Question 11.7: A consumer has the utility function U = 40A^0.5B^0.5 . The p......

o help solve this problem the relevant budget schedules and indifference curves are illustrated in Figure 11.2, although the indifference curves are not accurately drawn to scale. The original optimum is at X. The price fall for A causes the budget line to become flatter and swing round, giving a new equilibrium at Y.
Hicks’s method for splitting the total change in A into its income and substitution effects requires one to draw a ‘ghost’ budget line parallel to the new budget line (reflecting the new relative prices) but tangential to the original indifference curve. This is shown by the broken line tangential to indifference curve I at H. From X to H is the substitution effect and from H to Y is the income effect of the price change. This problem requires us to find the corresponding values of A and B for the three tangency points X, Y and H and then to comment on the direction of these changes.
The original equilibrium is the combination of A and B that maximizes the utility function U = 40 A^{0.5}B^{0.5} subject to the budget constraint 600 = 20A + 5B. These values of A and B can be found by deriving the stationary points of the Lagrange function

G = 40 A^{0.5}B^{0.5} + λ(600 − 20A − 5B)

Thus

\frac{\partial G}{\partial A } = 20 A^{-0.5}B^{0.5} − 20λ = 0   giving   A^{-0.5}B^{0.5} = λ        (1)
\frac{\partial G}{\partial B } = 20 A^{0.5}B^{-0.5} − 5λ = 0   giving   4 A^{0.5}B^{-0.5} = λ        (2)
\frac{\partial G}{\partial \lambda } = 600 − 20A − 5B = 0       (3)

Setting (1) equal to (2)

 A^{-0.5}B^{0.5} = 4 A^{0.5}B^{-0.5}
B = 4A            (4)

Substituting (4) into (3)

600 − 20A − 5(4A) = 0
                           600 = 40A
15  =  A

Substituting this value into (4)

B = 4(15) = 60

Thus, A = 15 and B = 60 at the original equilibrium at X.
When the price of A falls to 10, the budget constraint becomes

600 = 10A + 5B

and so the new Lagrange function is

G = 40 A^{0.5}B^{0.5} + λ(600 − 10A − 5B)

New stationary points will be where

\frac{\partial G}{\partial A } = 20 A^{-0.5}B^{0.5} − 10λ = 0   giving   2 A^{-0.5}B^{0.5} = λ      (5)
\frac{\partial G}{\partial B } = 20 A^{0.5}B^{-0.5} − 5λ = 0   giving   4 A^{0.5}B^{-0.5} = λ      (6)
\frac{\partial G}{\partial \lambda }  = 600 − 10A − 5B = 0          (7)

Setting (5) equal to (6)

2 A^{-0.5}B^{0.5} = 4 A^{0.5}B^{-0.5}
B = 2A                (8)

Substituting (8) into (7)

600 − 10A − 5(2A) = 0
600 = 20A
30 = A

Substituting this value into (8) gives

B = 2(30) = 60.

Thus, the total effect of the price change is to increase consumption of A from 15 to 30 unitsand leave consumption of B unchanged at 60.
There are several ways of finding the values of A and B that correspond to point H. We know that H is on the same indifference curve as point X, and therefore the utility function will take the same value at both points. We can find the value of utility at X where A = 15 and B = 60. This will be

U = 40 A^{0.5}B^{0.5} = 40(15)^{0.5}(60)^{0.5} = 40(900)^{0.5} = 40(30) = 1,200

Thus, at any point on the indifference curve I

40 A^{0.5}B^{0.5} = 1,200
 B^{0.5} = 30 A^{-0.5}
B = 900 A^{-1}       (9)

The slope of indifference curve I will therefore be

\frac{dB}{dA} = −900 A^{-2}        (10)

At point X, the indifference curve I is tangential to the new budget line whose slope will be

\frac{-P_A}{P_B} = \frac{-10}{5} = −2           (11)

Therefore, from (10) and (11)

−900 A^{-2} = −2
450 = A²
21.2132 = A

Substituting this value into (9)

B = 900 (21.2132)^{-1}= 42.4264

Thus the substitution effect of A’s price fall, from X to H, increases consumption of A from 15 to 21.2 units and decreases consumption of B from 60 to 42.4 units. This effect is negative (i.e. quantity rises when price falls) in line with standard consumer theory.
The income effect, from H to Y, increases consumption of A from 21.2 to 30 units and also increases consumption of B from 42.4 back to its original 60 unit level. As both income effects are positive, both A and B must be normal goods.