Question 14.1.4: Examine the general utility maximizing problem with two good......

The Lagrangian is \mathcal{L}(x,y)=u(x,y)-\lambda(p x+q y-m), the first-order conditions are

\mathcal{L}_{x}^{\prime}(x,y)=u_{x}^{\prime}(x,y)-\lambda p=0        (i)

\mathcal{L}_{y}^{\prime}(x,y)=u_{y}^{\prime}(x,y)-\lambda q=0         (ii)

p x+q y=m        (iii)

From equation (i) we get λ = u^{\prime}_{x}(x, y)/p, and from (ii), λ = u^{\prime}_{y}(x, y)/q. Hence, u^{\prime}_{x}(x, y)/p =u^{\prime}_{y}(x, y)/q, which can be rewritten as

{\frac{u_{x}^{\prime}(x,y)}{u_{y}^{\prime}(x,y)}}={\frac{p}{q}}       (14.1.5)

The left-hand side of the last equation is the marginal rate of substitution, or MRS, studied in Section 12.5. Utility maximization thus requires equating the MRS to the price ratio p/q.
A geometric interpretation of Eq. (14.1.5) is that the consumer should choose the point on the budget line at which the slope of the level curve of the utility function,−u^{\prime}_{x}(x, y)/u^{\prime}_{y}(x, y), is equal to the slope of the budget line, −p/q.^{6} Thus, at the optimal point the budget line is tangent to a level curve of the utility function, illustrated by point P in Fig. 14.1.1. The level curves of the utility function are the indifference curves, along which the utility level is constant by definition. Thus, utility is maximized at a point where the budget line is tangent to an indifference curve. The fact that λ = u^{\prime}_{x}(x, y)/p = u^{\prime}_{y}(x, y)/q  at point P means that the marginal utility per dollar is the same for both goods. At any other point (x, y) where, for example, u^{\prime}_{x}(x, y)/p > u^{\prime}_{y}(x, y)/q, the consumer can increase utility by shifting expenditure away from y toward x. Indeed, then the increase in utility per extra dollar spent on x would equal u^{\prime}_{x}(x, y)/p; this exceeds the decrease in utility per dollar reduction in the amount spent on y, which equals u^{\prime}_{y}(x, y)/q.
As in Example 14.1.3, the optimal choices of x and y can be expressed as demand functions of (p, q,m), which must be homogeneous of degree zero in the three variables together.

^{6}  See Section 12.3 to recall how to compute these slopes.