Suppose David Spends His Income (I) On Two Goods X And Y

Suppose David spends his income (I) on two goods x and y, whose market prices are p x and p y , respectively. His preferences are represented by the utility function u(x,y) = ln x + 2 ln y (MU x = 1 / x; MU y = 2 / y ). a. Derive his demand functions for x and y. Are they homogeneous in income and prices? b. Assuming I = $60 and p x = $1, graph his demand curve for y. c. Repeat part (b) for the case in which p x = $2

David's utility maximization problem involves allocating his income I between two goods, x and y, with prices p_x and p_y. The utility function given is u(x,y) = ln x + 2 ln y, which suggests that David derives utility from both goods, but with a preference weight skewed towards y.

The first step is to derive his demand functions for both goods. To optimize utility, David chooses x and y to maximize u(x,y) subject to the budget constraint:

p_x x + p_y y = I.

Demand Functions Derivation

Since the utility function is separable and concave, we can utilize the method of Lagrangian multipliers for constrained optimization:

L = ln x + 2 ln y + λ (I - px x - py y)

Set partial derivatives to zero:

∂L/∂x = 1/x - λ px = 0  =>  1/x = λ px (1)
∂L/∂y = 2/y - λ py = 0  =>  2/y = λ py (2)

From equations (1) and (2), eliminate λ:

(1/x) / (2/y) = (λ px) / (λ py) = px / py

which simplifies to:

(y / 2x) = px / py  =>  y = (2 x px) / py

Now substitute y into the budget constraint:

px x + py y = I

which becomes:

px x + py * (2 x px / py) = I

This simplifies to:

px x + 2 px x = I  =>  3 px x = I

and thus:

x* = I / (3 px)

Correspondingly:
y = (2 x px) / py = (2  (I / (3 px))  px) / py = (2 I) / (3 py)

Demand Functions

The demand functions are therefore:

• x*(p_x, p_y, I) = I / (3 p_x)
• y*(p_x, p_y, I) = 2 I / (3 p_y)

Homogeneity in Income and Prices

Demand functions are homogeneous of degree zero in prices and income if scaling all prices and income by the same positive factor δ leaves the demand unchanged:

Check scaling by δ:

x(δ p_x, δ p_y, δ I) = (δ I) / (3 δ p_x) = I / (3 p_x) = x

Similarly for y:

y(δ p_x, δ p_y, δ I) = 2 δ I / (3 δ p_y) = 2 I / (3 p_y) = y

Thus, the demand functions are homogeneous of degree zero in prices and income.

Graphical Analysis for Specific Prices

Part (b): I = $60, p_x = $1

Demand for y as a function of its price p_y with fixed I and p_x: y = 2 I / (3 p_y)

Plugging in the given values:

y = (2  60) / (3  py) = 120 / (3 py) = 40 / py

The demand curve for y is a rectangular hyperbola, indicating inverse relationship between y and p_y. As p_y increases, demand decreases proportionally, and vice versa. This demand function can be plotted to illustrate this inverse relationship, where y intercepts diminish as p_y grows.

Part (c): p_x = $2

The demand functions are unaffected by changes in p_x except for their own demand due to the demand equations' dependence solely on p_x and p_y. For this scenario, y remains y = 120 / (3 p_y) = 40 / p_y. The change in p_x does not impact y's demand directly, but it affects the choice of x, which remains at x = I / (3 p_x) = 60 / (3 * 2) = 10.

Thus, with higher p_x, the demand for x decreases, but y's demand remains the same given the fixed income and p_y.

Conclusion

David's demand functions are derived from utility maximization and show that demand for x and y depends inversely on their respective prices, scaled by income. The demand functions are homogeneous of degree zero in all prices and income, capturing a key property of demand behavior. Graphically, y's demand curve depicts a hyperbolic inverse relation with its price, illustrating fundamental principles of demand theory in microeconomics. These insights are vital for understanding consumer choice, price effects, and demand elasticity in economic analysis.

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