5. The government is considering increasing the tax on gasoline by $3 per gallon

Question

5. The government is considering increasing the tax on gasoline by $3 per gallon and has asked you to determine the impact on Janet’s consumer surplus. Janet spends 5% of her income on gasoline and her utility function is Cobb-Douglas.
(a) If her income is $1000 per week, what is her demand function for gasoline per week?
(b) If gasoline originally costs $2 per gallon, what is her demand for gasoline?
(c) What is the change in Janet’s consumer surplus when the tax is imposed, assuming that the full tax is passes onto consumers (that is, price rises to $5).
(d) How much tax revenue does the government collect from Janet?
(e) How much is the deadweight loss?

Explanation / Answer

Remember that the answer to (a) was G = (I/i)*(a/(a+b)) G = (1000/i)*(a/(a+b)) (b) The price is $2. So, Janet's demand is: G = (1000/i)*(a/(a+b)) G = (1000/2)*(a/(a+b)) G = 500*(a/(a+b)) Remember that Jane spends 5% of income on gasoline. So. G = 1000*0.05 = 50 G = 500*(a/(a+b)) = 50 (a/(a+b)) = 1/10 This helps us specify the demand function further: G = (1000/i)*(a/(a+b)) G = (1000/i)/10 G = 100/i So, if the price of gasoline is 2, G1 = 100/2 G1 = 50 (c) If the price is 5, then consumption is: G2 = 100/i G2 = 100/5 G2 = 20 The consumer surplus is the integral of the inverse demand curve minus the price from 0 to the point of consumption. The original consumer surplus was: The inverse demand curve is: i = 100/G CS1 = Int{50,0} (100*/G - 2)dG CS1 = (100*ln(G) - 2G) | [50,0] CS1 = 100*ln(50) - 2*50 CS1 = 291.20 The consumer surplus after the tax is imposed is: CS2 = Int{0,20} (100*/G - 5)dG CS2 = (100*ln(G) - 5G) | [20,0] CS2 = 100*ln(20) - 5*20 CS2 = 199.57 So, the change in consumer surplus is: CS1 - CS2 = 291.20 - 199.57 = 91.63 (d) T = t*G T = 3*20 T = 60 (e) The deadweight loss is calculated as the integral of the demand curve minus the original price from G2 to G1. DWL = Int{50,20}(100*/G - 2)dG DWL = 100*ln(G) - 2G | [50,20] DWL = 100*ln(50) - 2*50 - 100*ln(20) + 2*20 DWL = 31.62

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