5. Consider a provincial economy for which the demand side is represented by a c

Question

5. Consider a provincial economy for which the demand side is represented by a consumer with income m and utility function u(x, y) = xy where x is composite private good expenditure and y is public school expenditure. The public school system is considered a public good, which is provided by the provincial government through an income tax with rate t (0,1) such that public good expenditure is y = tm where (0,1).

a. Provide a brief interpretation of the parameter .

b. Derive the utility-maximizing tax rate t* and verify that preferences over the policy variable t are single-peaked.

c. Suppose a private school system - a private good - is introduced at a price of $1 per unit and is considered a perfect substitute for the public school system. Use of the private school system does not permit the consumer to avoid paying the income tax financing the public school system. Letting z be private school expenditure by the consumer, derive the utility-maximizing level of private school consumption z* if the consumer chooses the private school system.

d. Let u(t) represent utility under the public school system as a function of the tax rate t, and let v(t) be the analogous utility function under the private school system. Show that the consumer will choose the public school system over the private school system only if t 1 / (1+4) .

e. Show that preferences over the policy variable t are double-peaked in the presence of the private school system if > 0.25.

Explanation / Answer

The first dimension – represented by the horizontal axis in the diagram below – relates to the location where spending occurs. Spending on schools and universities, education ministries and other agencies directly involved in providing and supporting education is one component of this dimension. Spending on education outside these institutions is another. • The second dimension – represented by the vertical axis in the diagram below – classifies the goods and services that are purchased. Not all expenditure on educational institutions can be classified as direct educational or instructional expenditure. Educational institutions in many OECD countries offer various ancillary services – such as meals, transport, housing, etc. – in addition to teaching services to support students and their families. At the tertiary level, spending on research and development can be significant. Not all spending on educational goods and services occurs within educational institutions. For example, families may purchase textbooks and materials themselves or seek private tutoring for their children. • The third dimension – represented by the colours in the diagram below – distinguishes among the sources from which funding originates. These include the public sector and international agencies (indicated by light blue), and households and other private entities (indicated medium-blue). Where private expenditure on education is subsidised by public funds, this is indicated by cells in the grey colour. Spending on educational institutions (e.g. schools, universities, educational administration and student welfare services) Spending on education outside educational institutions (e.g. private purchases of educational goods and services, including private tutoring) Spending on core educational services e.g. public spending on instructional services in educational institutions e.g. subsidised private spending on books e.g. subsidised private spending on instructional services in educational institutions e.g. private spending on books and other school materials or private tutoring e.g. private spending on tuition fees Spending on research and development e.g. public spending on university research e.g. funds from private industry for research and development in educational institutions Spending on educational services other than instruction e.g. public spending on ancillary services such as meals, transport to schools, or housing on the campus e.g. subsidised private spending on student living costs or reduced prices for transport e.g. private spending on fees for anc

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