Question

How does the effect on work effort from a permanent proportional upward shift in the production function differ from a permanent parallel upward shift in the production function? Explain your answer

Answer 1

Robinson Crusoe is alone on an island, so he is an economy unto himself. He has preferences over consumption and leisure and can produce consumption goods by using labor and capital. We examine production first. Then we turn to preferences. Putting these two pieces together yields Crusoe’s optimal choices of labor, leisure, and consumption.

Crusoe uses factors of production in order to make output y. We can think of this output as being coconuts. Two common factors of production, and those we consider here, are capital k and labor l. Capital might be coconut trees, and labor is the amount of time Crusoe works, measured as a fraction of a day. How much Crusoe produces with given resources depends on the type of technology A that he employs. We formalize this production process via a production function. We often simplify our problems by assuming that the production function takes some particular functional form. As a first step, we often assume that it can be written: y = Af (k; l), for some function f (). This means that as technology A increases, Crusoe can get more output for any given inputs. It is reasonable to require the function f () to be increasing in each argument. This implies that increasing either input k or l will increase production. Another common assumption is that output is zero if either input is zero: f (0; l) = 0 and f (k; 0) = 0, for all k and l.

In the maximization problem we are considering, we have c in the objective, but we know that c = f (l), so we can write the max problem as: max l u[f (l); l]: We no longer have c in the maximand or in the constraints, so c is no longer a choice variable. Essentially, the c = f (l) constraint tacks down c, so it is not a free choice. We exploit that fact when we substitute c out. At this point, we have a problem of maximizing some function with respect to one variable, and we have no remaining constraints. To obtain the optimal choices, we take the derivative with respect to each choice variable, in this case l alone, and set that derivative equal to zero.2 When we take a derivative and set it equal to zero, we call the resulting equation a first-order condition, which we often abbreviate as “FOC”. In our example, we get only one first-order condition:(See the Appendix for an explanation of the notation for calculus, and note how we had to use the chain rule for the first part.) We use l? because the l that satisfies this equation will be Crusoe’s optimal choice of labor.3 We can then plug that choice back into c = f (l) to get Crusoe’s optimal consumption: cObviously, his optimal choice of leisure will be Under the particular functional forms for utility and consumption that we have been considering, we can get explicit answers for Crusoe’s optimal choices. Recall, we have been using. When we plug these functions into the first-order condition

In this section, we study the consumption/saving decision of an individual which has access to a bond market and can, thus, freely borrow and lend. We start with a setting in which we assume that the consumer lives for two time periods. Next, we assume that the the consumer lives for T time periods, where T can be any positive integer. Finally, we assume that the consumer lives forever.

In all three cases, decisions are based on perfect foresight of the future.

e., the condition that (−1) · MRS equals the slope of the budget line. In the case of a dynamic choice problem, this efficiency condition is called an intertemporal Euler equation. This Euler equation, which will recur in many guises, has a simple interpretation: At a utility maximum, the individual cannot gain from feasible shifts of consumption between periods. A one-unit reduction in first-period consumption lowers U by u (c1) (marginal utility cost). The consumption unit thus saved can be converted (by lending it) into 1 + r units of second-period consumption that raise U by (1 + r) βu (c2) (sure marginal utility gain). An intertemporally optimal consumption plan, thus, equates the cost of forgone consumption today and the benefits of increased future consumption. An important special case is the one in which ρ = r and thus β = 1/ (1 + r). In this case the Euler equation becomes u (c1) = u (c2), which implies that the consumer desires a flat lifetime consumption path, c1 = c2.

Finally, going out all the way to the last period we find that aT +1 = [(1 + rT ) (1 + rT −1) ··· (1 + r0)] (a0 + y0 − c0) (4) + [(1 + rT ) (1 + rT −1) ··· (1 + r1)] (y1 − c1) + [(1 + rT ) (1 + rT −1) ··· (1 + r2)] (y2 − c2) +... + [(1 + rT ) (1 + rT −1)] (yT −1 − cT −1) + [(1 + rT )] (yT − cT ) Next, let us define the market discount factor (or simply: price) for date t consumption on date 0 pt ≡ 1 (1 + r0) (1 + r1) ··· (1 + rt−1) With this discount factor at our disposal we can rewrite (4) as pT aT +1 = (a0 + y0 − c0) +p1 (y1 − c1) +p2 (y2 − c2) +... +pT (yT − cT ) pT aT +1 = a0 + y0 − c0 + T t=1 pt (yt − ct) From the terminal asset condition we get 1 pT a0 + T t=0 pt (yt − ct) aT +1 ≥ 0 a0 + T t=0 pt (yt − ct) ≥ 0 T t=0 ptct use of funds ≤ a0 + T t=0 ptyt source of funds We conclude that solving the asset difference equation and employing aT +1 ≥ 0 leads to the same type of present value lifetime budget constraint (PVBC) as we encountered in the two-period case. The dynamic optimization problem maxU = T t=0 βt u (ct)