Perfect Substitutes

Recall that perfect substitutes have indifference curves that are straight lines (constant MRS), and have a utility function of: \[u(x,y)=w_1x+w_2y\]

Since both the budget constraint and the indifference curves are straight lines, we will not have a point of tangency between them, but a point(s) where the lines intersect.

There are three possible cases,

1. if \(p_x>p_y\), the slope of the budget constraint is flatter than the slope of the indifference “curves.” The optimal bundle is spending all income on good \(x\).
2. if \(p_x<p_y\), the slope of the budget constraint is steeper than the slope of the indifference “curves.” The optimal bundle is spending all income on good \(y\).
3. if \(p_x=p_y\), the budget constraint and indifference “curves” are the same line, so any point on the lines is optimal!

The first two cases are known as “corner solutions,” where a consumer chooses all of one good, and none of another. Note perfect substitutes are not convex, and violate the assumption that “averages are preferred to extremes.”

We can therefore write the demand function for good x (and similarly for y):

\[q_d^x = \begin{cases} \frac{m}{p_x} & \text{When } p_x < p_y\\ \text{any number between 0 and } \frac{m}{p_x} & \text{When } p_x=p_y\\ 0 & \text{When } p_x > p_y\\ \end{cases}\]

On the graph above, since \(p_x < p_y\), the optimal consumption bundle is to spend all income on \(x\) at (8,0), point A.

Perfect Complements

Recall that perfect complements have indifference curves that are right angles, and have a utility function of: \[u(x,y)=\min \{w_1x,w_2y\}\]

The optimal choice must always be at the corner of the indifference “curves.”

The consumer is purchasing equal amounts of \(x\) and \(y\) in this example, no matter what the prices are of either, so \(x=y\). We must satisfy the budget constraint, that \(p_xx+p_yx=m\) (recall \(y=x\)). Solving this for \(x\) gives us the demand function for good x:

\[q_D^x=q_D^y=\frac{m}{p_x+p_y}\]

This should be intuitive: the consumer must always purchase equal amounts of \(x\) and \(y\), so it’s as if the consumer was spending all of their income on a single good \((x+y)\) that has a price of \((p_x+p_y)\).

EXAMPLE

Suppose a consumer has a utility function

\[u(x,y)=x^{0.5}y^{0.5}\]

and faces constraints of \(m=50\), \(p_a=10\), \(p_b=5\).

\[\max_{x,y} x^{0.5}y^{0.5}\] \[\text{subject to } 10x+5y=50\]

Write the Lagrangian (and taking logs of the objective function, for simplicity):

\[\mathbb{L} = 0.5 \: ln \: x+0.5 \: ln \: y-\lambda(10x+5y-50)\]

FOCs are:

\[\begin{align*} \frac{\partial \mathbb{L}}{\partial x} = \frac{0.5}{x}-10\lambda &= 0 \\ \frac{\partial \mathbb{L}}{\partial y} = \frac{0.5}{y}-5\lambda &= 0 \\ \frac{\partial \mathbb{L}}{\partial \lambda} = 10x+5y-50 &= 0\\ \end{align*}\]

Rearrange the first two FOCs:

\[\begin{align*} 0.5&=10 \lambda x\\ 0.5&=5 \lambda y\\ \end{align*}\]

Add then together, and solve for \(\lambda\):

\[\begin{align*} 1&=\lambda(10x+5y)\\ 1&=\lambda(50)\\ \frac{1}{50}&=\lambda\\ \end{align*}\]

Now substitute lambda into each of the first two FOCs, solving for \(x\) and \(y\), respectively:

\[\begin{align*} \frac{0.5}{x}&=10(\frac{1}{50})\\ 10x&=0.5(50)\\ x&=2.5\\ \end{align*}\]

\[\begin{align*} \frac{0.5}{y}&=5(\frac{1}{50})\\ 10x&=0.5(50)\\ y&=5\\ \end{align*}\]

This consumer buys 2.5 units of \(x\) at $10/each ($25) and 5 units of \(y\) at $5/each ($25), spending all $50.

Recall the utility function is

\[u(x, y)=\sqrt{xy}=x^{0.5}y^{0.5}\]

Note the Cobb-Douglas exponents on \(x\) and \(y\) are equal and sum to 1: \((a\) and \(1-a=0.5)\). So the consumer spends \(a=50\%\) of her income on \(x\), and \(1-a=50\%\) on \(y\).

To get the demand functions for \(x\) and \(y\), go back to the FOCs, and keep all budget constraint terms as variables \((p_x, p_y, m)\).

\[\begin{align*} \frac{\partial \mathbb{L}}{\partial x} = \frac{0.5}{x}-p_x\lambda &= 0 \\ \frac{\partial \mathbb{L}}{\partial y} = \frac{0.5}{y}-p_y\lambda &= 0 \\ \frac{\partial \mathbb{L}}{\partial \lambda} = p_xx+p_yy-m &= 0\\ \end{align*}\]

Rearrange the first two FOCs:

\[\begin{align*} 0.5&=p_x \lambda a\\ 0.5&=p_y \lambda y\\ \end{align*}\]

Add then together, and solve for \(\lambda\):

\[\begin{align*} 1&=\lambda(p_xx+p_yy)\\ 1&=\lambda(m)\\ \frac{1}{m}&=\lambda\\ \end{align*}\]

Substitute this back into each of the first two FOCs and solve for \(x\), and \(y\), respectively, to get:

\[\begin{align*} \frac{0.5}{x}-p_x\lambda&=0\\ \frac{0.5}{x}-p_x \left(\frac{1}{m}\right)&=0\\ \frac{0.5}{x}&=\frac{p_x}{m}\\ p_xx&=0.5m\\ x&=0.5\frac{m}{p_x}\\ \end{align*}\]

\[\begin{align*} \frac{0.5}{y}-p_y\lambda&=0\\ \frac{0.5}{y}-p_y \left(\frac{1}{m}\right)&=0\\ \frac{0.5}{y}&=\frac{p_y}{m}\\ p_yy&=0.5m\\ y&=0.5\frac{m}{p_y}\\ \end{align*}\]

These are the demand functions for \(x\) and for \(y\).³

Note if we want to graph this, we need to find the inverse demand function by solving for \(p_x\):

\[\begin{align*} x&=0.5\frac{m}{p_x} \\ p_xx&=0.5m\\ p_x&=0.5\frac{m}{x} \\ \end{align*}\]

Now we can graph a demand function for a given amount of income. As income changes, the demand curve shifts. Here are two examples, one where \(m=20\) and another where \(m=40\):