
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font-size: 0.75em;
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|||
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overflow: hidden;
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|||
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max-height: 4.5em;
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|||
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word-break: break-word;
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|||
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}
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.bookmark-href {
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|||
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font-size: 0.75em;
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|||
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margin-top: 0.25em;
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|||
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}
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|||
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.sans { font-family: ui-sans-serif, -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, "Apple Color Emoji", Arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol"; }
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|||
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.code { font-family: "SFMono-Regular", Menlo, Consolas, "PT Mono", "Liberation Mono", Courier, monospace; }
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|||
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.serif { font-family: Lyon-Text, Georgia, ui-serif, serif; }
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|||
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.mono { font-family: iawriter-mono, Nitti, Menlo, Courier, monospace; }
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|||
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.pdf .sans { font-family: Inter, ui-sans-serif, -apple-system, BlinkMacSystemFont, "Segoe UI", Helvetica, "Apple Color Emoji", Arial, sans-serif, "Segoe UI Emoji", "Segoe UI Symbol", 'Twemoji', 'Noto Color Emoji', 'Noto Sans CJK JP'; }
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</style></head><body><article id="0b3c331d-bc93-40d4-8b0c-736f2c58243e" class="page sans"><header><h1 class="page-title">Unit 5: Factor Markets</h1><table class="properties"><tbody><tr class="property-row property-row-created_time"><th><span class="icon property-icon"><svg viewBox="0 0 14 14" style="width:14px;height:14px;display:block;fill:rgba(55, 53, 47, 0.45);flex-shrink:0;-webkit-backface-visibility:hidden" class="typesCreatedAt"><path d="M7.01356 14.0001C8.8042 14.0001 10.5958 13.3107 11.9575 11.9324C14.681 9.21201 14.6808 4.7603 11.9571 2.04013C9.23336 -0.680043 4.77573 -0.680043 2.05199 2.04013C0.727519 3.36277 0 5.13301 0 6.99553C0 8.8764 0.727811 10.6285 2.05199 11.9509C3.43207 13.3106 5.22243 14.0001 7.01356 14.0001ZM3.72947 7.00914V8.461V8.65543H3.92382H5.34563H8.2794H8.4738V8.461V5.52541V3.37947V3.18502H8.2794H6.82747H6.63307V3.37947V6.81467H3.92382H3.72947V7.00914ZM1.83985 6.99553C1.83985 5.61698 2.38099 4.32597 3.36061 3.3477C5.36746 1.34337 8.64803 1.34062 10.6585 3.33944C10.6613 3.34219 10.6639 3.34494 10.6668 3.3477C12.676 5.3546 12.6763 8.63642 10.6668 10.6434C8.65705 12.6504 5.37031 12.6504 3.36061 10.6434C2.38099 9.66506 1.83985 8.37408 1.83985 6.99553Z"></path></svg></span>Created</th><td><time>@February 16, 2022 8:49 AM</time></td></tr><tr class="property-row property-row-multi_select"><th><span class="icon property-icon"><svg viewBox="0 0 14 14" style="width:14px;height:14px;display:block;fill:rgba(55, 53, 47, 0.45);flex-shrink:0;-webkit-backface-visibility:hidden" class="typesMultipleSelect"><path d="M4,3 C4,2.447715 4.447715,2 5,2 L12,2 C12.5523,2 13,2.447716 13,3 C13,3.55228 12.5523,4 12,4 L5,4 C4.447715,4 4,3.55228 4,3 Z M4,7 C4,6.447715 4.447715,6 5,6 L12,6 C12.5523,6 13,6.447716 13,7 C13,7.55228 12.5523,8 12,8 L5,8 C4.447715,8 4,7.55228 4,7 Z M4,11 C4,10.447715 4.447715,10 5,10 L12,10 C12.5523,10 13,10.447716 13,11 C13,11.55228 12.5523,12 12,12 L5,12 C4.447715,12 4,11.55228 4,11 Z M2,4 C1.44771525,4 1,3.55228475 1,3 C1,2.44771525 1.44771525,2 2,2 C2.55228475,2 3,2.44771525 3,3 C3,3.55228475 2.55228475,4 2,4 Z