CAPÍTULO III- INVENCIÓN DE COSAS POSITIVAS PARA LA SUBSISTENCIA
4. Celebración de la danza, distintos tipos de danzas y cánticos
Now, think about the contributions from the constraint term. Suppose Crusoe is at some choice ofcandlsuch that the budget is exactly met. If he wants to decrease laborlby a little bit, then he will have to cut back on his consumptionc. The constraint term in the Lagrangean is:[f(l) c]. The Lagrangean, our new objective, goes down by the required cut inctimes, which is the marginal utility of consumption. Essentially, the Lagrangean subtracts off the utility cost of reducing consumption to make up for shortfalls in budget balance. That way, the Lagrangean is an objective that incorporates costs from failing to meet the constraint.
2.4 Income and Substitution Effects
Barro uses graphs to examine how Crusoe’s optimal choices of consumption and labor change when his production function shifts and rotates. He calls the changes in Crusoe’s choices “wealth and substitution effects”. That discussion is vaguely reminiscent of your study of income and substitution effects from microeconomics. In that context, you con-sidered shifts and rotations of linear budget lines. Crusoe’s “budget line” is his production function, which is not linear.
This difference turns out to make mathematical calculation of income and substitution ef-fects impractical. Furthermore, the “wealth efef-fects” that Barro considers violate our as-sumption that production is zero when laborlis zero. Such a wealth effect is depicted as an upward shift of the production function in Barro’s Figure 2.8. This corresponds to adding a constant to Crusoe’s production function, which means that production is not zero whenlis.
Barro’s Figure 2.10 depicts a pivot of the production about the origin. This type of change to production is much more common in macroeconomics, since it is how we typically rep-resent technological improvements. If Crusoe’s production function isy = Al , then an increase inAwill look exactly like this. Given a specific functional form foru() as well, it is straightforward to compute how Crusoe’s choices of consumptioncand laborlchange for any given change inA.
For example, supposeu(c;l) = ln(c) + ln(1 l) as before. Above we showed that:
c
?=A
1 +
:
Determining how c? changes when A changes is called comparative statics. The typical exercise is to take the equation giving the optimal choice and to differentiate it with respect to the variable that is to change. In this case, we have an equation for Crusoe’s optimal choice ofc?, and we are interested in how that choice will change asAchanges. That gives us:
@c
?
@A
=
1 +
(2.7) :
The derivative in equation (2.7) is positive, so Crusoe’s optimal choice of consumption will increase whenAincreases.
The comparative statics exercise for Crusoe’s optimal labor choicel?is even easier. Above we derived:
l
?= 1 + :
There is noAon the right-hand side, so when we take the partial derivative with respect toA, the right-hand side is just a constant. Accordingly,@l?=@A= 0, i.e., Crusoe’s choice of labor effort does not depend on his technology. This is precisely what Barro depicts in his Figure 2.10.
The intuition of this result is as follows. WhenAgoes up, the marginal product of labor goes up, since the slope of the production function goes up. This encourages Crusoe to work harder. On the other hand, the increase inAmeans that for anylCrusoe has more output, so he is wealthier. As a result, Crusoe will try to consume more of any normal goods. To the extent that leisure 1 lis a normal good, Crusoe will actually work less.
Under these preferences and this production function, these two effects happen to cancel out precisely. In general, this will not be the case.
Variable Definition
y Income, in units of consumption
k Capital
l Labor, fraction of time spent on production
f(l) Production function
A parameter of the production function
A Technology of production
c Consumption
1 l Leisure, fraction of time spent recreating
L() Lagrangean function
Lagrange multiplier
Table 2.1: Notation for Chapter 2
Exercises
Exercise 2.1(Easy)
An agent cares about consumption and leisure. Specifically, the agent’s preferences are:
U = ln(c) + ln(l), wherecis the agent’s consumption, andlis the number of hours the agent
Appendix: Calculus Notation 19
spends per day on leisure. When the agent isn’t enjoying leisure time, the agent works, either for herself or for someone else. If she worksnshours for herself, then she produces
y = 4n0s:5units of consumption. For each hour that she works for someone else, she gets paid a competitive wagew, in units of consumption.
Write out the agent’s optimization problem.
Exercise 2.2 (Moderate)
Suppose Crusoe’s preferences are given by:u(c;l) =c (1 l)1 , for some between zero and one. His technology is: y = f(l) = Al , just like before. Solve for Crusoe’s optimal choices of consumptioncand laborl. (You can use either substitution or a Lagrangean, but the former is easier in this sort of problem.)
