# For the case of two goods, give an example of preferences that are represnted by a continuous utility function that allows for fat indifference curves

The question in the title sounds like a trick question, due to the monotonicity property that indifference curves have, such that for two goods x and y, strong monotonicity implies y > x.

Possible that you could answer the question by stipulating the area under a curve?

• You should specify every one of your assumptions/conditions more clearly. In particular, do you require strong monotonicity? – user18 Jan 13 '20 at 2:10

$$u(x, y) = 2$$ is a simple example of a continuous utility function with thick indifference curve.

$$\begin{eqnarray*} u(x, y)= \begin{cases} x & x < 1 \\ 1 & 1 \leq x < 2 \\ x - 1 & x \geq 2 \end{cases} \end{eqnarray*}$$

is another continuous utility function with thick indifference curve for $$u=1$$.

How about a utility function that only enjoys discrete quantities. This takes discrete values but allows for continuous inputs.

$$U(x,y) = (floor(x))^{\alpha} \cdot (floor(y))^{1-\alpha}$$


import numpy as np
import seaborn as sns
import pandas as pd
import matplotlib.pyplot as plt

x = np.arange(0, 10, 0.01)
y = np.arange(0, 10, 0.01)
alpha = 0.7

xv, yv = np.meshgrid(x, y)

def U(x1,y1, alpha1):
return(np.floor(x1)**alpha1 * np.floor(y1)**(1-alpha1))

uv = U(xv,yv, alpha)

fig, ax = plt.subplots()
CS = ax.contourf(xv, yv,uv, levels=np.unique(uv))
ax.set_xlabel('x')
ax.set_ylabel('y')
plt.colorbar(CS)
plt.show()


If it must be strongly monotonic and differentiable, then no, I believe this is impossible. An agent's preferences are said to be strongly monotonic if, given a consumption bundle $$x$$, the agent prefers all consumption bundles $$y$$ that have more of at least one good, and not less in any other good. That is $$y\geq x$$ and $$y\neq x$$ imply $$y\succ x$$. If this is a continuous and differentiable utility function then this means: $$\frac{\partial U}{\partial X_{i}} > 0 \: \forall i$$ This implies that the indifference curves cannot be fat. Consider being on a thin indifference curve and, holding fixed x, increasing y by an infinitesimal amount. This will increase utility by $$\frac{\partial U}{\partial y}$$, which we just stipulated is positive. So you must be moving onto a higher indifference curve and therefore it is not "fat" in the y direction. Similarly, taking an infinitesimal step in the x direction holding fixed y increase utility by $$\frac{\partial U}{\partial x}$$, which we also just stipulated is positive. So this utility function cannot have fat indifference curves.

I am unsure if this also is true for non-differentiable utility functions.

• But this again violates (strong) monotonicity. – user18 Feb 15 '20 at 4:28