M2,8 C1.44771525,8 1,7.55228475 1,7 C1,6.44771525 1.44771525,6 2,6 C2.55228475,6 3,6.44771525 3,7 C3,7.55228475 2.55228475,8 2,8 Z M2,12 C1.44771525,12 1,11.5522847 1,11 C1,10.4477153 1.44771525,10 2,10 C2.55228475,10 3,10.4477153 3,11 C3,11.5522847 2.55228475,12 2,12 Z"></path></svg></span>Tags</th><td></td></tr></tbody></table></header><div class="page-body"><h1 id="0a22e4be-87a1-48d7-bdb8-52a58bc76ffb" class="">5.1 - Introduction to Factor Markets</h1><h2 id="e1d7bfc6-f8f0-4b1a-9afa-a88afb780961" class="">Key Vocabulary</h2><h3 id="8b6ad6c3-83f7-43fc-a445-888e3b82c191" class="">Marginal Product</h3><p id="2eabce58-0a2c-4481-8b50-8375f54032d2" class="">The additional product that is produced by each additional input. We calculated this number by dividing the change in the total product by the change in the number of inputs.</p><figure id="6358e40a-1982-4feb-830f-4545c0ca9c2e" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>MP</mtext><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>Y</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>X</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\textrm{MP} = \frac{\Delta Y}{\Delta X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord textrm">MP</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.04633em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfra
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licensing. Here are some examples of this determinant:</p><ul id="fab70045-ebbf-41fa-be33-97ef4f053647" class="bulleted-list"><li style="list-style-type:disc">If the government establishes a certification process that makes it harder to be an electrician than we would see the supply of electricians decrease shifting the labor supply curve to the left.</li></ul><h3 id="24651a07-b4ad-42e1-9117-efc71527d049" class="">Change in Societal Values</h3><p id="853e6be6-15b0-4175-a43e-a82bb4735470" class="">The third determinant is personal values regarding leisure time and societal roles. Here are some examples of this determinant.</p><ul id="b86f2149-eee8-4a0a-b836-f697f358fbc0" class="bulleted-list"><li style="list-style-type:disc">The increase in the labor force, especially women, during WWII because people saw it as a patriotic duty to help produce the goods that would help in the war efforts. This would increase the labor supply and shift the curve right.</li></ul><ul id="98c48eca-a281-475e-a522-269d28a18993" class="bulleted-list"><li style="list-style-type:disc">Low-skill workers decide that working at minimum wage isn't worth their time. This would shift the labor supply curve to the left due to the decrease in the amount of workers.</li></ul><h3 id="3f775200-1210-4f30-8820-7fa2f92c31f7" class="">Graphing Change in Factor Supply</h3><figure id="f9dec9c3-dd83-4d20-a1e6-e43a3213630d" class="image"><a href="Unit%205%20Factor%20Markets%200b3c331dbc9340d48b0c736f2c58243e/Untitled%202.png"><img style="width:1156px" src="Unit%205%20Factor%20Markets%200b3c331dbc9340d48b0c736f2c58243e/Untitled%202.png"/></a></figure><h1 id="7db9b7c8-0b37-4283-b8a0-b63ac5db157e" class="">5.3 - <strong>Profit-Maximizing Behavior in Perfectly Competitive Factor Markets</strong></h1><p id="bfff3279-78fd-4a20-bc77-66f3bb581147" class="">There are two types of factor markets. The first type is what is known as a perfectly competitive factor/resource market. There is significant use of labor as the type of factor (resource) in describing this type of factor market in AP Micro.</p><h2 id="347968ae-ed8d-40bd-bcc3-d7d9f7d5a8be" class="">Characteristics of Perfectly Competitive Labor (Factor) Markets</h2><p id="cf1fbef0-11d9-40e6-b3f9-6ee3b7b93073" class="">Perfectly competitive labor (factor markets) are very similar the perfectly competitive market structure EXCEPT that we are dealing with resources instead of goods and services.