Final answer:
To maximize utility given the UMP U = min{7q1, 5q2} with the budget constraint 8q1 + 8q2 = 1, the ratio of the consumption of good 1 to good 2 (q1/q2) is 7/5, derived from setting 7q1 equal to 5q2, as per the utility function's implication.
Step-by-step explanation:
To solve the utility maximization problem (UMP) that involves a utility function of the form U = min{7q1, 5q2} subject to a budget constraint 8q1 + 8q2 = 1, we need to find the ratio of the consumption of good 1 relative to good 2 (q1/q2) that maximizes utility given the constraint.
To maximize utility, the ratios of marginal utility to price for each good must be equal. This can be deduced from the general economic principle stating that at the utility-maximizing point, the ratio of the marginal utility of good 1 divided by its price (MU1/P1) should equal the marginal utility of good 2 divided by its price (MU2/P2).
In this specific case, the utility function suggests that utility is maximized when 7q1 equals 5q2 since it is defined by the minimum of these two expressions. Thus, setting 7q1 = 5q2 gives the ratio 7/5 = q1/q2.
So, the consumption of good 1 relative to good 2, q1/q2, is 7/5 when the utility is maximized. The budget constraint tells us that the combined price of the consumed quantities equals the budget of 1, and this is used here simply to delineate the feasible set of goods the consumer could choose from.