Representative Consumer
The consumer in a oneperiod competitive equilibrium, and the labor leisure tradeoff.
In a competitive equilibrium model, the representative consumer (also called the “representative household”) is a single agent which represents the collective behavior of of a large number of small individual households or consumers. This representative consumer takes prices as given (because each individual consumer can’t affect the price level), and chooses a level of consumption and leisure that maximizes their utility.
Definition of the representative consumer's problem with lumpsum taxes
Taking prices, profits, taxes, and time available as given $\lbrace w,\pi,T,h\rbrace$, the consumer chooses labor supplied, leisure and consumption $\lbrace N_{s},l,c\rbrace$ to solve:
\[\begin{aligned} \max_{\lbrace N_{s},l,c\rbrace } & U(c,l) \\ \text{s.t. } & c\geq0,\ \ \ l\geq0,\ \ \ N_{s}\geq0 \\ & c\leq w\cdot(hl)+\piT \\ & N_{s}=(hl) \end{aligned}\](Note that $\lbrace w,\pi,T,h\rbrace$ are “exogenous” and $\lbrace N_{s},l,c\rbrace$ are “endogenous”)
We can simplify the consumer’s problem to just have $c,l$ as our endogenous variables:
\[\begin{aligned}\max_{\lbrace l,c\rbrace } & U(c,l) \\ \text{s.t. } & c\geq0,\ \ \ 0\leq l\leq h \\ & c\leq w\cdot(hl)+\piT \\ \end{aligned}\]Assuming an interior solution, the characterizing equations are:
\[c=w\cdot(hl)+\piT \\ MRS=\frac{MU_{l}}{MU_{c}}=w\]Details about the problem.
Any constrained optimization problem has a set of choice variables, constraints which limit what values those choices can take, and an optimand (the thing which is optimized).
Choices
In the simplest version (as in ch 4,5) the representative consumer decides how much to consume $c$ and how much to work $N_{s}$ (or equivalently how much time to spend on leisure $l$)
 Consumption represents some aggregate of goods and services which are purchased
 Labor is equivalent to the total amount of time worked
 leisure is meant to represent both relaxing but also home production (any time not spent working)
Constraints:
 The time constraint  You only have so many hours in the day to allocate to labor or leisure
 We have some variable h, representing the amount of time in the day

That time can be split between labor $N_{s}$ and leisure $$
\[N_{s}+l=h \\ N_{s}=hl\]
 The Budget Constraint  Consumption is limited by disposable income.

Most basic version:
\[\underbrace{c}_{consumption}\leq w\cdot N_{s}=\underbrace{w}_{realWage}\cdot\underbrace{(hl)}_{laborSupplied}\] 
A more complicated version that includes government and dividends
\[c\leq w(hl)+\underbrace{\pi}_{profit}\underbrace{T}_{lumpSumTax}\] 
Could also have, for example, a tax on labor:
\[c\leq w(hl)\cdot\underbrace{(1\tau)}_{laborTax}+\piT\]


The Nonnegativity Constraints: You can’t consume or work negative amounts.
\[c\geq0 \\ l\geq0 \\ N_{s}\geq0\]
Note that $N_{s}+l=h$, so
\[l\geq0\implies N_{s}\leq h \\ N_{s}\geq0\implies l\leq h\]

The Optimand (Utility)
The consumer is trying to maximize a utility function which represents their preferences.
\[U(c,l)\] For these simple models, we want a “representative consumer”, which is represented as a single agent but is meant to represent a large number of small agents. Rep Cons. is meant to represent aggregate decisions.
 To have a “representative consumer”, we need the utility function to have certain properties:
 More (consumption and leisure) is preferred to less (U is strictly increasing). Need this to ensure budget constraint always binds.
 The Rep. Cons. likes diversity (U is concave). This ensures a unique solution.
 Consumption and leisure are normal goods. (The optimal $c$ and $l$ both increase when we increase income $(\piT+wN_{s})$)

