Question

explain intuitively the role of preferences and constraints in determining the optimal choice of hours worked.

Answer 1

In microeconomic theory, preferences are captured using the utility function: the utility function u(Z) is such that consumption bundles which are strictly more preferred are assigned higher utility numbers.

i.e. u(Z1) > u(Z2)
if and only if Z1 is strictly preferred to Z2
where strict preference eliminates the possibility of indifference;

if the utility function satisfies the assumption of continuity and differentiability, then we can calculate the marginal utilities i.e. the incremental increase in utility for unit increase (ceterus paribus) in one of the commodities in the bundle Z.
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Let's take a simple example: Z is a bundle of two goods, so that Z = (x , y)
so we can rewrite the utility function as u = u ( x, y )

Marginal utility of X MUx is
[ u(x + 1, y-constant) - u (x , y-constant) ]
(intuitively, I'm not using calculus notation)
y-constant means amount of y in the bundle is held constant.

Similarly, Marginal utility of Y MUy is
[ u(x-constant, y + 1) - u (x-constant , y) ]
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The intuition behind the Equimarginal Principle says that agents will pursue value for money;

i.e. at equilibrium,
(Increment in utility for one unit expenditure on X)
= (Increment in utility for one unit expenditure on Y)

or, in more mathematical notation,
(MUx / px) = (MUy / py) || equation for Equimarginal Principle, EQUATION 1

where px and py are the unit prices of X and Y respectively.

Rearranging the above equation,
(MUx/MUy) = (px/py)
=> MRS = Slope of budget line

MRS is the slope of the indifference curve - it is the rate at which the agent is willing to trade X for Y
i.e. MRS gives minimum Y that the agent would accept for one unit sacrifice of X,
so that their utility level is held constant

Slope of the budget line (aka relative price ratio) is the rate at which the market is willing to trade X for Y.

At equilibrium, the rate at the agent would like to trade is exactly equal to the rate at which the market will trade
=> slope of IC = slope of budget line
=> (in standard microeconomics) the optimal bundle is given by the point of tangency between the budget line and the highest IC which can be reached
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Highest IC which can be reached is constrainted by the exogeneous endowment of money income (though the endowment need not be "money", more on that in a bit)

i.e. (x*, y*) have to be choosen using the Equimarginal Principle
so that the budget constraint
(px)x* + (py)y* = M || Budget constraint, EQUATION 2
is also satisfied.

Therefore we have two equations EQUATION 1 and EQUATION 2
to solve for two unknowns (x*, y*).

THis is the standard structure of a microeconomic consumer-side optimization problem.

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Now, the case for labor-leisure tradeoff is not that different.
the utility function can be given by
u = u(c, l)
where c is Consumption and l is leisure hours.

If price level = 1, then consumption = income.
Given exogeneous wage W,
consumption c = income = w*H
where H is hours worked.

=> H = C/w, we'll use this in a bit

The budget constraint can be written as
H + l = 16
since a person only has 16 waking hours a day to either work or consume leisure.

But notice the utility function features variables C and l, whereas budget constraint features H and l. Not a problem if we substitute H into the budget equation to get

(1/w)*C + l = 16
u = u(C, l)

Compare the above with the standard microeconomic forms:
(px)X + (py)Y = M
u = u(X, Y)

THus we can interpret (1/w) as the price of consumption C,
the price of leisure as 1
and the "money" endowment as 16

(like I said previously, M in the general microeconomic set up represents exogeneous endowment which may or may not be "money income";
but then time is money isn't it?)

So now that we've "fit" the labor-leisure tradeoff problem into the standard microeconomic setup, we can proceed (almost blindly) like we usually do:
use Equimarginal principle (EQUATION 1)
and Budget Equation (EQUATION 2)
to calculate (X*, Y*)
which here will be interpreted as (C*, l*) or optimal (Consumption, Leisure) bundle.
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