The Reservation Wage Curve Equation (Utility-Based)

An individual's reservation wage ((w_r)) can be expressed using a planning horizon formula. This formula can be algebraically rearranged into a weighted-average form. The steps below show this mathematical derivation. Arrange these steps in the correct logical order, starting from the initial planning horizon formula.

Interpreting Reservation Wage Formulations

An analyst is attempting to algebraically rearrange the planning horizon formula for the reservation wage into its weighted-average form. The goal is to express the reservation wage, (w_r), as a weighted average of the utility from unemployment and the utility from employment, using (\tau) as the proportion of time unemployed.

Below are the steps the analyst took:

Initial Formula: (w_r = \frac{j(b+a^M) + (h-j)v}{h}) Where:

• (j) = expected weeks unemployed
• (h) = total weeks in planning horizon
• (b+a^M) = weekly utility while unemployed
• (v) = weekly utility from the new job

Derivation Steps:

1. Split the fraction: (w_r = \frac{j(b+a^M)}{h} + \frac{(h-j)v}{h})
2. Isolate the time proportions: (w_r = \frac{j}{h}(b+a^M) + \frac{h-j}{h}v)
3. Define the proportion of time unemployed as (\tau = j/h).
4. Substitute (\tau) into the first term: (w_r = \tau(b+a^M) + \frac{h-j}{h}v)
5. Substitute for the second term's weight: (w_r = \tau(b+a^M) + \tau v)

Which step contains the logical error that prevents the derivation from reaching the correct weighted-average form?

Justifying a Key Substitution in the Reservation Wage Derivation

True or False: In the algebraic rearrangement of the reservation wage equation from its planning horizon form, (w_r = \frac{j(b+a^M)+(h-j)v}{h}), to its weighted-average form, the expression (\frac{h-j}{h}) is simplified to (1 - \tau) because (\tau) is defined as the proportion of time spent employed. (Where (j) is weeks unemployed and (h) is total weeks in the planning horizon).

An individual's reservation wage ((w_r)) can be expressed with a 'planning horizon' formula: (w_r = \frac{j(b+a^M)+(h-j)v}{h}). This can be algebraically rearranged into an equivalent 'weighted-average' form: (w_r = \tau(b+a^M) + (1-\tau)v). Match each mathematical expression from the derivation process with its correct conceptual or symbolic equivalent.

The reservation wage equation can be rearranged from its 'planning horizon' form, (w_r = \frac{j(b+a^M)+(h-j)v}{h}), to its 'weighted-average' form. The derivation begins by splitting the fraction and isolating the time proportions: (w_r = \frac{j}{h}(b+a^M) + \frac{h-j}{h}v). Given that (\tau) is defined as the proportion of time unemployed ((\tau = j/h)), the first term (\frac{j}{h}(b+a^M)) becomes (\tau(b+a^M)). To complete the final weighted-average equation, (w_r = \tau(b+a^M) + (______)v), what mathematical expression, in terms of (\tau), must be placed in the blank?

Analyzing a Conceptual Misinterpretation of the Reservation Wage Equation

An economist starts with the 'planning horizon' formula for a reservation wage:

[w_r = \frac{j(b+a^M) + (h-j)v}{h}]

They rearrange it into the 'weighted-average' form:

[w_r = \tau(b+a^M) + (1-\tau)v]

In this derivation, (\tau) represents the proportion of the planning horizon ((h)) that an individual expects to be unemployed ((j)).

If a new government policy is expected to increase the number of weeks an individual is unemployed ((j)), while the total planning horizon ((h)) remains constant, how does this change affect the weights in the final weighted-average equation?

The algebraic rearrangement of the reservation wage equation from a 'planning horizon' form to a 'weighted-average' form is conceptually significant. In the final equation, (w_r = \tau(b+a^M) + (1-\tau)v), what is the primary economic interpretation of the terms (\tau) and ((1-\tau))?