Two-period neoclassical model, far simpler than the models of GLS Chapter 12. In this question, in period t you only use labor to make period t 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:

Yt=N, Y=zK

You get log utility from consumption in the first period, but square root in the second period (I chose that for a reason-notice that log has stronger diminishing marginal utility than square root!). Of course, you don't like work, so extra hours of work hurt your utility:

U = ln(Ct) mNt + 2(Ct+1)1/2

[I chose to make present and future equally important to focus attention on other issues: the marginal misery of labor (-m, more work is worse) and the productivity (z) of future capital]

In this simple economy with no externalities and perfect property rights, Adam Smith'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 same answer. Since the "social planner problem" is simpler to solve mathematically than the Smith-Hayek decentralized market outcome, we'll do what neoclassicals do so often-and we'll just solve the social planner problem.

Again to simplify, capital doesn't depreciate. So all the potatoes you grow but don't 2 eat in time t become K, and they are stored to become zK delicious French fries in the next period.

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)0.5

The two choice variables are N and K: everything else on the right hand side is an exogenous fact of life from the social planner's point of view. Notice how N shows up as both a good thing (part of current consumption) and a bad thing (work). Also notice 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 doing economics-tradeoffs are everywhere.

You're going to try to maximize well-being (U), so this is going to involve taking some first derivatives and setting them equal to zero: That's the usual way to search for a maximum. This is a little more calculus than we're used to-it's true multivariate calculus, with two choice variables, N and K. How on earth can we mathematically "solve" a problem that has one equation and two variables? You can't. But "doing the best you can," translated into calculus, means taking two first order conditions (F.O.C.s):

1. F.O.C. of U with respect to N: 1/(N-K) = m

2. F.O.C. of U with respect to K: 1/(N-K) = (z/K)0.5

I did set this up so that it yields a simple solution-an answer to the question, "How do the marginal utility of leisure early in life and the level of technology late in life shape the capital stock? Notice that in each F.O.C., the left hand side the marginal utility of consumption in time t: More work (N) increases consumption in time t and hence reduces the marginal utility of consumption, and more savings (K) in time t reduces consumption at time t and hence increases the marginal utility of consumption. As Cicero said, "Hunger is the best sauce," which captures the idea that at low levels of consumption, the marginal utility of consumption is higher-an extra bite of food is more enjoyable.

a. Your 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 each other in order to eliminate the 1/(N-K), and then solve for K.]

b. Your 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 function of exogenous parameters z and m, it's probably easiest to find N by plugging your solution for K into the first F.O.C.]

c. Your third question is this: If the marginal utility of leisure rises, what happens to second-period consumption: Does it rise, fall, or remain un-changed? [Leisure is the opposite of labor (N), so m is the marginal utility of leisure. In calculus, this question is asking, What is the sign of dC /dm ? It's just asking for a sign, you may be able to think it t+1 through just based on the economics.

d. Your fourth question is this: If future productivity (z) rises, will current labor supply rise as result, fall, or remain unchanged?

e. Your fifth question is this: In your answer to the fifth question, which is winning out, the substitution effect (make hay while the sun shines) or the income effect (you're gonna be rich someday, take it easy)?

I could change the correct answers to many of these questions with innocuous-looking changes to the utility function. A lot of macroeconomic outcomes, as we've seen al- ready, turn on the question, "What are people really like?" Square-root utility of consumption here tends to yield big substitution effects compared to log utility, for instance: Square-rooters don't tire quickly of more consumption compared to their log utility counterparts, so square-rooters are relatively more willing to "make hay while the sun shines." But even aside from the precise functional forms, you should notice a broader is- sue here: that a taste parameter (m) and a technology parameter (z) deeply shape economic outcomes in this toy economy. If supply and demand are central to under- graduate economics, then tastes and technology are central to graduate neoclassical economics. And notice: the supply and demand curves we see in textbooks are in large part the products of taste and technology parameters. When we think in terms of tastes and technology, we're going behind the curtain of supply and demand, and beginning to find out where supply and demand come from.

In this world (but not every world) higher productivity increases equilibrium (and optimal) labor inputs-and to many of us that sounds like a labor demand shock.

But if I'd had much stronger diminishing marginal utility of consumption, you'd end up with a backward-bending labor supply curve: and in that world, a rise in productivity would reduce equilibrium (and optimal) labor inputs. What kind of world is that? It'd a world where, yes, labor demand increases, but when the labor supply curve bends backwards, news of higher productivity makes a rational person choose a big vaction. So the real-world outcomes we see are (at a minimum) inter- actions of supply and demand.

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