Chapter 4: Problem 10

Suppose individuals require a certain level of food (X) to remain alive. Let this amount be given by $\mathrm{X}_{0} .$ Once $\mathrm{X}_{\mathrm{o}}$ is purchased, individuals obtain utility from food and other goods $$U(X, Y)=\left(X-X_{0}\right) < < Y_{i}^{\prime}$$ where $a+(3-1$ a. Show that if $\rangle P_{x} X_{o}$ the individual will maximize utility by spending $a\left(I-P_{x} X_{o}\right)+P_{x} X_{o}$ on good Xand $/ 3\left(/-P_{X} X_{o}\right)$ on good $Y$ b. How do the ratios $P_{x} X / I$ and $P_{Y} Y / I$ change as income increases in this problem? (See also Extension E4.2.)

Short Answer

Expert verified

Question: Determine the optimal consumption of goods X and Y that maximizes the utility function given by $U(X, Y) = (X - X_{0})^{\alpha}Y^{\beta}$, subject to the budget constraint $P_{X}X + P_{Y}Y = I$. Then, analyze how the ratios $P_{x}X/I$ and $P_{Y}Y/I$ change as income increases. Answer: The optimal consumption of goods X and Y are given by $X=\alpha(I-P_{X}X_{0})+P_{X}X_{0}$ and $Y=\beta(I-P_{X}X_{0})$, respectively. As income increases, the ratios $P_{x}X/I$ and $P_{Y}Y/I$ approach $\alpha$ and $\beta$, respectively.

Step by step solution

Write down the utility function and budget constraint

The utility function is given by $$U(X, Y) = (X - X_{0})^{\alpha}Y^{\beta}$$ and the budget constraint is $$P_{X}X + P_{Y}Y = I$$

Find the Lagrangian for the maximization problem

Setting up the Lagrangian, we have $$\mathcal{L}(X, Y, \lambda) = (X - X_{0})^{\alpha}Y^{\beta} + \lambda(I - P_{X}X - P_{Y}Y)$$

Calculate the first-order conditions

Differentiate the Lagrangian with respect to $X$, $Y$ and $\lambda$ and set the results equal to zero. $$\frac{\partial \mathcal{L}}{\partial X} = \alpha(X - X_{0})^{\alpha-1}Y^{\beta} - \lambda P_{X} = 0 \quad (1)$$ $$\frac{\partial \mathcal{L}}{\partial Y} = \beta(X - X_{0})^{\alpha}Y^{\beta-1} - \lambda P_{Y} = 0 \quad (2)$$ $$\frac{\partial \mathcal{L}}{\partial \lambda} = I - P_{X}X - P_{Y}Y = 0 \quad (3)$$

Solve the system of equations

Divide equation (1) by equation (2): $$\frac{\alpha(X-X_{0})^{\alpha-1}Y^{\beta}}{\beta(X-X_{0})^{\alpha}Y^{\beta-1}} = \frac{P_{X}}{P_{Y}}$$ Simplifying, we get $$\frac{\alpha}{\beta}\frac{Y}{X-X_{0}} = \frac{P_{X}}{P_{Y}}$$ Now using the budget constraint (3), substitute $Y=\frac{I-P_{X}X}{P_{Y}}$ into this equation to solve for $X$: $$\frac{\alpha}{\beta}\frac{I-P_{X}X}{P_{Y}(X-X_{0})}= \frac{P_{X}}{P_{Y}}$$ Solving for $X$, we find: $$X=\alpha(I-P_{X}X_{0})+P_{X}X_{0}$$ Now, substitute this result back into the budget constraint (3) to get the optimal consumption of good $Y$: $$Y=\beta(I-P_{X}X_{0})$$ For part b:

Calculate the ratios PxX/I and PyY/I

Using the results from part a, calculate the ratios: $$\frac{P_{x}X}{I}=\frac{P_{x}(\alpha(I-P_{X}X_{0})+P_{X}X_{0})}{I}=\alpha(1-\frac{P_{X}X_{0}}{I})+\frac{P_{X}X_{0}}{I}$$ $$\frac{P_{Y}Y}{I}=\frac{P_{Y}(\beta(I-P_{X}X_{0}))}{I}=\beta(1-\frac{P_{X}X_{0}}{I})$$

Analyzing the change in these ratios as income increases

As income increases, the term $\frac{P_{X}X_{0}}{I}$ becomes smaller since $I$ is in the denominator. Thus, the ratios will approach: $$\lim_{I \to \infty}\frac{P_{x}X}{I} = \alpha$$ $$\lim_{I \to \infty}\frac{P_{Y}Y}{I} = \beta$$ So, as income increases, the ratios $\frac{P_{x}X}{I}$ and $\frac{P_{Y}Y}{I}$ approach $\alpha$ and $\beta$, respectively.