Question

Exercise 2 Suppose un(x) = x(1/2), the positive square-root function. For this function, the derivative function, u(x), is the function (1/2)1). In the same graph, sketch x(1/2) and (1/2)i) For this case, express the slope of the indifference through the point (Cu1, Gna) in terms of tuo ratios (T1/T2) and (c2/n1). Interpret the way the slope of the indifference through the point (cn1, Cn2) depends on those two ratios.

Answer 1

The graph of U_n(x) and U_n'(x) w.r.t. x would like:

The graph of U_n (x) and U prime_n (x) w. r .t x would like: Assuming zero costs, profits through a consumption bundle c_n is given by u(c_n). thus, pi_1/pi_2 = u_n (c_n1)/ u_n (c_n2) = (c_n1/c_n2) 1/2 = (c_n2/c_n1) -1/2 Slope of the indifference curve is given by: m = -partial differential u_n (c_n1)/partial differential c_n1/ partial differential u_n (c_n2)/partial differential c_n2 = -u prime_n (c_n1)/u prime_n (c_n2) = -(c_n1/c_n2) -1/2 = -(c_n2/c_n1) 1/2 Slope f the indifference curve m = -1/pi_1/pi_2 slope of the indifference curve is orthogonal to the ratio of the profits through the two points

Assuming zero costs, profit through a consumption bundle c_n is given by u(c_n). Thus,

$\frac{\pi_1}{\pi_2} = \frac{u_n(c_{n1})}{u_n(c_{n2})} = (\frac{c_{n1}}{c_{n2}})^{1/2} = (\frac{c_{n2}}{c_{n1}})^{-1/2}$

Slope of the indifference curve is given by:

$m = - \frac{\frac{\partial u_n(c_{n1}) }{\partial c_{n1}}}{\frac{\partial u_n(c_{n2}) }{\partial c_{n2}}}= - \frac{u_n'(c_{n1})}{u_n'(c_{n2})} = -(\frac{c_{n1}}{c_{n2}})^{-1/2} = - (\frac{c_{n2}}{c_{n1}})^{1/2}$

Slope of the indifference curve

$m = \frac{-1}{\frac{\pi_1}{\pi_2}}$

Slope of the indifference curve is orthogonal to the ratio of the profits through the two points.