Question

Suppose coke (c) and hamburgers (h) provide a consumer utility of U(c,h)= (c∗h)^1/2

(a) If coke costs 1 TL and hamburger costs 25 TL, how should this consumer spend 100 TL that his mother gives him to maximize his utility?

(b) Suppose government wants to discourage coke consumption by taxing coke by 3 TL. From the point of consumer is it better to tax coke or to tax income?

Answer 1

Answer : a) Given, U(c,h) = (c*h)^1/2 = (c*h)^0.5h

U = c^0.5h^0.5

Per unit cost of coke = 1 TL. This means total cost of coke is 1*c = c

Per unit cost of hamburger =25 TL. This means total cost of hamburger is 25*h = 25h.

Total budget is 100 TL.

Therefore, the budget constraint becomes

100 = c + 25h

The lagrangean function becomes

L = U + ( 100 - c - 25h )

=> L = c^0.5h^0.5 + ( 100 - c - 25h)

The first order conditions for utility maximization are

L / c = 0.5c ^(-0.5)h^0.5 - = 0 => = 0.5c ^(-0.5)h^0.5 ...... (i)

L / h = 0.5c^0.5h ^(-0.5) - 25 => 25 = 0.5c^0.5h ^(-0.5)

=> = 0.02c^0.5h ^(-0.5) ........ (ii)

L / = 100 - c - 25h => 100 = c + 25h ....... (iii)

By equalizing equations (i) and (ii), we have

0.5c ^(-0.5)h^0.5 = 0.02c^0.5h ^(-0.5)

=> c ^(-0.5)/c^0.5 = 0.02/0.5 * h ^(-0.5)/h^0.5

=> c^(-0.5)*c ^(-0.5) = 0.04*h ^(-0.5)*h ^(-0.5)

=> c^-1 = 0.04h^-1 => c = 0.04h .... (iv)

Now putting the value of c in equation (iii), we have,

100 = 0.04h + 25h = 25.04h

=> h = 100/25.04 = 3.993

From equation (iv),

c = 0.04*3.993 = 0.159 => c = 0.2

Total Cost on coke = 1*0.2 = 0.2

Total Cost on hamburger = 25*3.993 = 99.8

Thus 100 TL budget is spend on coke and hamburger.

b) Now if per unit coke cost = 3 TL then total cost of coke becomes (3*0.2) = 0.6

Total Cost both coke and hamburger is (0.6 + 99.8) = 100.4 TL which is higher than the budget and hence from consumer point of view tax on coke is not better.