a)by use of diagrams where appropriate differentiate between Marshallian and Hicksian demand curves.
b)The utility function for a consumer is given as follows:
U=X0.3Y0.7.
i)derive the Hicksian demands for X and Y.
ii)give economic interpretation of the Lagrangian multipliers
iii)use the result from part (i)and (ii)for good X to show that the sky equestio hold
a) Marshallian and Hicksian demand curves represent different perspectives on consumer behavior.
Marshallian demand refers to the relationship between the quantity demanded of a good and its own price, holding other factors constant. It reflects the consumer's preferences and budget constraint in terms of the market price.
To construct a Marshallian demand curve, start by assuming that the consumer's income and the prices of other goods remain constant. Then, plot the quantity demanded of the good on the horizontal axis and its own price on the vertical axis. By adjusting the price of the good and observing the corresponding quantity demanded, you can create the demand curve.
Hicksian demand, on the other hand, represents the relationship between the quantity demanded of a good and its own price, assuming that the consumer's utility remains constant. It reflects the substitution effect of a price change.
To construct a Hicksian demand curve, start by determining the consumer's initial utility level and then adjust the price of the good while keeping their utility constant. Plot the quantity demanded of the good on the horizontal axis and its own price on the vertical axis. The resulting curve shows how the quantity demanded changes as the price of the good fluctuates, while the consumer's utility remains the same.
b) i) To derive the Hicksian demands for X and Y, we need to solve the consumer's utility maximization problem subject to the budget constraint.
The utility function is given as U = X^0.3 * Y^0.7, where X represents good X and Y represents good Y.
The consumer's problem can be written as:
Maximize U = X^0.3 * Y^0.7
subject to the budget constraint: Px * X + Py * Y = I
ii) The Lagrangian multipliers in this case represent the marginal utility of income or the shadow price of the budget constraint. They reflect how much additional utility the consumer gains by having an extra unit of income.
iii) Using the results from part (i) and (ii) for good X, we can show that the sky equation holds. The sky equation states that the total effect of a price change on the quantity demanded of a good can be decomposed into the substitution effect and the income effect.
To demonstrate this, we need to calculate the compensated (Hicksian) demand for good X and differentiate it with respect to the price of X while holding utility constant. Then, we can express the derivative as the sum of the price substitution effect and the income effect.
By comparing the results from part (i) and the decomposition in part (iii), we can verify if the sky equation holds for good X.
Note: To provide a more detailed mathematical explanation and calculation, please provide specific values for prices and income.