2. Let's create a simple, two-period neoclassical modelIn this question, in period t you only use labor to make periodt output, and then you decide how much of that output to convert into capital thatcan be used later to help make period t+1 output. That means the "representativeagent" has to decide how much of period t output to consume rather than save, butthere's no such choice in period t+1. After all, t+1 is the end of time. Output ismade in this very simple, constant returns to scale manner, so in each period doublethe input creates double the output:Y₁ = N₁ Y zK=You get log utility from consumption in the first period, but square root in the secondperiod Of course, you don't like work, so extra hours of work hurt your utility:U = ln(Ct) — mNt + 2(Ct+1)¹/²In this simple economy with no externalities and perfect property rights, AdamSmith's Invisible Hand Theorem will hold in its most optimistic form: A benevo-lent social planner and a decentralized private market will yield exactly the sameanswer. Since the "social planner problem" is simpler to solve mathematically thanthe Smith-Hayek decentralized market outcome, we'll do what neoclassicals do sooften and we'll just solve the social planner problem.Together, all of this means we can combine the utility function and the two produc-tion functions to yield this utility function, which sums up all of the cruel tradeoffs:U = ln(N − K) − mN + 2(zK)⁰.5The two choice variables are N and K: everything else on the right hand side is anexogenous fact of life from the social planner's point of view. Notice how N showsup as both a good thing (part of current consumption) and a bad thing (work). Alsonotice that K shows up as a bad thing (current saving, ugh sacrifice) and some-thing good (it helps create future consumption). That's how you know you're doingeconomics-tradeoffs are everywhere.You're going to try to maximize well-being (U), so this is going to involve taking somefirst derivatives and setting them equal to zero: That's the usual way to search fora maximum.1. F.O.C. of U with respect to N: 1/(N-K)2. F.O.C. of U with respect to K: 1/(N-K) = (z/K)⁰.5= mQUESTIONS1Your first question to answer: What is the socially optimal level of K,solely as a function of z and m? [I'd set the two F.O.C.s equal to eachother in order to eliminate the 1/(N-K), and then solve for K.]2Your second question is this: What is the socially optimal level of N,solely as a function of z and m? [Once you know K solely as a functionof exogenous parameters z and m, it's probably easiest to find N byplugging your solution for K into the first F.O.C.]Your third question is this: If the marginal utility of leisure rises, whathappens to second-period consumption: Does it rise, fall, or remain un-3 changed? [Leisure is the opposite of labor (N), so m is the marginalutility of leisure. In calculus, this questionasking, What is the signof dC+1/dm ? It's just asking for a sign, you may be able to think itthrough just based on the economics.4Your fourth question is this: If future productivity (z) rises, will currentlabor supply rise as result, fall, or remain unchanged?5Your fifth question is this: In your answer to the fifth question, which iswinning out, the substitution effect (make hay while the sun shines) orthe income effect (you're gonna be rich someday, take it easy)?

Production function in 2 periodsYt=NY=zKUtility function in 2 periodsU=ln ct-mNt+2(Ct+1)0.5 By using…

2. Let's create a simple, two-period neoclassical model In this question, in period t you only use labor to make period output, and then you decide how much of that output to convert into capital that can be used later to help make period t+1 output. That means the "representative agent" has to decide how much of period t output to consume rather than save, but there's no such choice in period t+1. After all, t+1 is the end of time. Output is made in this very simple, constant returns to scale manner, so in each period double the input creates double the output: Y₁ = N₁ Y = 2K You get log utility from consumption in the first period, but square root in the second period Of course, you don't like work, so extra hours of work hurt your utility: