Price-consumption curve

Suppose a consumer whose income is $$b$$ has a utility function given by $$U(x,y) = 2xy+y^2$$ with the price of $$x$$ being $$p_x$$ and the price of $$y$$ being $$p_y$$.

Draw the price-consumption curve assuming $$y$$ is an inferior good (keep the price of $$x$$ constant).

If we set up the Lagrangian $$L = 2xy+y^2 + \lambda \cdot (b - p_xx-p_yy)$$

and solve the maximization problem, we'll end up with

$$x(p_x,p_y,b) = \left( \frac{p_y-p_x}{2p_xp_y - p_x^2} \right) \cdot b, \,\,\,\, y(p_x,p_y,b) = \left( \frac{p_x}{2p_xp_y - p_x^2} \right) \cdot b$$

If $$y$$ is to be inferior, then we must have (considering income and prices to be strictly positive)

$$\frac{\partial y(p_x,p_y,b)}{\partial b} < 0 \implies \left( \frac{p_x}{2p_xp_y - p_x^2} \right) < 0 \Leftrightarrow 2p_xp_y - p_x^2 < 0$$

However, in order for the amount consumed of $$y$$ to be non-negative, we must have

$$\left( \frac{p_x}{2p_xp_y - p_x^2} \right) \cdot b \geq 0 \Leftrightarrow 2p_xp_y - p_x^2 \geq 0$$

Thus, we can conclude that if the demand for $$y$$ is to be non-negative at strictly positive prices and income, then $$y$$ can't be an inferior good.

However, if we respect the condition given by the non-negativity of $$y$$, we can still plot the price-consumption curve. Taking the following pairs of prices

$$(p_x,p_y) = \{ (1,1),(1,3),(1,5) \}$$

and letting $$b = 48$$, we'll have:

$$\begin{cases} x(1,1,48) = 0, y(1,1,48) = 48 \\ x(1,3,48) = 9.60, y(1,3,48) = 9.60 \\ x(1,5,48) \approx 21.34, y(1,5,48) \approx 5.34 \end{cases}$$

Plotting these values on a graph, we have:

Thanks!

• You seem to have inferred the shape of the curve from just 3 points. It would have been safer to test whether the U-shape is correct by also calculating $x,y$ at $(1,4,48)$. – Adam Bailey May 15 at 11:23
• You are completely right! If had I done that I'd have noticed that the price-consumption is piece-wise linear! – Pedro Cunha May 15 at 15:16

It's true that given the utility function the $$y$$-good is a normal good, so the question is quite odd. Ignoring this, your calculations are correct, but you could simplify to $$y(p_x,p_y,b)=\frac{b}{2p_y-p_x}$$.
Your curve contains a mistake, since $$x(1,3,48)=19.2$$. Indeed by substituting your expressions for $$x$$ and $$y$$ you can show that $$2x+y=\frac{b}{p_x}$$ as long as $$p_y\ge p_x$$, so the price-consumption curve is piecewise linear.