The golden rule of utility maximization is: $\frac{MU_T}{P_T} = \frac{MU_R}{P_R}$. In English: the extra joy from the last taco, divided by its price, should equal the extra joy from the last ramen, divided by its price. If tacos give more joy per buck, you’d buy more tacos until the balance resets—like a see-saw of deliciousness.
How To Derive A Demand Function From A Utility Function | Detroit Chinatown
Plug in your marginal utilities: $\frac{0.5 T^{-0.5} R^{0.5}}{P_T} = \frac{0.5 T^{0.5} R^{-0.5}}{P_R}$. See those 0.5s? They cancel out! (Math fairies love when that happens.) Now you get: $\frac{R^{0.5}}{T^{0.5} P_T} = \frac{T^{0.5}}{R^{0.5} P_R}$.
Cross-multiply to get: $R \times P_R = T \times P_T$. That’s simpler than it looks—it means you’ll spend the same amount on tacos and ramen. Deep down, your budget is balanced like a zen garden of snacks.