Another nice property to have: Marginal utility of a good is infinite if you don’t have that good:
\[\lim_{c\to0}\frac{\partial U}{\partial c}=+\infty\] \[\lim_{l\to0}\frac{\partial U}{\partial l}=+\infty\] This isn’t needed to have a representative consumer, but it makes the math easier because it makes it so the nonnegativity conditions will always be nonbinding.
Example
Example: Let $U(c,l)=\ln c+\ln l$
This utility function satisfies all all requirements for a rep. consumer problem:
\[\begin{aligned} \max_{\lbrace l,c\rbrace } & (\ln c+\ln l) \\ \text{s.t. } & c\geq0,\ \ \ l\geq0,\ \ \ h\geq l \\ & c\leq w\cdot(hl)+\piT \\ \end{aligned}\]Assuming an interior solution, this consumer’s optimum is characterized by the following:
\[MRS=\frac{MU_{l}}{MU_{c}}=\frac{\left(\frac{1}{l}\right)}{\left(\frac{1}{c}\right)}=w=MRT\] \[c=w(hl)+\piT\]Solving this example for optimal $c$ and $l$
\[l^{*}=\frac{wh+\piT}{2w}\] \[c^{*}=\frac{wh+\piT}{2}\]Examples of testing the interior solutions assumption
If $w=1$, $h=60$, $\pi=30$, $T=10$, then we have an interior solution
\[l^{*}=\frac{60+20}{2}=40\] \[c^{*}=\frac{60+20}{2}=40\]If $w=0.5,h=30,\pi=30,T=10,$ if we try to plug in the above
\[l^{*}=\frac{15+20}{1}=35\] \[c^{*}=\frac{15+20}{2}=35/2\]So for these parameters, the assumption of an interior solution cannot be correct. In this particular case, the actual solution is found at the “corner” where the consumer doesn’t work and only consumes their exogenous income.
Shocks to exogenous variables
Lumpsum taxes goes up:
Exogenous outcome $\piT$ goes down, then because $l$ and $c$ are normal goods, the solution will have less of each
\[\frac{\partial}{\partial T}l^{*}=\frac{\partial}{\partial T}\frac{wh+\piT}{2w}=\frac{\partial}{\partial T}\frac{T}{2w}=\frac{1}{2w} < 0\] \[\frac{\partial}{\partial T}c^{*}=\frac{\partial}{\partial T}\frac{wh+\piT}{2}=\frac{1}{2} < 0\]Time endowment goes up:
Because $c+wl\leq wh+\piT$, the time endowment is like an endowment of potential income, then because $l$ and $c$ are normal goods, the optimum have more of each
Profits goes up:
Exogenous outcome $\piT$ goes up, then because l and c are normal goods, the optimum have more of each
\[\frac{\partial}{\partial\pi}l^{*}=\frac{\partial}{\partial\pi}\frac{wh+\piT}{2w}=\frac{1}{2w} > 0\] \[\frac{\partial}{\partial\pi}c^{*}=\frac{\partial}{\partial\pi}\frac{wh+\piT}{2}=\frac{1}{2} > 0\]Real wage goes up:
Because the consumer’s decisions are bound by the budget constraint $c+wl\leq wh+\piT$, an increase in the real wage has two effects:

Income effect: $wh$ goes up, increases potential labor income, which increases $l^{}$ and $c^{}$

Substitution effect: $wl$ goes up, increasing opportunity cost of $l$ is higher, causing substitution with $l$ going down and $c$ going up
For consumption, both effects move in the same direction, so $c$ goes up when $w$ goes up.
\[\frac{\partial}{\partial w}c^{*}=\frac{\partial}{\partial w}\frac{wh+\piT}{2}=\frac{h}{2}\]For leisure, when $w$ goes up, $l$ could go up or down, depending on which effect is stronger.
\[\frac{\partial}{\partial w}l^{*}=\frac{\partial}{\partial w}\frac{wh+\piT}{2w}\]