Quiz 6: Demand Relationships Among Goods

Business

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: img 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: img By using above information, the following lagragian function is obtained which is shown below: img Taking first order condition: img By using above, the following result is obtained which is shown below: img img Taking ratio of equation (1) and equation (2), the following result is obtained which is shown below: img Now, substituting the value of M in the budget constraint, the following result is obtained which is shown below: img Thus, from above value of s the following result is obtained which is shown below: img 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: img Taking first order derivative, the following result is obtained which is shown below: img By using above function, the following result is obtained which is shown below: img img Taking ratio of equation (1) and equation (2) the following result is obtained which is shown below: img Substitute the M value in budget constraint: img Thus, from above value of M the following result is obtained which is shown below: img 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: img Where, Uncompensated demand is represented by img , Compensated demand or Hickson demand is represented by img . To generate img create the expenditure function and take the first derivative: img The Hickson demand function is obtained by taking the derivative of expenditure function with respect to their prices as shown below: img Now, taking the derivative of img with respect to their prices the following result is obtained which is shown below: img Since, img and img It shows that img Hence proved. d ) The Marshallian demand function shows changes in the price of y do not affect x purchases. That is, img Thus, by using this the following result is obtained which is shown below: img Hence Proved.

Hard Times buys only rotgut whiskey and jelly donuts. Rought whiskey and jelly donuts are Hicksian substitute goods in customary sense. Since rotgut whiskey is an inferior good that exhibits giffen's paradox. Law of demand does not work in case of giffen good. As we know that in case of giffen good, demand of the good is increased as price of the good increases and vice-versa. As price of rotgut whiskey increases, thus demand of rotgut whiskey is also increase. Hard Times' income remains the same. Now Hard Times expend more income to buy whiskey than that of earlier. Simultaneously he reduces his expenditure to buy jelly donuts. Therefore, an increase in the price of rotgut whiskey must cause fewer jelly donuts to be bought.

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: img Where, img b. Mr. D spends exactly half of his meager stipend on coffee (c) and the other half on buttered toast (bt). img Differentiating value of bt and c with respect to each other prices as shown below: img img 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. img img It is true that img and img 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.