Consider a consumer who is choosing between cake (c) and ice-cream (i). Their utility function is given by u(x_c, x_i) = x_cx²_i. Initially their income is m = 10, and prices are given by p_c = 2 and p_i = 4.

a. Solve for the consumer's demand functions for c and i, and calculate their demand at the prices given above.

b. Right before a bank holiday, the price of cake increases to $4. Calculate the consumer's demand at these new prices.

c. Calculate the income and substitution effect of this price change on demand for cake.

d. With the aid of a diagram, explain why the weak axiom of revealed preferences implies that the substitution effect for x_c must be negative.