Shift from risk neutrality to risk aversion: savers value both expected return and certainty.
Portfolio choice between a safe asset with risk-free real return $r_f$ and a risky asset with two outcomes $r_1<r_f<r_2$ (bad/good) with probabilities $t$ and $1-t$ .
Choice variable: share $x$ in the safe asset (savings normalized to 1); consumption occurs at a known future date.

Preferences: risk aversion ⇒ indifference curves convex to the origin in the $(c_1,c_2)$ space; 45° line $c_2=c_1$ is certainty.
Expected consumption constant along lines with slope $-t/(1-t)$ .
Feasible set: straight budget line joining all-safe $(1+r_f,1+r_f)$ and all-risky $(1+r_1,1+r_2)$ ; no short selling truncates the line at these endpoints.
Risk premium: $r^e-r_f>0$ needed for risky assets to be held; trade-off measured by $(r^e-r_f)/(r_f-r_1)$ .

Payoffs: $c_1=1+x r_f+(1-x)r_1$ , $c_2=1+x r_f+(1-x)r_2$ .
Budget slope: $-\frac{r_2-r_f}{r_f-r_1}$ .
Decomposition: $-\frac{r_2-r_f}{r_f-r_1}=-\Big[\frac{t}{1-t}+\frac{1}{1-t}\cdot\frac{r^e-r_f}{r_f-r_1}\Big]$ , where $r^e=t r_1+(1-t) r_2$ .
Tangency: hold some risky asset iff budget is steeper than $-t/(1-t)$ , i.e., $r^e-r_f>0$ .

With $r^e-r_f>0$ : interior mix (some safe, some risky); if $r^e-r_f\le 0$ : corner at all-safe; if even $r_1>r_f$ : all-risky (ruled out for meaningful trade-off).
Higher risky-asset price (lower $r_1,r_2,r^e$ ) flattens the budget line ⇒ smaller risky share by substitution; income effect ambiguous but assumed dominated by substitution.

Risk aversion favors certainty (movement toward 45° line); dispersion between $c_1,c_2$ is disliked.
The risk premium compensates for downside exposure $r_f-r_1$ ; steeper budget than expected-consumption lines induces risk taking.
Market imperfection: no short selling limits feasible $(c_1,c_2)$ combinations to the segment between the two endpoints.

Risk-averse savers hold risky assets only when a positive risk premium makes the budget line steeper than the expected-consumption trade-off, yielding an interior portfolio whose risky share falls as the premium declines.

Competition and Welfare

Market Power and Efficiency