7. Consider the model where an individual has wealth k which they can either save or consume. If they save it, they receive a fixed and exogenous return r. The instantaneous utility function is given by: u(c, k) = c + a(k) where c is consumption, k is wealth, and a(k) is a function that defines the utility that an individual gets from holding wealth. The growth in wealth is given as the returns on wealth rk, plus income from working z(t), minus consumption c(t). a. Write out the differential equation for wealth. b. For an infinite time model, set up the optimal control problem with discounting at a rate 8. c. Write the current-valued Hamiltonian of this problem. d. Derive the steady-state level of consumption.

Problem 4 - Costless Magical MacGuffin Consider a consumer that lives only for two periods. He works in period 1 (and gets income ₁) and moves up the corporate ladder in period 2 (and gets income ₁ 0 in the first period, and is expected to trade at P₂ in the second period. 3. Write down the new budget constraint. 4. Under what conditions on P₂ would a consumer borrow and then sell a MacGuffin? Would they be better off with this option?

How does savings change with changes in y2? Provide some intuition behind this result.

Suppose you have a monthly income of $1000, $850 in monthly expenses, and you can put money in a savings account that yields a monthly interest rate of 4%. Create a budget constraint showing the trade-off between present consumption (horizontal axis) and future consumption (vertical axis). How much will you have in the future if you choose to consume $850 now? Show this point on your budget constraint.

2. Suppose Jill derives utility from not only consuming goods, but also from enjoying leisure time. Let her utility function be defined as follows: U=C.25.R.75 where C is a consumption good that can be bought at a price of $1 and R is hours of leisure (relaxation) consumed per day. There are 24 hours in a day and leisure is defined as time spent not working. Jill has a job that pays $w per hour, a trust fund that pays her $M per day, and she can work any number of hours per day, L, she desires. C, consumption good; R, Leisure (relaxation); L, labor M, non-wage income; w, wage rate. a. Derive her labor supply function? b. Assume M = $100, at what wage is her quantity supplied of hours = 0?

If you draw the cash flows from any investment, you would have negative cash flows at the beginning, and then you would receive a stream of positive cash flows thereafter. So why do we need the separate concept of a J-curve? In other words, what is the difference between a J-curve and the cash profile of any other investment?

Suppose the Super Bowl is this week, and Raphael is in need of a television to watch the big game. As a college student, Raphael knows that he can either buy his flat-screen television at the local electronics store, or he can shop online for a better deal but have to wait four days for the television to arrive. The following problem uses the economic concept of rate of time preference to help determine which decision is better for Raphael. Throughout the question, assume that Raphael pays for the good the day he buys it, so his wealth is affected in the initial time period no matter where he buys the good. Also, assume the shipping cost and cost to travel to the store are incorporated into their respective given prices. Finally, assume the goods are identical, and there's no cost to gaining information about prices-in other words, he knows the best price online and in the store without having to search. Suppose Raphael receives a utility of 35.11 utils once he actually receives his…

3. Consider a parent who is altruistic towards her child, but also cares about her own consumption. The parent's utility over her own consumption and that of her child is up = log(co) +a log(ci) where c is the child's consumption, and a > 0 is the degree of parental altruism. Suppose that the parent can invest in the child's human capital by spending money (e) on her education; education generates human capital h /() and human capital is paid at rate w. The parent has a total income of (a) Write down an expression for the child's future consumption in terms of the parent's choice of e. (b) Now write down the Lagrangian for the parent's decision problem.

Consider the problem of an individual that has Y dollars to spend on consuming over two periods. Let c, denote the amount of consumption that the individual would like to purchase in period 1 and c2 denote the amount of consumption that the individual would like to consume in period 2. The individual begins period 1 with Y dollars and can purchase c1 units of the consumption good at a price P and can save any unspent wealth. Use sı to denote the amount of savings the individual chooses to hold at the end of period 1. Any wealth that is saved earns interest at rate r so that the amount of wealth the individual has at his/her disposal to purchase consumption goods in period 2 is (1+r)s1. This principal and interest on savings is used to finance period 2 consumption. Again, for simplicity, we can assume that it costs P2 dollars to buy a unit of the consumption good in period 2. 2 The individual's total happiness is measured by the sum of period utility across time, u(cı) + u(c2). Let u(c)…

6. If intertemporal preferences are consistent and the lifetime utility function is additive, then the discount function 8(t) must be (a) bounded (b) exponential (c) hyperbolic (d) linear (e) logarithmic