In the competitive equilibrium discussed in chapters 4 and 5 of the book, The consumer's problem looks something like this:
On this page, I show how to find the algebraic solution to this constrained optimization problem in its general form, and then work through the problem with several specific forms of the utility function.
Firstly, the problem will be easier to solve if we can narrow down which of the constraints are binding.
We know that if \(U\) is strictly increasing, then the budget constraint will hold with equality. And for a wide array of concave utility functions, we can find arguments for why setting either \(C=0\) or \(l=0\) would be suboptimal.
But the constraint that \(l \leq h\) is more tricky to deal with. If we have parameters such that \(\pi-T > 0\), then in principle, the consumer's optimal response could be to choose \(l=0,C = \pi-T\). In practice, we can just solve the problem assuming that this constraint is non-binding, and if we get a contradictory result when we plug in values for our parameters, then we know we were mistaken and that the consumer's optimal response is instead at \(l=h\).
So if the budget constraint above is the only binding constraint in the consumer's constrained optimization problem, then we can find the solution by setting up a Lagrangian like so:
\[ \mathcal{L} = U(C,l) - \lambda \cdot \left[ c - w\cdot(h-l) - \pi + T \right] \]Now we take the partial derivatives and set them equal to zero to get:
\begin{align} 0 = \mathcal{L}_c^\prime &= U^\prime_C - \lambda \\ 0 = \mathcal{L}_l^\prime &= U^\prime_l - \lambda w \\ 0 = \mathcal{L}_\lambda^\prime &= - \left[ c - w\cdot(h-l) - \pi + T \right] \\ \end{align}If we rearrange this, then we get the following system of equations:
\begin{align} \lambda &= U^\prime_C \\ \lambda w &= U^\prime_l \\ c &= w\cdot(h-l) + \pi - T \\ \end{align}Combine the first two and we get that \(U^\prime_l = w U^\prime_C\), and so the consumer's problem has a solution characterized by the following two equations:
\[\boxed{\begin{gather} w = {U^\prime_l \over U^\prime_C} = MRS_{lC} \\ c = w\cdot(h-l) + \pi - T \\ \end{gather}}\]The first equation is saying that the utility isoquant through the optimum is parallel to the budget constraint at the optimum. And the second equation is saying that the optimum is on the budget constraint. Together, these two conditions establish that the optimum is at a point where the utility isoquant is tangent to the budget constraint.
Depending on the utility function, it may be possible to find an explicit formula for \(l\) and \(C\).
When \(\pi\) is increased or \(T\) is decreased, we expect that both \(C\) and \(l\) will increase from the increased income. The textbook assumes that this is always the case.
When the real wage \(w\) is increased, there will be both an income effect and a substitution effect. The income effect will increase both \(C\) and \(l\), while the substution effect will increase \(C\) and decrease \(l\). Thus the effect of a change in wages on leisure may depend on the choice of utility function or even the specific parameters used.
(Here, \(\ln()\) is the natural log and \(\gamma\) is just some positive constant.)
First note that the marginal utilities are like so:
\begin{gather} U^\prime_C = \frac{1}{C} \\ U^\prime_l = \frac{\gamma}{l} \\ \end{gather}Both of these are positive for \(C,l\geq 0\), so U is strictly increasing, and the budget constraint will bind with equality.
Note that as \(C \to 0^{+}\), \(U^\prime_C \to +\infty\), and so \(C\) will be strictly positive at the optimum. A similar argument can be used for \(l\).
Assuming that \(l < h\), we can set up the Lagrangian:
\[ \mathcal{L} = \ln(C) + \gamma \ln(l) - \lambda \cdot \left[ c - w\cdot(h-l) - \pi + T \right] \]Take the partial derivatives and set equal to \(0\) to get the first-order conditions:
\begin{align} 0 = \mathcal{L}_c^\prime &= \frac{1}{C} - \lambda \\ 0 = \mathcal{L}_l^\prime &= \frac{\gamma}{l} - \lambda w \\ 0 = \mathcal{L}_\lambda^\prime &= - \left[ c - w\cdot(h-l) - \pi + T \right] \\ \end{align}Rearrange and combine to get:
\[\boxed{\begin{gather} w = {\left(\frac{\gamma}{l}\right) \over \left(\frac{1}{C}\right)} = \frac{\gamma C}{l} = MRS_{lC} \\ c = w\cdot(h-l) + \pi - T \\ \end{gather}}\]For this utility function, we can also explicitly solve for the optimal allocation. Rearrange to get \(C=\frac{w}{\gamma} l \), then plug this into the budget constraint and solve for \(l\):
\begin{gather} \frac{w}{\gamma} l = w\cdot(h-l) + \pi - T \\ \left(1+\frac{1}{\gamma}\right)lw = wh + \pi - T \\ l^* = \frac{\gamma}{1+\gamma}\frac{wh + \pi - T}{w}\\ l^* = \frac{\gamma}{1+\gamma}\left(h+\frac{ \pi - T}{w}\right)\\ \end{gather}And plug this into \(C=\frac{w}{\gamma} l \) to get:
\[C^* = \frac{wh + \pi - T}{1+\gamma}\]Now we have explicit equations for the optimal choice of \(C\) and \(l\). If we take the derivatives of these formula with respect to \(w\), we get:
\[{\partial C^* \over \partial w} = \frac{h}{1+\gamma}\] \[{\partial l^* \over \partial w} = - \frac{\gamma}{1+\gamma}\frac{ \pi - T}{w^2}\]When the real wage is increased, there will be both an income effect and a substitution effect. The income effect will increase both \(C\) and \(l\), while the substution effect will increase \(C\) and decrease \(l\).
So consumption increases as the real wage goes up, while the effect on leisure depends on the value of \(\pi-T\).
TODO: more examples (perfect complements, no income effect, perfect substitutes, cobb douglass?...)