Find the utility maximizing bundle [Sundaram, P.169, Q.7 (Kuhn-Tucker Theorem) ]

A consumer with a utility function given by $$u(x_1, x_2) = \sqrt{x_1} + x_1x_2$$ has an income of $$100$$. The unit prices of $$x_1$$ and $$x_2$$ are $$4$$ and $$5$$, respectively.

(a) Compute the utility-maximizing commodity bundle, if consumption must be nonnegative.

The necessary condition for this optimization is that $$\partial L/\partial x_1 = \frac1{2\sqrt{x_1}} + x_2 -4\lambda_1+\lambda_2 = 0$$ $$x_1 - 5\lambda_1 + \lambda_3 = 0$$ $$100 - 4x_1 - 5x_2 = 0$$ $$x_1 \ge 0, \quad x_2 \ge 0$$

There are two cases we need to consider $$\{x_1 >0, x_2 >0\}$$ and $$\{x_1 >0, x_2 =0\}$$. For the first case, if I solve the above system of equations, I have $$100\sqrt{x_1} = 8x_1^{3/2} - 5.$$ In this case, how can we obtain the bundle $$(x_1^*, x_2^*)$$?

• Is the question how to solve $100\sqrt{x_1} = 8x_1^{3/2} - 5$ for $x_1$? – Michael Greinecker Sep 12 '20 at 17:45
• yes............ – shk910 Sep 12 '20 at 18:13

The equation $$100\sqrt{x_1} = 8x_1^{3/2} - 5$$ is cubic in $$\sqrt{x_1}$$. Unless you have been asked for an analytic solution (which is possible but complex), the approximate method I suggest is first to ignore the minus $$5$$, enabling division by $$\sqrt{x_1}$$ yielding:
$$100 = 8x_1$$
The first approximation is then $$x_1 = 100/8 = 12.5$$. Since $$100\sqrt{12.5} \approx 354$$ is much greater than $$5$$, it can be inferred that the exact solution will be very close to $$12.5$$. So try successive approximation, perhaps starting with $$x_1=12.6$$.