Appendix: Calculus Notation
Suppose we have a function: y =f(x). We can think of differentiation as an operator that acts on objects. Write d
dx
as the operator that differentiates with respect tox. We can apply the operator to both sides of any equation. Namely,
d
dx y
= d
dx f(x)
:
We often write the left-hand side as dy
dx
, and the right-hand side asf0(x). These are just notational conventions.
When we have functions of more than one variable, we are in the realm of multivariate calculus and require more notation. Suppose we havez = f(x;y). When we differenti-ate such a function, we will take partial derivatives that tell us the change in the function from changing only one of the arguments, while holding any other arguments fixed. Par-tial derivatives are denoted with curly dees (i.e., with@) to distinguish them from normal derivatives. We can think of partial differentiation as an operator as before:
@
@x z
= @
@x
f(x;y)
:
The left-hand side is often written as@z
@x
, and the right-hand side asf1(x;y). The subscript 1 onf indicates a partial derivative with respect to the first argument off. The derivative off with respect to its second argument,y, can similarly be written:f2(x;y).
The things to remember about this are:
Primes (f0) and straight dees (df) are for functions of only one variable.
Subscripts (f1) and curly dees (@f) are for functions of more than one variable.
Chapter 3
The Behavior of Households with Markets for Commodities and
Credit
In this chapter we move from the world in which Robinson Crusoe is alone on his island to a world of many identical households that interact. To begin, we consider one particular representative household. When we add together the behaviors of many households, we get a macroeconomy.
Whereas in Chapter 2 we looked at Crusoe’s choices between consumption and leisure at one point in time, now we consider households’ choices of consumption over multiple periods, abstracting from the labor decisions of households. Section 3.1 introduces the basic setup of the chapter. In Section 3.2 we work out a model in which households live for only two periods. Households live indefinitely in the model presented in Section 3.3. Both these models follow Barro fairly closely, but of course in greater mathematical detail. The primary difference is that Barro has households carry around money, while we do not.
3.1The General Setup
The representative household cares about consumptionctin each period. This is formal-ized by some utility functionU(c1;c2;c3;:::). Economists almost always simplify intertem-poral problems by assuming that preferences are additively separable. Such preferences look like: U(c1;c2;c3;:::) = u(c1) + u(c2) + 2u(c3) +. Theu() function is called the period utility. It satisfies standard properties of utility functions. The variable is called
the discount factor. It is just a number, say 0.95. The fact that it is less than 1 means that the household cares a little more about current consumption than it cares about future consumption.
The household gets exogenous income ytin each period. This income is in terms of con-sumption goods. We say that it is exogenous because it is independent of anything that the household does. Think of this income as some bequest from God or goods that fall from the sky.
At timet, the household can buy or sell consumption goodsctat a price ofPper unit. (As in Barro, the price levelPdoes not change over time.) For example, if the household sells 4 units of consumption goods to someone else, then the seller receives $4P for those goods.
The household is able to save money by buying bonds that bear interest. We usebt to denote the number of dollars of bonds that the household buys at periodt, for which it will collect principal and interest in periodt+ 1. If the household invests $1 this period, then next period it gets back its $1 of principal plus $Rin interest. Hence, if the household buysbtin bonds this period, then next period the principal plus interest will bebt(1 +R).
The household comes into the world with no bonds, i.e.,b0= 0.
Since each $1 of investment in bonds pays $Rof interest,R is the simple rate of interest on the bonds. If the bonds payR “next period”, then whether the interest rate is daily, monthly, annual, etc., is determined by what the length of a “period” is. If the “period” is a year, then the interest rateRis an annual rate.
The household can either borrow or lend, i.e., the household can issue or buy bonds, what-ever makes it happier. Ifbtis negative, then the household is a net borrower.
At periodtthe household’s resources include its incomeytand any bonds that it carries from last period, with interest. The dollar value of these resources is:
Py
t+bt 1(1 +R):
At periodtthe household allocates its resources to its current consumption and to invest-ment in bonds that it will carry forward to the next period. The dollar cost of these uses is:
Pc
t+bt:
Putting these together gives us the household’s period-tbudget equation:
Py
t+bt 1(1 +R) =Pct+bt:
In a general setup, we would have one such budget equation for every period, and there could be arbitrarily many periods. For example, if a period were a year, and the household
“lived” for 40 years, then we would have forty budget constraints. On the other hand, a period could be a day, and then we would have many more budget constraints.