2. Hicksian Demands
Hicksian demands are the quantities that an individual demands of goods and services given their budget constraints and the relative prices of those goods and services. In order to find the Hicksian demands, we need to know the budget constraint for the given expenditure function. We can rewrite the expenditure function as E(Px,Py,U) = −U/[(Px + Py)2], where U is the utility function. To find the budget constraint, we need to find the slope of the expenditure function with respect to Px and Py. We can do this using the formula for the derivative of a composite function, which is the derivative of the inner function multiplied by the derivative of the outer function with respect to the relevant variable.
Here, the inner function is −[U/(Px + Py)2], and the outer function is E(Px,Py,U). Taking the derivative with respect to Px, we get:
−(−[U/(Px + Py)2])/(Px + Py) = [−U/[(Px + Py)3] /(1 + Py/Px)]
Similarly, taking the derivative with respect to Py, we get:
−(−[U/(Px + Py)2])/(Px + Py) = [−U/[(Px + Py)3] /(1 + Px/Py)].
Solving these equations for x and y, we can get the price and quantity Hicksian demands.
3. Indirect Utility
Indirect utility is the change in utility that occurs when the individual changes one of the goods or services in the budget constraint. The budget constraint changes due to the change in prices, so the indirect utility is the change in utility due to the new budget constraint.
To find the indirect utility, we need to find the effect of the price change on the budget constraint. This can be found using the budget constraints above or by differentiating the expenditure function with respect to Px and Py.
4. Marshallian Demands
Marshallian demands are the quantities demanded of goods and services given a change in the price of one good or service. To find the Marshallian demands, we need to differentiate the expenditure function with respect to Px and Py while holding all other prices constant. This can be done using the formula for the derivative of a function, which