Question

utility function u(x,y;t )= (x-t)^ay^1-a

x>=t, t>0, 0<a<1

u(x,y;t )=0 when x<t

does income consumption curve is y=[(1-a)(x-t)p_x]/ap_y ?(my result, i used lagrange, not sure about it)

how to draw the income consumption curve?

Answer 1

Answer: The above problem can be solved as follows:

First, let the budget contraint to the utility function is

M = p_xx + p_yy

Gievn the budget, we should maximize the utility:

We form the consumption function as follows:

θ = U + λ(M - p_xx - p_yy) where λ = lagrangian multiplier

First we derive the derivative with respect to x, y.

?θ/?x = a (x-t)^a-1 y^1-a - λp_x = 0 ----- i

?θ/?y = (1-a) y^-a (x-t)^a - λpy = 0 ------- ii

Now deviding equation i by ii, we get,

λp_x / λp_y = a (x-t)^a-1 y^1-a / (1-a) y^-a (x-t)^a

Solving the equation we get,

p_x / p_y = ay / (1-a)(x-t)

or, y = [(1-a)(x-t)p_x] / ap_y (proved)

Hence the consumption curve is derived.

For the newxt question we can derive the Income consumption curve as follow:

If we consider the values of prices and coefficients as constant, say a = .5, p_x = 1, p_y = 1, then for t = 1 and x = 2, (since x>=t)

we get

y = 1,

for t = 2 and x = 3

y = 2

Hence we see that y incease at half of the rate of x. Plotting such values in the graph, we get the above income consumption curve OA.