The slope of the budget constraint represents the rate at which the consumer can trade one good for another while keeping the same level of utility. In this case, the price of good x is p and the price of good y is 1. The slope of the budget constraint is determined by the ratio of their prices, so the answer is A. The slope of the budget constraint is p.
To find the marginal utility with respect to good x, we need to take the partial derivative of the utility function U(x, y) with respect to x. Differentiating U(x, y) = 2x^0.5 * y^0.5 with respect to x gives us:
dU/dx = 1 * 0.5 * 2x^(-0.5) * y^0.5 = y^0.5 / x^0.5
So the answer is E. Marginal Utility with respect to good x is (y^0.5) / (x^0.5).
The marginal rate of substitution (MRS) of y for x represents the rate at which the consumer is willing to substitute good y for good x while keeping utility constant. It is calculated as the negative ratio of the marginal utilities of the two goods:
MRS_y,x = -(MU_x / MU_y)
Using the derivative from the previous question, MU_x = (y^0.5) / (x^0.5), and MU_y is found by differentiating U(x, y) with respect to y:
MU_y = 0.5 * 2x^0.5 * y^(-0.5) = x^0.5 / y^0.5
Substituting these values into the MRS equation, we get:
MRS_y,x = -[(y^0.5) / (x^0.5)] / [x^0.5 / y^0.5] = -(y^0.5 / x^0.5) * (y^0.5 / x^0.5) = -1
So the answer is C. The marginal rate of substitution of y for x is -1.
To find the demand function for good x, we need to maximize the consumer's utility subject to the budget constraint. Since the utility function is a Cobb-Douglas function, the optimal demand for x will depend on the prices of the goods and the consumer's income. Assuming an interior solution, where positive amounts of both goods are consumed, the demand function for good x can be found by solving the following problem:
Maximize U(x, y) = 2x^0.5 * y^0.5 subject to p*x + y = m
Using Lagrange multipliers, the first-order conditions for maximizing utility subject to the budget constraint are:
dU/dx - λ * dp/dx = 0
dU/dy - λ * dp/dy = 0
p * x + y = m
Differentiating U(x, y) with respect to x and y:
dU/dx = y^0.5 / x^0.5
dU/dy = x^0.5 / y^0.5
Setting these equal to λ * dp/dx and λ * dp/dy:
y^0.5 / x^0.5 = λ * p
x^0.5 / y^0.5 = λ
Dividing the two equations:
(y^0.5 / x^0.5) / (x^0.5 / y^0.5) = (λ * p) / λ
Simplifying:
1 = p
Therefore, the demand function for good x is x = m / (2p). So the answer is D. x = p / (2m).
Whether good x is a normal good or an inferior good depends on the income elasticity of demand. The income elasticity of demand for good x can be calculated as:
Elasticity_x = (% change in quantity demanded of x) / (% change in income)
Since the utility function U(x, y) = 2x^0.5 * y^0.5 is homogenous of degree zero, the income elasticity of demand for good x is zero. This means that good x is a normal good, which implies that as income increases, the quantity demanded of good x also increases. Therefore, the answer is A. Good x is a normal good.
Whether goods x and y are complements or substitutes depends on the cross-price elasticity of demand. The cross-price elasticity of demand between goods x and y can be calculated as:
Elasticity_xy = (% change in quantity demanded of x) / (% change in price of y)
However, the question does not provide information about the cross-price elasticity or the relationship between the two goods, so the answer is A. Not enough information to determine if goods x and y are complements or substitutes.