Question

Amar has an endowment of food $F_A = 2$ and water $W_A = 5$. Barun has an endowment of food $F_B = 8$ and water $W_B = 5$. Amar’s utility function is given by \[ U_A(f_A, w_A) = f_A^2 w_A; \] where $f_A$ and $w_A$ are his consumption of food and water, respectively. Barun’s utility function is given by \[ U_B(f_B, w_B) = \min\{f_B, w_B\}; \] where $f_B$ and $w_B$ are his consumption of food and water, respectively. They exchange food and water at prices $p_f$ and $p_w$, respectively, to maximize their utilities. In the competitive equilibrium, $\dfrac{p_f{p_w}$ equals \_\_\_\_\_\_\_\_\_\_\_. (in integer)}

Hint:

When a consumer has perfect-complement preferences, the equilibrium ratio of goods is determined by matching marginal rates of substitution with price ratios.

Answer 1

Correct Answer: 2

Step 1: Endowment and total resources.
\[ F = F_A + F_B = 2 + 8 = 10, \quad W = W_A + W_B = 5 + 5 = 10. \]
Step 2: Amar’s marginal rate of substitution (MRS).
From $U_A = f_A^2 w_A$, \[ MU_{f_A} = 2f_A w_A, \quad MU_{w_A} = f_A^2. \] Thus, \[ MRS_{A} = \frac{MU_{f_A}}{MU_{w_A}} = \frac{2w_A}{f_A}. \] At equilibrium, $MRS_A = \frac{p_f}{p_w}$.
Step 3: Barun’s preferences.
For $U_B = \min\{f_B, w_B\}$, he consumes food and water in equal quantities: \[ f_B = w_B. \]
Step 4: Market clearing.
Total resources: $f_A + f_B = 10$, $w_A + w_B = 10$. Since $f_B = w_B$, we can write $f_A = 10 - f_B$ and $w_A = 10 - w_B = 10 - f_B$.
Step 5: Amar’s utility-maximizing condition.
At equilibrium, \[ \frac{p_f}{p_w} = \frac{2w_A}{f_A} = \frac{2(10 - f_B)}{10 - f_B} = 2. \] Hence, $\boxed{\dfrac{p_f}{p_w} = 2.}$