Income and substitution effect: why MRS=relative prices?

I'm trying to understand how to solve an exercise about income and substitution effects. I got the theory, I guess, but can't get the maths.

One of the steps would be to put in a system the equation MRS=relative prices with the utility function to calculate the coordinates of the substitution effect. Why do you do that, in plain terms?

• Welcome to SE. Could you explain better your question? You are asking just why MRS= relative prices, as in the title, or you want help to solve a specific exercise about income and substitution effects? In this case, you'd better post your exercise, to allow contributors to answer. Nov 16, 2022 at 15:29
• Hi @BakerStreet, sorry for not explaining my question better. I wanted to understand the reasons why that specific method is used, in general. More specifically, I'd like to know what it means from a geometric point of view. Nov 17, 2022 at 15:56
• You are welcome. I said this just to allow you to have the right answer you are looking for. Nov 17, 2022 at 16:08

Just to integrate the very good answer of Nicolas Torres, maybe a picture is the better way to see the geometrical meaning of the condition MRS=relative prices.

Remember that relative prices are (the absolute value of) the slope of the budget line.

And that the MRS is the slope of the indifference curves.

The graphic representation can be as follows:

The condition MRS=relative prices, therefore, says that the consumer achieves her optimal choice, given her income, when she chooses a bundle of the two goods for which the two slopes are equal, that is the indifference curve is tangent to the budget line.

In the graph, at point $$F$$.

From an intuitive point of view, this is clear: an indifference curve as $$C$$ cannot be reached, as the consumer has not enough income.

A curve like curve $$A$$ is not convenient, as the consumer can reach a higher utility, with her income, on curve $$B$$.

• Thank you, and thanks to Nicolas too. Now I got it! Nov 17, 2022 at 20:22
• You are welcome! Nov 17, 2022 at 20:25

We want to solve the optimization problem

• $$\max U(x,y)$$ subject to $$p_x x + p_y y = I$$

To set up the Lagrangian, we reorder the constraint so that it looks like “something $$=0$$”:

• $$I - p_x x - p_y y = 0$$

The Lagrangian is

• $$\mathcal{L}(x,y,\lambda) = U(x,y) + \lambda (I - p_x x - p_y y)$$

Our first order conditions are

• $$\frac{\partial \mathcal{L}}{\partial x} = \frac{\partial U}{\partial x} - \lambda p_x = 0 \implies \lambda = \frac{MU_x}{p_x}$$

• $$\frac{\partial \mathcal{L}}{\partial y} = \frac{\partial U}{\partial y} - \lambda p_y = 0 \implies \lambda = \frac{MU_y}{p_y}$$

• $$\frac{\partial \mathcal{L}}{\partial \lambda} = I - p_x x - p_y y = 0 \implies p_x x + p_y y = I$$

Since $$\lambda = \lambda$$, from the first two equations we have

$$\frac{MU_x}{p_x} = \frac{MU_y}{p_y} \implies MRS = \frac{MU_x}{MU_y} = \frac{p_x}{p_y}$$

Note: $$\frac{p_x}{p_y}$$ is actually the absolute value of the slope of the budget constraint line.

• Hi @Nicolas Torres, and thank you for taking the time. What I am trying to understand, though, is why later on you use MRS = px/py (with new prices) in a system with the utility function. What is the first equation (MRS = px/py), in geometric terms? Is it a representation of the budget constraint equation? Or of its slope? Or is it something else? Nov 17, 2022 at 15:58