To find the optimal consumption bundle for the consumer given the utility function (u(x, y) = 2\sqrt{x} + y), a budget constraint, and prices, we can set up the Lagrangian for the constrained optimization problem.
The consumer’s problem is to maximize utility subject to the budget constraint:
[ \max_{x, y} 2\sqrt{x} + y ]
subject to the budget constraint:
[ p_x \cdot x + p_y \cdot y = I ]
where:
- ( p_x ) is the price of good x,
- ( p_y ) is the price of good y,
- ( I ) is the consumer’s income.
In this case, the budget constraint is given by:
[ 1 \cdot x + 4 \cdot y = 20 ]
Now, set up the Lagrangian:
[ \mathcal{L}(x, y, \lambda) = 2\sqrt{x} + y – \lambda(1 \cdot x + 4 \cdot y – 20) ]
Find the first-order conditions by taking partial derivatives with respect to x, y, and λ, and setting them equal to zero:
- (\frac{\partial \mathcal{L}}{\partial x} = 0):
[ \frac{1}{\sqrt{x}} – \lambda = 0 ] - (\frac{\partial \mathcal{L}}{\partial y} = 0):
[ 1 – 4\lambda = 0 ] - (\frac{\partial \mathcal{L}}{\partial \lambda} = 0):
[ 1 \cdot x + 4 \cdot y – 20 = 0 ]
Now, solve the system of equations to find the values of x, y, and λ.
- From the first equation: (\frac{1}{\sqrt{x}} – \lambda = 0)
- Solve for x: (x = \frac{1}{\lambda^2})
- From the second equation: (1 – 4\lambda = 0)
- Solve for λ: (\lambda = \frac{1}{4})
- Use λ in the budget constraint: (1 \cdot x + 4 \cdot y – 20 = 0)
- Substitute x and λ: (\frac{1}{\lambda^2} + 4y – 20 = 0)
- Solve for y: (y = 3)
Now that you have the values for x, y, and λ, you can compute the optimal consumption bundle:
- (x = \frac{1}{\lambda^2} = 16)
- (y = 3)
- (λ = \frac{1}{4})
So, the optimal consumption bundle is ( (x, y) = (16, 3) ) when the price of x is equal to 1.
