1. In this exercise, we will explore what happens when we include taxes and government spending (public investment) in the Becker-Tomes model that you learnt in class. In this economy, there exists / family dynasties; denote by i = 1,2,..., 1. In each family, there is only one parent and one child; i.e., there is no population growth and reproduction is asexual. Finally, each individual lives for just 2 periods: (i) childhood (c), and (ii) adulthood (a). The index denotes the time of birth; i.e., denotes the \"birth cohort\" that each individual belongs to. 1(a) Suppose that the parent pays taxes that are proportional to her income. The tax rate is given by 1. The taxes are used to fund public investment in children's human capital. From the point of view of the parent, the level of public investment that accrues to her child is taken as exogenously given. Write down and explain the parent's budget constraint. 1(b) As with the model you learnt in class, the human capital production function is linear, but now employs both private investment by the parent in her child as well as public investment (Gicr): Ei..=06 log(li,c,t + Gf,c,r) + i, e (1) The child's income (when she becomes an adult) also depends on her human capital linearly so that: log(yf,a,t+1) =a+WwE; ;+&q041 (2) Finally, the parent's utility function is given by: Upae = (1 m) log(ci,a,t) + nlog(yi,a,t+1) (3) The decision problem of the parent is to maximize utility given by (3) subject to the budget constraint you obtained in 2(a) given her knowledge of (1) and (2). Characterize the parent's optimization problem paying particular attention to what are her choice variables (and what are not) and derive an expression for the optimal level of parent's investment in child's human capital (i.e., I7 . ;). Interpret the expression you obtained. 1(c) Using your solution to 1(b), characterize the equilibrium law of motion for income. 1(d) Now, assume that public investments in children's human capital are made according to the following rule: Gi,c,t (1 - T)Yi,a,t = k - y log(Yi,a,t) Assume that y > 0. What is the LHS quantity? What does the above equation say ("in English")? Can you provide an economic interpretation for the parameter y?1(e) Using your solution to 1(c) and (4), can you write the income process in the following (linear) form?: log(yf,a,t+1) =a+p log(yi,c,t) +&ae1 (O) Can you provide an economic interpretation for f? [Hint: Note that if x is small, then log(1+x) = X, approximately.] 1(f) You are told that if we assume that there is intergenerational persistence in latent abilities, then, the OLS estimate of B above converges in probability to the following quantity: BOLS -P a+ (1 -y)we 1+ (1-y)wea (6) What does the (tax augmented) Becker-Tomes model predict that intergenerational mobility is dependent on and how