Lagrangean for the utility maximizing problem

) Suppose that a couple receives utility from their family’s consumption, c, and their child’s private education, z (think of z as quality, not years of education). The couple’s actual utility function is u c z c z ( , ) ( ), 0 . The price of consumption is $1/unit and that of z is p. The couple’s income is I. Consequently, their budget constraint is given by:///////Let the utility function over two goods be given by the following: 0.5 0.5 1 2 1 2 u x x x x ( , ) = State the Slutsky equation for the effect of a change in 1 p on the amount demanded of 1 x . Identify income and substitution effects

1. (24 points) Suppose that a couple receives utility from their family’s consumption, c,
and their child’s private education, z (think of z as quality, not years of education).
The couple’s actual utility function is u(c, z) c(z ), 0 . The price of
consumption is $1/unit and that of z is p. The couple’s income is I. Consequently,
their budget constraint is given by:
I c pz
a) Write down the Lagrangean for the utility-maximizing problem, including nonnegativity
constraints that may be needed.
b) Find the couple’s optimal choice of * c = c( p, I ) and * z = z( p, I ) (the Marshallian
demand functions for consumption and schooling). Are there any values of I for
which * 0 z = . Illustrate with a diagram.
Suppose that 0  = for all couples, but income varies across couples over the range
[ , ] II . Now the government offers public education to all couples at a zero price. The
amount of public education is g z . If a couple chooses public education, they cannot
purchase private education, and vice versa. Let g I pz = , and 10 g I = pz .
c) Illustrate how free government schooling changes a couple’s choice set.
d) Show that parents with 4 g I  pz do not choose public education for their child,
but parents with 4 g I pz  do. Explain?
e) What impact does the introduction of public education have on the amount of
education received by children of parents with incomes in the range
2 4 g g pz  I  pz ? What impact does the introduction of public education have on
the amount of education received by children of parents with incomes in the range
2 g g pz  I  pz ?
f) The above assumes that household incomes were unaffected by the need to
finance public education. If all households were taxed an amount proportional to
their income to finance public education, leaving after tax income at (1−t)I ,
where t is the tax rate, what further impact would this have on the amount of
education received by children?
g) Based on the above, can you say whether the introduction of free public
education will increase the amount of education children receive?
2. (6 points) Let the utility function over two goods be given by the following:
0.5 0.5
1 2 1 2 ( , ) u x x x x =
State the Slutsky equation for the effect of a change in 1p on the amount demanded
of 1x . Identify income and substitution effects.

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