Should Mister Hungerman eat Thai or Mexican?

Question

Mr. Hungerman’s preferences on Thai (T) and Mexican (M) food are given by

$$ U(x_T,x_M) = a x_T + b x_M $$

Let $ p_T $ and $ p_M $ denotes the prices of a Thai dinner and a Mexican dinner, respectively. He wants you to help him figure out how many Thai dinners and Mexican dinners he should consume on a food budget of £70. Suppose $a = 5$, $b=3$, $p_T$ = 10 and $p_M$ = 5.

1. How many Thai dinners and Mexican dinners should Mr. Hungerman consume?

2. How would you describe such a solution in mathematical language?

So I figured out the constraint is

$$ x_t p_t + x_m p_m \leq 70 $$

And then $\max x_t = 7$ and $x_m$=14. Then I calculate the slope which is $-p_t/p_m$= -1/2. And equate it to -MU$x_t$/MU$x_m$ but get -5/3=-1/2. Am I doing something wrong?

Answer 1

You missed another condition. You have written that $x_t^{\texttt{max}}=7$ and $x_m^{\texttt{max}}=14$. That´s true since $x_t,x_m\geq 0$. Amounts are never negative. But this conditions have to be included into your calculation. Since they are inequalities you may apply the KKT-conditions method (The method of lagrange multipliers cannot be applied). This KKT-conditions method is a little bit time-consuming. You are lucky that the objective function and the constraint are linear. So you can appy the simplex method. The problem is

$\texttt{max} \ 5x_t+3x_m$

$10x_t+5x_m\leq 70$

$x_t,x_m\geq 0$

Now we introduce a slack variable $s_1\geq 0$ to get an equality. $10x_t+5x_m+s_1= 70$

The following red marked numbers are the pivot elements. The initial table is

$$\begin{array}{|c|c|c|c|} \hline x_t&x_m&s_1&RHS \\ \hline -5&-3&0&0 \\ \hline \color{red}{10}&5&1&70 \\ \hline \end{array} $$

$$\begin{array}{|c|c|c|c|} \hline x_t&x_m&s_1&RHS \\ \hline 0&-\frac12&\frac12&35 \\ \hline 1&\color{red}{\frac12}&\frac1{10}&7 \\ \hline \end{array} $$

$$\begin{array}{|c|c|c|c|} \hline x_t&x_m&s_1&RHS \\ \hline 1&0&\frac6{10}&42 \\ \hline 2&1&\frac1{5}&14 \\ \hline \end{array} $$

Since all coefficient of the first line (objective function) are all non-negative we have found the optimal solution which is $(x_t^*,x_m^*)=(0,14)$. And the optimal utility is $42$. This is the answer to the “ultimate question of life, the universe, and everything.“ (Douglas Adams)