Hicks's "second law" of demand states that the predominant relationship among goods is net substitutability (see footnote 3 of Chapter 6 ). To prove this result: a. Show why compensated demand functions $$X ;=h,\left(P_{U} \ldots, P_{n}, V\right)$$ are homogeneous of degree zero in \(P_{x} \ldots P_{n}\) for a given level of \(V\) b. Use Euler's theorem for homogeneous functions (for a statement of this theorem, see footnote 5 of Chapter 7 ) to show that \(=0 \text { (for all } i=1, n)\) c. Use the "first law of demand" to conclude that is, net substitution must prevail, on average.

To prove that compensated demand functions \(X_i=h_i(P_x,\ldots,P_n,V)\) are homogeneous of degree zero, we need to show that for any \(\lambda > 0\), $$h_i(\lambda P_x, \ldots, \lambda P_n, \lambda V) = h_i(P_x, \ldots, P_n, V).$$ Since the demand function is compensated, the consumer's utility \(U\) is constant, and we have: $$U=h^{-1}(P_x,\ldots,P_n,V).$$ Now, let's analyze the effect of scaling all prices and expenditure by a factor \(\lambda\), that is, computing the utility when all prices and expenditure are multiplied by \(\lambda\): $$U=h^{-1}(\lambda P_x, \ldots, \lambda P_n, \lambda V).$$ Since the consumer's utility does not change (\(U\) is constant), we have: $$h^{-1}(P_x, \ldots, P_n, V)=h^{-1}(\lambda P_x, \ldots, \lambda P_n, \lambda V).$$ Taking the inverse of both sides, we obtain: $$h_i(\lambda P_x, \ldots, \lambda P_n, \lambda V) = h_i(P_x, \ldots, P_n, V).$$ Therefore, the compensated demand functions are homogeneous of degree zero.

Euler's theorem for homogeneous functions states that, for a homogeneous function \(g\) of degree \(k\), we have: $$\sum_{i=1}^n x_i\frac{\partial g}{\partial x_i} = kg(x_1, \ldots, x_n).$$ Since the compensated demand functions \(h_i\) are homogeneous of degree zero, we can apply Euler's theorem with \(k=0\), and sum over all goods \(i = 1,\ldots, n\). For each \(i\), we will compute the partial derivative of \(h_i\) with respect to expenditure \(V\) and multiply it by the corresponding price \(P_i\): $$\sum_{i=1}^n P_i\frac{\partial h_i}{\partial V} = 0.$$

The first law of demand states that, as the price of a good increases, the quantity demanded of that good decreases, holding utility constant. By applying the first law of demand to the expression from step 2, we can conclude that net substitution must prevail on average. If we increase the price of one good in the set, the first law of demand implies that the quantity demanded of that specific good will decrease, while the quantities demanded of the remaining goods will increase, maintaining the overall expenditure constant. To sum up, we have proven that the compensated demand functions are homogeneous of degree zero, and we have used Euler's theorem and the first law of demand to show that net substitution must prevail in these functions on average, proving Hicks's second law of demand.