{"id":203873,"date":"2025-03-21T02:27:17","date_gmt":"2025-03-21T02:27:17","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=203873"},"modified":"2025-03-21T02:27:19","modified_gmt":"2025-03-21T02:27:19","slug":"optimal-quantity-of-public-goods-that-the-government-should-provid","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/21\/optimal-quantity-of-public-goods-that-the-government-should-provid\/","title":{"rendered":"optimal quantity of public goods that the government should provid"},"content":{"rendered":"\n

optimal quantity of public goods that the government should provide, and interpret your results. Make sure you show all of the relevant cases. What happens when b changes, or when q changes?<\/p>\n\n\n\n

(b) Repeat part (a), except with perfect complements preferences; that is, for the case where the representative consumer always wishes to consume private consumption goods and public goods in fixed proportions, or C = aG, with a > 0.<\/p>\n\n\n\n

Consider the model of public goods in the last section of this chapter.<\/p>\n\n\n\n

(a) Suppose that preferences over private consumption C<\/strong> and public goods G<\/strong> are such that these two goods are perfect substitutes; that is, the marginal rate of substitution of public goods for private goods is a constant b > 0<\/strong>. Determine the optimal quantity of public goods that the government should provide, and interpret your results. Make sure you show all of the relevant cases. What happens when b<\/strong> changes, or when q<\/strong> changes?<\/p>\n\n\n\n

(b) Repeat part (a), except with perfect complements preferences; that is, for the case where the representative consumer always wishes to consume private consumption goods and public goods in fixed proportions, or C = aG,<\/strong> with a > 0.<\/strong><\/p>\n\n\n\n

The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Optimal Quantity of Public Goods Provision by the Government<\/strong><\/h3>\n\n\n\n

(a) Perfect Substitutes Case<\/strong><\/h4>\n\n\n\n

When private consumption CC and public goods GG are perfect substitutes, the consumer\u2019s utility function implies a constant marginal rate of substitution (MRS) between the two goods: MRS=bMRS = b<\/p>\n\n\n\n

where b>0b > 0 represents the rate at which the consumer is willing to trade public goods for private consumption.<\/p>\n\n\n\n

The government\u2019s budget constraint is: C+qG=YC + qG = Y<\/p>\n\n\n\n

where qq is the marginal cost of providing one unit of GG and YY is total income.<\/p>\n\n\n\n

Since the consumer views CC and GG as perfect substitutes, they will only consume public goods if: b\u2265qb \\geq q<\/p>\n\n\n\n

In this case, the government should allocate all resources to GG, providing G=YqG = \\frac{Y}{q} and C=0C = 0.<\/p>\n\n\n\n

If b<qb < q, the consumer values private consumption more, so the government should provide no public goods (G=0G = 0) and allocate all resources to C=YC = Y.<\/p>\n\n\n\n

Comparative Statics<\/strong><\/h4>\n\n\n\n
    \n
  • When bb increases<\/strong>: The consumer places a higher value on public goods, so the government will allocate more resources to GG.<\/li>\n\n\n\n
  • When qq increases<\/strong>: The cost of public goods rises, so if q>bq > b, the government provides fewer public goods.<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    (b) Perfect Complements Case<\/strong><\/h4>\n\n\n\n

    If consumers require private goods and public goods in a fixed proportion: C=aG,a>0C = aG, \\quad a > 0<\/p>\n\n\n\n

    the government\u2019s budget constraint is: C+qG=YC + qG = Y<\/p>\n\n\n\n

    Substituting C=aGC = aG: aG+qG=YaG + qG = Y G(a+q)=YG(a + q) = Y G\u2217=Ya+q,C\u2217=aG\u2217G^* = \\frac{Y}{a + q}, \\quad C^* = aG^*<\/p>\n\n\n\n

    Comparative Statics<\/strong><\/h4>\n\n\n\n
      \n
    • When aa increases<\/strong>: Consumers demand more private goods relative to public goods, reducing GG.<\/li>\n\n\n\n
    • When qq increases<\/strong>: The cost of public goods rises, reducing the optimal GG.<\/li>\n<\/ul>\n\n\n\n

      Now, I\u2019ll generate an image illustrating these cases.<\/p>\n\n\n\n

      Here is the conceptual economic diagram illustrating the optimal provision of public goods under both perfect substitutes and perfect complements cases. Let me know if you need any modifications!<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      optimal quantity of public goods that the government should provide, and interpret your results. Make sure you show all of the relevant cases. What happens when b changes, or when q changes? (b) Repeat part (a), except with perfect complements preferences; that is, for the case where the representative consumer always wishes to consume private […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-203873","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/203873","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=203873"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/203873\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=203873"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=203873"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=203873"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}