Consider the following utility functions where the goods are “good” goods. ?
(i) U(x1, x2 ) = x1? x1?
2 , 0 < ? < 1 ; (ii) U ( x1 , x2 ) = x1 + x2 ; (iii) U ( x1 , x2 ) = min{x1 , x2 } . The prices and income are { p1 , p2 , m}.
(1) Determine the optimal consumption bundle ( x1* , x2* ) for each utility function. Draw
diagrams for each of the cases. Determine the indirect utility function for each of the utility
functions.
diagrams for each of the cases. Determine the indirect utility function for each of the utility
functions.
(2) Suppose there is a change in income m. Say it increases from m1 to m2. Determine the new
consumption bundles. From this analysis determine whether the utility functions are
homothetic or not. Are they homogeneous? If yes, determine the degree. Show that Euler’s
?
equation holds for U(x1, x2 ) = x1? x1?
and U ( x1 , x2 ) = x1 + x2 . Why did not I ask the same for
2
U ( x1 , x2 ) = min{x1 , x2 } ?
consumption bundles. From this analysis determine whether the utility functions are
homothetic or not. Are they homogeneous? If yes, determine the degree. Show that Euler’s
?
equation holds for U(x1, x2 ) = x1? x1?
and U ( x1 , x2 ) = x1 + x2 . Why did not I ask the same for
2
U ( x1 , x2 ) = min{x1 , x2 } ?
(3) A quantity tax is a tax on per unit of consumption. For example a petrol tax of t for every
litre of petrol consumed. An income tax is simply a tax on income.
(i) Suppose there is a quantity tax of t on x1 . Determine the new consumption bundle
denoted as ( x1t* , x2t* ) . This will allow you to determine the tax revenue collected from the
quantity tax.
(ii) Now consider an income tax that allows you to collect the same tax revenue as in the
case of the quantity tax. Determine the optimal consumption bundle denoted as ( x1I * , x2I * ) .
Determine whether ( x1t* , x2t* ) or ( x1I * , x2I * ) lies on a higher indifference curve which will allow
you to determine whether the quantity tax or the income tax is superior. Explain your answer
clearly using the concepts of Income Effect, Substitution Effect and the properties homothetic
utility functions.
(iii) Of the three utility functions which one will allow the consumer to be just as well-off
facing a quantity tax as an income tax? Explain your answer.
litre of petrol consumed. An income tax is simply a tax on income.
(i) Suppose there is a quantity tax of t on x1 . Determine the new consumption bundle
denoted as ( x1t* , x2t* ) . This will allow you to determine the tax revenue collected from the
quantity tax.
(ii) Now consider an income tax that allows you to collect the same tax revenue as in the
case of the quantity tax. Determine the optimal consumption bundle denoted as ( x1I * , x2I * ) .
Determine whether ( x1t* , x2t* ) or ( x1I * , x2I * ) lies on a higher indifference curve which will allow
you to determine whether the quantity tax or the income tax is superior. Explain your answer
clearly using the concepts of Income Effect, Substitution Effect and the properties homothetic
utility functions.
(iii) Of the three utility functions which one will allow the consumer to be just as well-off
facing a quantity tax as an income tax? Explain your answer.
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