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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, with marginal utilities MU_x = 1 / x and 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.

Paper For Above instruction

David's utility maximization problem involves determining the combination of goods x and y that maximizes his utility given his budget constraint. This is a classical problem in consumer theory where optimization methods, particularly the use of Lagrangian multipliers, provide the demand functions for the goods.

To start, the utility function is given as u(x,y) = ln x + 2 ln y, reflecting the consumer's preferences. The budget constraint is I = p_x x + p_y y. The goal is to maximize u(x,y) subject to this constraint.

Formulating the Lagrangian function:

L = ln x + 2 ln y + λ ( I - p_x x - p_y y ).

Taking the partial derivatives and setting them equal to zero:

∂L/∂x = 1/x - λ p_x = 0 => λ = 1/(x p_x),

∂L/∂y = 2/y - λ p_y = 0 => λ = 2/(y p_y).

Equating the two expressions for λ yields:

1/(x p_x) = 2/(y p_y) => y p_y = 2 x p_x => y = (2 p_x / p_y) x.

This relationship indicates that the consumer's optimal choices lie along a line where y is proportional to x, with the proportionality factor 2 p_x / p_y.

Substituting y into the budget constraint:

I = p_x x + p_y y = p_x x + p_y * (2 p_x / p_y) x = p_x x + 2 p_x x = 3 p_x x.

Solving for x:

x* = I / (3 p_x),

and plugging back into the relation for y:

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

Therefore, the demand functions are:

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

These demand functions are homogeneous of degree zero in income and prices because if all prices and income are scaled by the same positive factor t, then:

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

which remain unchanged. This confirms the demand functions possess homogeneity of degree zero, a key property in consumer theory, indicating consistent demand behavior with respect to proportional changes in income and prices.

Now, considering the specific cases:

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

Plugging values into the demand functions:

  • x = 60 / (3 1) = 20 units
  • y = (2 60) / (3 * p_y) = 120 / (3 p_y) = 40 / p_y

Thus, y varies inversely with p_y. To graph y's demand curve, plot y against different values of p_y, for example: p_y = $1, $2, $4, $8:

  • At p_y = $1, y* = 40 / 1 = 40
  • At p_y = $2, y* = 40 / 2 = 20
  • At p_y = $4, y* = 40 / 4 = 10
  • At p_y = $8, y* = 40 / 8 = 5

This illustrates the typical downward-sloping demand curve where y decreases as p_y increases.

Part (c): p_x = $2

Recompute demand for y with p_x = $2, still with I = $60 and, say, p_y = $1 for illustration:

  • x = 60 / (3 2) = 10 units
  • y = 120 / (3 p_y) = 40 / p_y

Similarly, at p_y = $1, y = 40; at p_y = $2, y = 20; at p_y = $4, y = 10; at p_y = $8, y = 5.

The demand curve for y now remains inversely proportional to p_y, but x increases due to the change in p_x, maintaining the total expenditure within the consumer's income. These examples confirm the demand functions' homogeneity and the principle that consumers optimize utility subject to their budgets, leading to predictable demand responses to price changes.

Conclusion

Through utility maximization using Lagrangian methods, we derived demand functions demonstrating how consumers allocate income between goods to maximize their satisfaction. The demand functions for both goods are homogeneous of degree zero in income and prices, reflecting consistent demand behavior under proportional price or income changes. Graphical analysis confirms typical downward-sloping demand curves for y as its price varies. These results exemplify fundamental principles of consumer theory and demonstrate the practical application of elasticity and demand analysis in understanding consumer behavior in real markets.

References

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  • Mas-Colell, A., Whinston, M. D., & Green, J. R. (1995). Microeconomic Theory. Oxford University Press.
  • Mankiw, N. G. (2020). Principles of Economics. Cengage Learning.

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