Find Indifference curve/s and Marginal Rate/s of Substitution given only one point

Question

Arka likes fries. She wants to consume as much as possible. She consumes either regular (1 oz) or large sizes (5 oz).

1. Draw her indifference curve through $(x_R, x_L) = (10,0)$ and her indifference curve through the bundle $(x_R, x_L) = (10,2)$.

2. Determine her $MRS_{R,L}$ at $(x_R, x_L) = (10,2)$

3. Give an example of a utility function that captures her preferences.

---

What I tried:

1. $(10,0)$: Corner solution?

$(10,2)$: I'm thinking some indifference curve that does not intersect the indifference curve passing through $(10,0)$. Is that unique?

---

2. How can I do this without prices, utility or marginal utility? The formula from Wiki:

$$MRS_{R,L}(10,2) = \frac{MU_R(10,2)}{MU_L(10,2)} = -\frac{p_R(10,2)}{p_L(10,2)}$$

or in some strange books

$$MRS_{R,L}(10,2) = \color{red}{-}\frac{MU_R(10,2)}{MU_L(10,2)} = \color{red}{+}\frac{p_R(10,2)}{p_L(10,2)}$$

Well if we are at a point of tangency then (following Wiki)

$$MRS_{R,L}(10,2) = \frac{MU_R(10,2)}{MU_L(10,2)} = -\frac{p_R(10,2)}{p_L(10,2)} = -\frac{dR(10,2)}{dL}$$

I got nothing.

---

3. Some $U(x_R, x_L)$

s.t.

1. Domain includes $(10,0)$ and $(10,2)$

2. $$U(10,0) < U(10,2)$$

3. $$x_R' + 5x_L' < x_R' + 5x_L'' \to U(x_R',x_L') < U(x_R',x_L'') \ \text{(thus making #2 redundant?)}$$

4. $$x_R' + 5x_L' < x_R'' + 5x_L' \to U(x_R',x_L') < U(x_R'',x_L')$$

5. Domain is all pairs of nonnegative numbers (thus making #1 redundant?)

Is that right?

Answer 1

In general, you are right to be mystified: specifying a point (consumption bundle) isn't enough to compute MRS and indifference curves.

However, in this problem, I would suggest you take the first sentence seriously as a description of her preferences. She likes fries. (She doesn't care about what box they come in!)

Let $v(f)$ represent her utility as function of $f$, the quantity of fries she consumes (in oz).

Mechanically, $f = 1R + \frac{1}{5} L$, where $R$ and $L$ are the quantities of regular and large orders of fries she consumes. So, her utility as a function of these is:

$$u(R,L) = v(1R + \frac{1}{5} L).$$

Now you can compute marginal utilities $u_R \equiv \frac{du}{dR}$ and $u_L \equiv \frac{du}{dL}$ and the marginal rate of substitution $MRS_{L,R} \equiv \frac{u_L}{u_R}$. You will see that in MRS, the $v'(\cdot)$ terms cancel out, and you are left with an MRS of $5$, at every bundle $(R,L)$. These are linear preferences over bundles, and the indifference curves are straight lines with slope $-1/5$ (if you put $R$ on the horizontal axis).

This makes intuitive sense in two ways.

First, the canceling tells us that preferences are independent of the specific functional form of $v(\cdot)$, so long as it is an increasing function. This is because utility functions only encode ordinal preferences.

Second, the MRS gives the rate at which she would be indifferent to trading those two goods. If you offered her five 1 oz boxes for one 5 oz box, she wouldn't mind, since it's all the same to her. (If you offered her a penny more, she would gladly make the trade, and if you offered a penny less, she would refuse.)

As an aside, we could say that the relationship $f = 1R + \frac{1}{5}L$ encodes the household production function: how she transforms things she can buy (boxes of various sizes) into the thing she really likes (fries). This approach is associated with the work of Gary Becker and is commonly used in family and health economics.