Question

2. Education: In this question we will explore some of the factors that influence how much education a person may get. Suppose that a person's income2 is determined by his or her years of education (S) and level of ability (a) ; as follows: Furthermore, the cost of schooling, c, varies with ability (which is always strictly positive, a > 0): where b and ф > 0, (The parameter ф captures how quickly schooling costs increase with the level of schooling, that is, how convex the cost function is.) Assume parents choose schooling for their children (and pay the costs), so they maximize where m> 0 captures the strength of parents' motivation to increase their children's income. what happens to the left-hand side of equation 1 (that is, ln(v)), when schooling schooling increase log income, In(vu)? (Your answer should give a value-either a number or a variable-and a goes up by one year? That is, how much does one extra year of one sentence explanation of how to interpret this value.) (b) In words, what is the interpretation of the fact that abilty, a, shows up in the denomi- nator of equation 2? Will someone with higher ability find it more or less costly to get more schooling in this model? Does this seem like a reasonable assumption to you? Why or why not? (2 sentences) (c) Plug equations (1) and (2) into oquation (3) to write the parents' objective function (U) as a function only of m,к, ai, b, S, ф. (That is, gỉ and c should not appear directly any more.) Call this new equation Equation 4 (d) Parents take their child's ability, ai, and the other parameters (m,K, b,4) as given. The variable they choose is their child's schooling, S,. Take the derivative of Equation 4 from part c with respect to S. (e) Use the derivative from part d to find the optimal level of S, as a function of ay,b,φ 1 that for a given choice of schooling to be an optimum, the derivative of the (Recal 'or actually, the log of income objective function with respect to schooling should be zero.) Call the parents' optimal value S. Based on your answer to part e, what at the comparative statics of S* with respect to m,b, p and az? That is, does S'increase or decrease when each of these variables increase? (In each case, when thinking about the effecet of one parameter, assume the others stay fixed.) For each of the 4 parameters, give a one sentence explanation of why the parameter has the effect it does. And, for each parameter give one ezample of a reason wh within a family, or due to a policy change (f) y this parameter could vary, for instance across families, or across children t is one aspect of the way people (for instance, you and your famil) choose schooling in the real world that is absent from this model? (24 sentences)

Answer 1

ln y_i = k + a_i + b times s_i where a_i = level of ability of individual i s_i = year of education of individual i. a) d ln y_i/ds_i = b where b > 0 i.e. if years of education in school increase by one more year, ln y_i increases by b times. b) cat of c(s, a) = phi/2a s^2 (phi, a > 0). Level of ability appears in the denominator implies higher the ability of a person, level in the cost of shooting. In this model, someone with higher ability will find it lens costly to get education. This is a reasonable assumption or can be sufficed with the fact that and are awarded to student with higher ability. \r\n c) u (y, s(a)) = m (ln y_i) - c(l, a) = m (k + a_i + b s_i) - phi/2a_i s^2_i v(s_i) = mk + ma_i + mb s_i - phi/2a_i s^2_i