Consider a Robinson Crusoe economy in one period without an investment schedule. Prove and interpret the fact that \(d^{2} y / d x^{2}\) is positive at the optimal point \(\left(x_{0}^{*}, Y+X-x_{\circ}^{*}\right)\) satisfying the equivalent of equation (1.3) in the presence of time preference.

Data From Robinson Crusoe economy:-

Equation 1.3:-

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Suppose that there is an individual who, just like Robinson Crusoe, has been left alone on an island. He lives in a one-period economy, defined by two points in time: now and the future (say, one year from now). The fact that he is alone means, in essence, that there is no one to trade with. He is given endowments in a nonperishable good. Let X denote the initial endowment and Y denote the promised future endowment. Given this situation, Robinson faces a problem-namely, to choose how much to consume today, x, as compared to how much to consume in the second period, y. Clearly, one must have y Y+X-x (1.1) together with 0 x X and 0 y Y+X. This region of admissible consumptions is illustrated in Figure 1.1.1 Let us assume that his time preferences for consumption are fully characterized by a utility function U(x, y) defined in R2 and that the