1. **Stating the problem:** Given a budget constraint $p_x x + p_y y = M$ and a set of indifference curves $K_1, K_2, K_3$, find the maximum consumer satisfaction (utility) represented by the highest indifference curve tangent to the budget line. 2. **Formula and concept:** The consumer maximizes utility by choosing a bundle $(x,y)$ such that the budget constraint is met and the indifference curve is tangent to the budget line. The tangency condition is: $$\frac{MU_x}{MU_y} = \frac{p_x}{p_y}$$ where $MU_x$ and $MU_y$ are the marginal utilities of goods $x$ and $y$ respectively. 3. **Explanation:** - The budget line $p_x x + p_y y = M$ represents all affordable combinations of $x$ and $y$. - Indifference curves $K_1, K_2, K_3$ represent levels of utility; higher curves mean higher satisfaction. - The maximum satisfaction is at the point $E$ where the budget line is tangent to the highest indifference curve $K_3$. 4. **Finding the maximum satisfaction:** - At point $E$, the slope of the budget line equals the slope of the indifference curve: $$-\frac{p_x}{p_y} = -\frac{MU_x}{MU_y}$$ - This ensures the consumer cannot increase utility by moving to another affordable bundle. 5. **Coordinates of $E$:** - The point $E$ has coordinates $(X_E, Y_E)$ on the graph. - These satisfy the budget constraint: $$p_x X_E + p_y Y_E = M$$ 6. **Conclusion:** The maximum consumer satisfaction is achieved at the bundle $(X_E, Y_E)$ where the budget line is tangent to the highest indifference curve $K_3$. This point satisfies both the budget constraint and the tangency condition, ensuring optimal consumption. **Final answer:** The maximum satisfaction is at $E = (X_E, Y_E)$ such that $$p_x X_E + p_y Y_E = M$$ and $$\frac{MU_x}{MU_y} = \frac{p_x}{p_y}$$