# Quiz 6: Demand Relationships Among Goods

Utility function is a mathematical function which represent the individual preference for the goods. Substitution effect: It measures the change in demand due to change in relative prices keeping purchasing power constant. Income effect: It measures the change in demand due to change in due to change in purchasing power keeping relative price constant. a ) The given utility function is shown below: Utility function: Where, m denotes the goat's milk and s denote the strudel. The condition of consumer equilibrium can be easily understood with the help of Lagrange multiplier. Budget constraint can be for this problem is shown below: By using above information, the following lagragian function is obtained which is shown below: Taking first order condition: By using above, the following result is obtained which is shown below:  Taking ratio of equation (1) and equation (2), the following result is obtained which is shown below: Now, substituting the value of M in the budget constraint, the following result is obtained which is shown below: Thus, from above value of s the following result is obtained which is shown below: This shows that increases in the price of goat's milk does not affected the quantity of strudel. b ) The lagragian function for the given problem is shown below: Taking first order derivative, the following result is obtained which is shown below: By using above function, the following result is obtained which is shown below:  Taking ratio of equation (1) and equation (2) the following result is obtained which is shown below: Substitute the M value in budget constraint: Thus, from above value of M the following result is obtained which is shown below: The above result shows that increase in the price of strudel does not affect the quantity of goat's milk. c ) In two goods case the income and substitution effects from the change in the price of one good on the demand for another good usually work in opposite directions. Its substitution effect is positive but its income effect is negative. Slutsky equation: Where, Uncompensated demand is represented by , Compensated demand or Hickson demand is represented by . To generate create the expenditure function and take the first derivative: The Hickson demand function is obtained by taking the derivative of expenditure function with respect to their prices as shown below: Now, taking the derivative of with respect to their prices the following result is obtained which is shown below: Since, and It shows that Hence proved. d ) The Marshallian demand function shows changes in the price of y do not affect x purchases. That is, Thus, by using this the following result is obtained which is shown below: Hence Proved.
a. Mr. D uses two pats of butter for each piece of toast. The price of buttered toast can be shown in terms of price of butter and price of toast as given below: Where, b. Mr. D spends exactly half of his meager stipend on coffee (c) and the other half on buttered toast (bt). Differentiating value of bt and c with respect to each other prices as shown below:  is zero because demand of coffee is not a function of price of butter. It means that change in price of butter toast does not affect the demand of coffee. c. Mr. D uses two pats of butter for each piece of toast.  It is true that and are equal to zero as demand for coffee doesn't depend on the prices of toast and butter. Mr. D spends a fixed proportion of his income on the coffee. So price of toast and butter will not affect his demand for coffee.