Question

a. Mr. Odde Ball enjoys commodities Xand \(Y\) according to the utility function $$U(X, Y)=\mathrm{V} \mathrm{X}^{2}+\mathrm{P}$$ Maximize Mr. Ball's utility if \(P_{x}=\$ 3, P_{Y}=\$ 4,\) and he has \(\$ 50\) to spend. Hint: It may be easier here to maximize \(U^{2}\) rather than \(U .\) Why won't this alter your results? b. Graph Mr. Ball's indifference curve and its point of tangency with his budget constraint. What does the graph say about Mr. Ball's behavior? Have you found a true maximum?

Short answer

Answer: Maximizing U² instead of U will not change the optimal consumption quantities of X and Y because the relative preferences of the consumer, Mr. Ball, remain unchanged. Only the scaling of the utility function is affected, but it does not change the optimal consumption bundle that maximizes his utility given the budget constraint and the prices of commodities X and Y.

Step-by-step solution

Set up the Lagrange function

Let's set up the Lagrange function for this utility maximization problem: $$\mathcal{L}(X,Y,\lambda) = VX^2 + P - \lambda (3X + 4Y - 50)$$ Here, \(\lambda\) is the Lagrange multiplier representing the relative importance of the budget constraint.

Find the partial derivatives

Now, find the partial derivatives of the Lagrange function with respect to X, Y, and \(\lambda\): $$\frac{\partial \mathcal{L}}{\partial X} = 2VX - 3\lambda$$ $$\frac{\partial \mathcal{L}}{\partial Y} = 4 - 4\lambda$$ $$\frac{\partial \mathcal{L}}{\partial \lambda} = 3X + 4Y - 50$$

Solve the system of equations

Equate the partial derivatives to zero and solve for X, Y, and \(\lambda\): $$2VX - 3\lambda = 0$$ $$4 - 4\lambda = 0$$ $$3X + 4Y = 50$$ From the second equation, we have: $$\lambda = 1$$ Substitute \(\lambda\) back into the first equation: $$2VX - 3(1) = 0 \Rightarrow X = \frac{3}{2V}$$ Finally, substitute X and \(\lambda\) back into the third equation: $$3(\frac{3}{2V}) + 4Y = 50 \Rightarrow Y = \frac{50 - \frac{9}{2V}}{4}$$

Explain why maximizing U² doesn't alter the results

Maximizing \(U^2\) instead of \(U\) will not change the results because the optimal X and Y values are still the same and only the function's interpretation changes (it becomes the square of the original utility). The change does not affect the relative preferences of the consumer, only the scaling of the utility. #b. Graphing Indifference Curve and Budget Constraint#

Find the slope of the indifference curve and budget constraint

The slope of the indifference curve is equal to the negative of the marginal rate of substitution (MRS), which is the ratio of the marginal utilities of X and Y: $$MRS = -\frac{MU_X}{MU_Y} = -\frac{2VX}{4} = -\frac{VX}{2}$$ The slope of the budget constraint is the ratio of the negative price of commodity X to the price of commodity Y, given by: $$-\frac{P_X}{P_Y} = -\frac{3}{4}$$

Find the point of tangency

Equate the slopes of the indifference curve and the budget constraint to find the point of tangency: $$-\frac{VX}{2} = -\frac{3}{4}$$ Using the optimal X value found in part a: $$-\frac{V(\frac{3}{2V})}{2} = -\frac{3}{4}$$ $$\frac{3}{4} = \frac{3}{4}$$ The point of tangency is indeed at the optimal values \((\frac{3}{2V}, \frac{50 - \frac{9}{2V}}{4})\).

Analyze Mr. Ball's behavior and the true maximum

The graph of the indifference curve and the budget constraint shows that Mr. Ball's utility is maximized at the calculated optimal quantities of X and Y. The indifference curve is tangent to the budget constraint which indicates a true maximum since the consumer can't achieve a higher utility level with the given budget constraint, i.e., Mr. Ball has chosen the best consumption bundle to maximize his utility given his budget constraint and the prices of commodities X and Y.

Key concepts

Budget Constraint

A budget constraint represents the combination of goods and services a consumer can purchase given their income and the prices of those goods. Imagine you have a strict shopping budget; this is exactly what a budget constraint is. It keeps you from buying everything your heart desires and instead focuses your spending on what you can actually afford.

The budget constraint is often depicted as a line on a graph where each point represents a different combination of two commodities that you can buy. The formula for a budget constraint when dealing with two goods, X and Y, is given by:

• 3X + 4Y = 50

Here, each of the prices, 3 and 4, represent how much money you spend depending on the quantity of X and Y respectively. Simplistically, if all your money went to X, you’d purchase fewer of Y due to this limit.

Understanding your budget constraint helps you make smart consumer choices. It visually shows the trade-off between different goods. Since you've only got $50 to spend, sticking within the constraint ensures you maximize your satisfaction without overspending.

Indifference Curve

An indifference curve represents all the combinations of two goods that provide a consumer with the same level of satisfaction, or utility. You'll notice that each curve arches because as you consume more of one good, you typically need less of another to maintain the same satisfaction level.

The important thing here is that no matter where you are on the curve, you have the same amount of happiness. However, the further from zero the curve is (further from the origin), the higher the utility it represents. It's like having equal enjoyment from different menu combinations at a restaurant and being indifferent to which meal you choose so long as you stay on that curve.

In Mr. Ball's case, his indifference curve tells us something specific about his preferences concerning commodities X and Y. The slope of this curve, known as the Marginal Rate of Substitution (MRS), indicates how Mr. Ball is willing to give up some of X to get more of Y while staying equally satisfied. Plan your consumption at tangency points, where it not only touches but aligns with your budget curve, to maximize utilities efficiently.

Lagrange Multiplier

The Lagrange multiplier is a handy tool in economics used to solve optimization problems with constraints, like our friend Mr. Ball trying to maximize his utility while keeping to his budget. When you picture someone trying to squeeze every last bit of satisfaction from their limited resources, the Lagrange multiplier is the math that makes it work.

For our problem, the Lagrange function is:

• \[ \mathcal{L}(X,Y,\lambda) = VX^2 + P - \lambda (3X + 4Y - 50) \]

Here, \( \lambda \) symbolizes the Lagrange multiplier, helping us by representing the change in the maximum value of the utility for unit changes in the right side of the constraint. It tells you how much extra utility Mr. Ball could gain if he could spend just one more dollar.

By setting up the function this way, we find the point where Mr. Ball gets the highest utility without breaching the budget. It shows the sensitivity of the utility function to relaxing the constraint by one unit. This approach guides economists towards efficient resource allocations under defined limitations.