Final answer:
Josh maximizes his utility by equating the marginal utility per dollar spent on goods A and B given his budget constraint. An increase in the price of good A causes both substitution and income effects, shifting consumption and utility. Compensating and equivalent variations measure the change in welfare due to the price change of good A.
Step-by-step explanation:
To maximize Josh's utility subject to his budget constraint using the Lagrangean method, we first need to set up the Lagrangian function. This would include the utility function U(A,B) = 5A1/4B3/4 and a constraint that represents Josh's income limit, which is his budget, equating expenditures on goods A and B to his total income. By differentiating the Lagrange function with respect to A, B, and the Lagrange multiplier, and then equating these derivatives to zero, we can find the optimal quantities of A and B that Josh should consume to maximize his utility.
When the price of good A (PA) increases, we can analyze the impact graphically by showing shifts on a typical consumer choice graph with good A on one axis and good B on the other. The budget line rotates inwards, resulting in a new tangency point with a lower indifference curve, indicating a loss in consumer satisfaction.
The substitution effect occurs as Josh shifts consumption from the relatively more expensive good A to the relatively cheaper good B, moving from the initial tangency point to a hypothetical point that lies on the initial higher indifference curve. This effect illustrates a change in consumption due to the change in relative prices. The income effect comes into play as a result of the effective reduction in Josh's purchasing power due to the price increase, further changing the consumption bundle along the new budget constraint to reach the final utility-maximizing point.
The compensating variation (CV) and equivalent variation (EV) represent different monetary measures of the welfare change due to a price increase. The CV is the amount of additional income Josh would need to reach the original utility level after the price increase, while the EV is the amount of income Josh would be willing to pay to avoid the price increase while remaining on the original indifference curve. These can be graphically shown as the vertical distance between the initial and the new budget constraints at the original and new optimal consumption points, respectively.