A consumer\'s utility is defined by the function u(x_1, x_2) = x_1^1/3 x_2^1/2 A

Question

A consumer's utility is defined by the function u(x_1, x_2) = x_1^1/3 x_2^1/2 Assume the prices of x_1 and x_2 are respectively defined by p_1 and P_2, and the consumer has in dollars in disposable income. (a) What type of utility function is u(x_1, x_2) = x_1^1/3 x_2^1/2 (b) Using Lagrange's method of constrained optimization, find the consumer's individual demand curve for both goods, x_1(p_1, p_2, w) and x_2(p_1, p_2, w). Show your work! Tess derives all her utility from eating ham and cheese sandwiches, but she will eat only sandwiches made with two pieces of ham and one piece of cheese (she is a very picky eater!) (a) Draw Tess' indifference map over ham and cheese, and illustrate her utility maximizing choice of the two products. (b) Find Tess' demand for ham and cheese as a function of the price of cheese (p_c) and the price of ham (p_h).

Explanation / Answer

5)a. we know the cobb douglas utility function form: U(x1,x2) = x1^a*x2^b where x1,x2>0

in this question the equation is u(x1,x2) = x1^1/3*x2^1/2

We can compare these two equation,we get

a = 1/3 which is >0 and b =1/2 which is >0.

so this is a cobb-douglas function.

(b) utility function is:u(x1,x2) = x1^1/3 * x2^2/3

budget equation is: m = p1x1 + p2x2

where p1 and p2 is a price of x1 and x2

now we can use the Lagrange equation

L = x1^1/3 * x2^2/3 - w(m -p1x1 -p2x2)

where w = lagrange multiplier

now we differenciate with respect to x1 and x2

dl/dx1 = 1/3*x1^(-2/3)*x2^2/3 + wp1 = 0 ...........(1)

or, 1/3*x1^(-2/3)*x2^2/3 = -wp1

or, 1/3* (x2^2/3 / x1^2/3) = -wp1

or, (x2/x1)^2/3 = - (3) * wp1

or, (x2/x1) = - (3)^3/2 * (wp1)^3/2

or, x1 = - x2 * (3)^-3/2 * (wp1)^3/2

now

dl/dx2 = 2/3 * x1^1/3 * x2^(-1/3) + wp1 = 0

or, 2/3 * x1^1/3 * x2^(-1/3) = -wp2

or, 2/3 * (x1/x2)^1/3 = -wp2

or, (x1/x2)^1/3 = 3/2 * (-wp2)

or,  (x1/x2) =  (3/2)^3 * (-wp2)^3

or, x2 =  (3/2)^(-3) * (-wp2)^(-3) * x1

now we get dl/dw = m-p1x1-p2x2 = 0 ...... (2)

now we can put the value of x1 in this equation,we get

m-p1x1-p2x2 = 0

or, m - p1[x2 * (1/3)^(-3/2) * (wp1)^(-3/2)] - p2x2 = 0

or m - p1^(-1/3) * x2 * (1/3)^(-3/2) * (w)^(-3/2) - p2x2 = 0

or, m = p1^(-1/3) * x2 * (1/3)^(-3/2) * (w)^(-3/2) + p2x2

or, m = x2 * [ p1^(-1/3) * (1/3)^(-3/2) * (w)^(-3/2) + p2 ]

or, x2 = m/  [ p1^(-1/3) * (1/3)^(-3/2) * (w)^(-3/2) + p2 ]

this the individual demand function of x2

also we can put the value of x2 in equation2,we get

m-p1x1-p2x2 = 0

or, m-p1x1-p2 * [(3/2)^(-3) * (-wp2)^(-3) * x1] = 0

or, m-p1x1-(p2)^(-2) * (3/2)^(-3) * (-w)^-3 * x1 = 0

or, m = p1x1 + (p2)^(-2) * (3/2)^(-3) * (-w)^-3 * x1

or, m = x1 * [p1 + (p2)^(-2) * (3/2)^(-3) * (-w)^-3]

or, x1 = m / [p1 + (p2)^(-2) * (3/2)^(-3) * (-w)^-3]

this is the individual demanfd function for x1.

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