Persistent Inequality

1. Persistent Inequality By- DilipMookherjee and Debraj Ray

2. The Model • There is a continuum of agents indexed by i on [0,1] • Each agent lives for one period, and has one child who inherits the same index. • Assume preferences are fullyaltruistic. • Each individual enjoy consumption of a single good c , with one period utility u. • It is assumed that u is increasing , smooth and strictly concave. • Given altruistic preferences, the payoff to generation t is given by the “tail sum” :

3. Professions • There is some set of professions which individuals in each generation select from. • It is assumed to be a compact set of the Euclidean space. • A population distribution of professions is simply a measure • The technology combines a production sector with an educational/training sector. • The technology is represented by means of a set which contains various combinations of the form where there is at least one profession that requires no training.

4. Prices and Behavior • The price of the consumption good is normalized to unity. • There are two sets of prices relevant at each stage : is the wage function summarizing the returns to the profession , and is the cost function that gives the costs incurred by the parents to train their offspring for different professions

6. Equilibrium

7. Defining a Steady State

12. Characterizing Steady States

13. Two Profession Scenario • Let there be only two professions – Skilled and Unskilled. • For the Unskilled labour take the training cost to be zero. • For Skilled labour assume that there is an exogenous training cost x, which is the units of the consumption good used in the training process. • Let denote the fraction of population at any date that is skilled. • Let f be a well-behaved production function satisfying INADA conditions. • Then, where the subscripts denote appropriate partial derivatives

14. A fraction o of skilled people is said to be compatible with a steady state if and only if The LHS of the above equation represents the utility sacrifice of a skilled parent in educating its child. Denote it as The RHS of the above equation represents the utility sacrifice of an unskilled parent in educating its child. Denote it as The term in the middle is the present value benefit of all successive descendants being skilled rather than unskilled. Denote it as

18. Persistent Inequality (continued) DilipMookherjee Debraj Ray

19. Review • H – Compact set of professions. • Individuals indexed on [0, 1] as dynasties with fully altruistic preferences on descendants. • Population • The technology combines a production sector with an educational/training sector. • Wage function of returns to profession -

20. Review • Cost incurred by parent to train child in profession h – • Technology is given by set T with elements of the form • Firms and households solve for professions’ and consumption goods’ demands and supplies by maximizing profit and discounted utilities. • Equilibrium is a collection which is obtained from this exercise along with market clearing conditions.

21. Review • Steady State – • Proposition 1 – There is no mobility across professions in steady states. • Proposition 2 – occupying distinct professions in steady state (ie professions with different training costs) implies having different consumpitions and utilities.

22. Review • Characterize steady states through two necessary and sufficient conditions. • Profit maximisation – • No ‘one-shot deviation’ –

23. A Case of Two Professions • Assume training cost to be exogenous. • Two professions – skilled and unskilled. • Training cost of skilled is x, 0 for unskilled. • Proportion of skilled population is λ. • Assume a well-behaved production function for consumption good.

24. A Case of Two Professions • Compatibility with steady state iff – • Utility sacrifice of skilled parent educating child - • Utility sacrifice of unskilled parent educating child - • Benefit if successors are skilled -

25. A Case of Two Professions

26. A Case of Two Professions • First part follows from continuity of the curves and concavity of utility function. • Continuum of steady states, but may be disconnected. • For second part, consider maximising –

27. A Continuum of Professions • H = [0,1] • We impose the restriction of a well-defined unit cost function for each profession. • This requires assumption T.1: • Well defined production function for profession h : • Production function for consumption good :

28. A Continuum of Professions • Thus, T is generated by production functions such that and subject to • T.1 implies the existence of a well defined unit cost function for training an ‘h’ professional: • In competitive equilibrium, this unit cost will equal x(h).

29. A Continuum of Professions • T.2 assumes :

30. A Continuum of Professions • Claim: • Range of x is [0,X]. There is function W, where w(h) = W(x(h)). • Full support implies W is continuous.

31. A Continuum of Professions • For an interior x: • Hence, Claim proved.

32. A Continuum of Professions • We need to prove that this is the unique steady state wage function.

33. A Continuum of Professions • We need to prove that this is the unique steady state wage function.

34. A Continuum of Professions • We need to prove that this is the unique steady state wage function.

35. A Continuum of Professions • We need to prove that this is the unique steady state wage function.

36. A Continuum of Professions • Examples for the previous result: • A ‘Fixed-coefficient recursive technology: • A Cobb-Douglas Technology

37. A Continuum of Professions • Examples for the previous result: • A Cobb-Douglas Technology

38. A Continuum of Professions

39. Summary and Research Directions • Long-run inequality is inevitable in any steady state with occupational diversity. • If indivisibilities in investment options are removed, we get a unique steady state, with population distributions, unlike when indivisibilities are present. • Non-steady state dynamics, financial bequests, where countries are agents etc. are potential research areas.