(a) How does the concept of wholesale price index work? Describe the major components of wholesale price index. Explain the methodology of index numbers of area, production and yield in agriculture. (15 marks)

(b) Explain G/M/1 model and show that t

Question

(a) How does the concept of wholesale price index work? Describe the major components of wholesale price index. Explain the methodology of index numbers of area, production and yield in agriculture. (15 marks)

(b) Explain G/M/1 model and show that the steady-state arrival point system has a geometric distribution. (20 marks)

(c) If e(x) is the average number of complete years of life lived by each of l(x) persons in life table population after attaining age x, and q(x) is the probability of dying within one year following the attainment of age x, prove that

q(x) = (1 - (e(x) - e(x+1))) / (1 + e(x+1)) (15 marks)

UPSC Answer Check · Accepted Answer

The directive 'explain' demands clear exposition with logical reasoning and supporting evidence. Structure your answer with: (a) ~30% time/space (15 marks) — define WPI, list its three major components (primary articles, fuel & power, manufactured products), then detail the Laspeyres/Fisher methodology for agricultural index numbers with base year selection; (b) ~40% time/space (20 marks) — define G/M/1 assumptions, derive the embedded Markov chain, use generating functions or recursive relations to prove the geometric steady-state distribution πₙ = (1-σ)σⁿ; (c) ~30% time/space (15 marks) — start with life table definitions, express e(x) and e(x+1) in terms of T(x) and l(x), manipulate the identities algebraically to reach the required expression. Conclude each part with a brief summary of significance.
- For (a): WPI definition as base-weighted index measuring wholesale price movements; three commodity groups with their weights (primary articles ~23%, fuel & power ~13%, manufactured products ~64% in India's WPI)
- For (a): Agricultural index methodology — fixed base vs chain base, Laspeyres formula P₀₁ = Σp₁q₀/Σp₀q₀, area and yield indices as relatives with geometric/ arithmetic mean aggregation
- For (b): G/M/1 model specification — general inter-arrival distribution, exponential service (rate μ), single server; embedded Markov chain at arrival epochs with transition probabilities
- For (b): Proof of geometric distribution — derive root σ of generating equation A*(μ(1-z)) = z where σ ∈ (0,1), show πⱼ = (1-σ)σʲ satisfies balance equations and normalization
- For (c): Life table relationships — T(x) = ∫ₓ^ω l(t)dt, e(x) = T(x)/l(x), L(x) = l(x+1) + ½d(x); algebraic manipulation using l(x+1) = l(x)(1-q(x)) to derive the identity
- For (c): Verification that the derived expression satisfies boundary conditions and consistency with complete expectation of life definitions

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	For (a): correctly identifies WPI as base-weighted index with 2011-12 base year and three commodity groups; for (b): properly specifies G/M/1 with general G, exponential M, single server, defines embedded chain; for (c): accurately defines e(x), T(x), L(x), and q(x) with standard life table notation	Basic definitions present but misses specific weights for WPI components, or confuses G/M/1 with M/G/1, or has incomplete life table symbol definitions	Fundamental errors like treating WPI as consumer price index, or G/M/1 as continuous-time Markov chain throughout, or incorrect life table relationships
Method choice	20%	10	For (a): selects appropriate index formulas (Laspeyres/Fisher) with justification; for (b): chooses embedded Markov chain approach with generating function or recursive method; for (c): selects algebraic manipulation of T(x) integrals with clear substitution strategy	Correct general approach but suboptimal choices like using simple aggregative method for agricultural indices, or direct steady-state equations without embedded chain, or roundabout derivation for (c)	Wrong methodological choices such as Paasche for WPI, treating G/M/1 as birth-death process directly, or attempting demographic approximations instead of exact algebraic proof
Computation accuracy	20%	10	For (b): correctly solves A*(μ(1-σ)) = σ for the unique root in (0,1), verifies σ = λ/μ for M/M/1 special case; for (c): precise algebraic steps showing T(x) - T(x+1) = L(x) = l(x+1) + ½d(x), exact cancellation leading to target formula	Correct final expressions but with skipped steps or minor algebraic slips in the derivation, or incomplete verification of geometric distribution normalization	Major computational errors like wrong root selection for σ, incorrect generating function setup, or algebraic mistakes in life table manipulations leading to wrong final form
Interpretation	20%	10	For (a): explains economic significance of WPI as inflation indicator and monetary policy tool, notes limitations; for (b): interprets geometric distribution as memoryless property at arrival epochs, relates σ to traffic intensity; for (c): interprets result as relationship between mortality probability and life expectancy curvature	Some interpretation present but superficial, missing policy relevance of WPI, or physical meaning of geometric parameter, or demographic insights of the identity	Purely mechanical presentation with no interpretation of why WPI matters, what geometric steady-state implies for queue behavior, or what the life table identity reveals about mortality patterns
Final answer & units	20%	10	Clear boxed/concluded answers: for (a) summarized WPI methodology; for (b) explicit πₙ = (1-σ)σⁿ with σ defined; for (c) clean derivation ending with required identity, all symbols consistently defined	Answers present but poorly highlighted, missing explicit final forms, or inconsistent notation between parts	Missing final answers, incomplete derivations stopping before conclusion, or contradictory expressions across the three parts

Q8

Directive word: Explain

How this answer will be evaluated

Approach

Key points expected

Evaluation rubric

Practice this exact question

More from Statistics 2022 Paper II