The preferences of a consumer are represented by the utility function 

The preferences of a consumer are represented by the utility function 

U = X + 2(Y)1/2

a) In the initial situation, the prices of the commodities and the income of the 

consumer are 

P

X = 5 , PY = 1 and I = 150 

Determine the optimal consumption vector (X*

, Y*

) and compute the 

maximum utility (U*

) of the consumer. 

b) Assume the government implements a commodity tax (t). In this situation, 

we have 

P

X = 5 + t , PY = 1 and I = 150 

and 

t = 5 

i. Determine the optimal consumption vector, the maximum utility of the 

consumer, and (iii) the government revenue from the tax. 

ii. Given the new prices induced by the commodity tax, determine the 

additional income the consumer would need to reach the same level of 

utility (U*) as in the initial situation. How does this additional income 

compare to the government revenue from the tax? Explain.