Problem 9 - Advanced Macroeconomics

Question

The Harris-Todaro model. (Harris and Todaro, \(1970 .\) ) Suppose there are two sectors. Jobs in the primary sector pay \(w_{p} ;\) jobs in the secondary sector pay \(w_{s}\) Each worker decides which sector to be in. All workers who choose the sec. ondary sector obtain a job. But there are a fixed number, \(N_{p},\) of primary-sector jobs. These jobs are allocated at random among workers who choose the primary sector. Primary-sector workers who do not get a job are unemployed, and receive an unemployment benefit of \(b\). Workers are risk- neutral, and there is no disutility of working. Thus the expected utility of a primary-sector worker is \(q w_{p}+(1-q) b\) where \(q\) is the probability of a primary-sector worker getting a job. Assume that \\[ b

Step-by-Step Solution

Answer

Equilibrium unemployment is determined by comparing expected utilities. Increasing \(N_p\) might initially increase applicant numbers, leading to more unemployed by crowding effect. Higher unemployment benefits may reduce willingness to risk primary sector unemployment.

Step 1: Understanding Equilibrium Unemployment

In equilibrium, workers are indifferent between choosing the primary and secondary sector. This implies that the expected utility from choosing either sector should be equal. For the primary sector, the expected utility is \( q w_{p} + (1-q) b \), where \( q = \frac{N_{p}}{n} \) is the probability of getting a job, and \( n \) is the number of workers who choose the primary sector. For the secondary sector, it's simply \( w_{s} \). From the condition given (equilibrium condition), we equate these utilities:\[ q w_{p} + (1-q) b = w_{s} \] Solving for \( q \), we get:\[ q = \frac{w_{s} - b}{w_{p} - b} \] This represents the proportion of workers employed in the primary sector.

Step 2: Deriving Equilibrium Condition

Substitute \( q = \frac{N_{p}}{n} \) into the equilibrium condition for \( q \):\[ \frac{N_{p}}{n} = \frac{w_{s} - b}{w_{p} - b} \] Solving for \( n \), the number of workers choosing the primary sector, yields:\[ n = \frac{N_{p}(w_{p} - b)}{w_{s} - b} \] So, equilibrium unemployment as a fraction of the total labor force \( \overline{N} \) is given by:\[ u = 1 - \frac{n}{\overline{N}} = 1 - \frac{N_{p}(w_{p} - b)}{\overline{N}(w_{s} - b)} \]

Key Concepts

Equilibrium Unemployment

Equilibrium unemployment occurs when workers are equally satisfied by staying in either the primary sector or the secondary sector. In the Harris-Todaro model, this equilibrium is reached when the expected income from joining either sector is the same. For the primary sector, which pays a higher wage but has fewer jobs available, the expected utility is calculated by considering both the probability of getting a job and the unemployment benefits received if not employed. The equation used is:

• For primary sector: \[ q w_{p} + (1-q) b \] where \( q \) is the probability of employment and \( b \) represents unemployment benefits.
• For secondary sector:\[ w_{s} \]

The equilibrium condition equates these expected utilities:\[ q w_{p} + (1-q) b = w_{s} \]When this condition is satisfied, workers are indifferent between sectors, and the level of unemployment stabilizes.

Primary Sector

The primary sector offers wages \( w_{p} \), which are higher than those in the secondary sector. However, jobs are limited, making this sector attractive but also competitive. Only a fixed number of jobs, \( N_{p} \), are available, and they are allocated randomly among those who choose to work here. Because not everyone gets employed, those who are not offered jobs enter the pool of unemployed workers but still receive unemployment benefits, denoted as \( b \). This sector's function is vital in understanding why unemployment exists in equilibrium despite many workers preferring these high-paying jobs. The equilibrium unemployment occurs because not all workers choosing the primary sector find employment due to limited job availability.

Secondary Sector

The secondary sector offers wages \( w_{s} \) which are guaranteed for any worker choosing this sector, leading to zero unemployment for those who opt for it. This sector is less lucrative than the primary one, but it provides job security. Workers who do not want to face unemployment sometimes opt for the secondary sector. The decision-making process between sectors involves weighing the risk of being unemployed in the primary sector yet getting higher wages if employed, versus the certainty of employment in the secondary sector with lower wages. The secondary sector acts as a safety net, ensuring some level of employment availability in the economy, contrasting the uncertainty of primary sector employment.

Unemployment Benefits

Unemployment benefits \( b \) are payments made to workers who were unable to secure a job in the primary sector. These benefits serve as a cushion for workers who risk higher monetary gains by aiming for the primary sector. The rate of unemployment benefits impacts the choice between primary and secondary sectors. If the benefits are relatively high, workers might be more inclined to risk not getting a job in the primary sector, leading them to choose it over the otherwise secure secondary sector. Increases in unemployment benefits can thus affect the equilibrium by altering the attractiveness of the primary sector versus guaranteed employment in the secondary sector. These benefits are essential in maintaining a sense of financial security for unemployed workers and influencing labor allocation choices.