</p><p id="2c762d50-34de-4d52-8423-78fb34245141" class="">The characteristics of this type of factor (resource) market include:</p><ul id="595d8bd2-5d66-4626-abe8-d3b3141a8cc0" class="bulleted-list"><li style="list-style-type:disc">Many, small firms hiring workers</li></ul><ul id="78a28094-d88c-40cd-b6c9-838ebb3bac2e" class="bulleted-list"><li style="list-style-type:disc">Workers are "wage takers"</li></ul><ul id="cd8064b6-5222-4ffa-a69d-1bb841479ffc" class="bulleted-list"><li style="list-style-type:disc">Skill level of workers is identical (i.e. workers are perfect substitutes)</li></ul><ul id="7f7ca4de-a2d5-43aa-b4fe-f3531bd9d03d" class="bulleted-list"><li style="list-style-type:disc">Firms can hire as many workers as they need or want at the wage set in the market</li></ul><ul id="4d1405b1-d9fb-4a45-880b-df1b7f85f03b" class="bulleted-list"><li style="list-style-type:disc">Firms will hire workers as long as <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>MRP</mtext><mo>></mo><mtext>MRC</mtext></mrow><annotation encoding="application/x-tex">\textrm{MRP} > \textrm{MRC}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="mord text"><span cla
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<style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>MRC</mtext><mo>=</mo><mtext>wage</mtext></mrow><annotation encoding="application/x-tex">\textrm{MRC} = \textrm{wage}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord textrm">MRC</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord textrm">wage</span></span></span></span></span></span><span></span></span> in this type of factor market.</li></ul><h2 id="be526bd3-3154-4fb0-8d45-058ddc22390b" class="">Perfectly Competitive Labor Market Graphs</h2><p id="bc773913-d079-4b34-baa6-81a66ee95825" class="">In the perfectly competitive labor market, there is a downward-sloping demand curve because of the law of diminishing marginal returns. This means that each additional worker generates less revenue (MRP), and, therefore, is worth less to the firm. The supply curve for the labor market graph is upward-sloping because of the incentive to earn higher wages and greater income. If there are higher wages, it gives workers the incentive to give up leisure time and offer more of their time as workers. The same can be said for lower wages, which will deter workers from wanting to work more.</p><p id="9b78eb1e-63b8-406e-8ac3-6853079298f8" class="">
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</p><figure id="89221a34-923c-4fb5-88bc-6a4bf387eb5b" class="image"><a href="Unit%205%20Factor%20Markets%200b3c331dbc9340d48b0c736f2c58243e/Untitled%203.png"><img style="width:336px" src="Unit%205%20Factor%20Markets%200b3c331dbc9340d48b0c736f2c58243e/Untitled%203.png"/></a></figure><figure class="block-color-gray_background callout" style="white-space:pre-wrap;display:flex" id="6990c14e-d622-4d18-9a86-1309b2b05522"><div style="font-size:1.5em"><span class="icon">💡</span></div><div style="width:100%">Notice that <em><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">S_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span></em><em> </em>and <em><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">D_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><span></span></span></em><em> </em>are used to describe the supply and demand of labor. It is important to use the subscript <style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><span data-token-index="0" contenteditable="false" class="notion-text-equation-token" style="user-select:all;-webkit-user-select:all;-moz-user-select:all"><span></span><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">L</span></span></span></span></span><span></span></span> when you are drawing graphs.</div></figure><h3 id="5e8a13df-4111-4b09-9d62-a4973f55290f" class="">Firm Graph
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</p><figure id="2e575c50-1259-40ec-8ea7-5b4d1fba5f8e" class="equation"><style>@import url('https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css')</style><div class="equation-container"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><msub><mtext>MP</mtext><mi>x</mi></msub><msub><mi>P</mi><mi>x</mi></msub></mfrac><mo>=</mo><mfrac><msub><mtext>MP</mtext><mi>y</mi></msub><msub><mi>P</mi><mi>y</mi></msub></mfrac></mrow><annotation encoding="application/x-tex">\frac{\textrm{MP}_x}{P_x} = \frac{\textrm{MP}_y}{P_y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.19633em;vertical-align:-0.8360000000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.3139999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord text"><span class="mord textrm">MP</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.332438em;vertical-align:-0.972108em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><
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