Macro Module 3 Cobb-Douglas production function practice exam questions
An economy has a Cobb-Douglas production function: Y = AKα L(1 – α).
A is the technology level, K is capital; L is labor; and Y is income.
● In 20X1, the technology level A is 143, capital K = 236, labor L = 458, income Y = 8,808
● In 20X2, the technology level A is 149.05, capital K = 248.74, labor L = 472.98, income Y = 9,620.56
● In 20X3, the technology level A is 155.59, capital K = 263.12, labor L = 490.71
Question 3.1: Percentage changes
What are the percentage changes for A, K, L, and Y from 20X1 to 20X2?
Answer 3.1: percentage change = (20X2 value – 20X1 value) / 20X1 value:
● technology level (A): (149.05 – 143) / 143 = 4.23%
● capital (K): (248.74 – 236) / 236 = 5.40%
● labor (L): (472.98 – 458) / 458 = 3.27%
● income (Y): (9,620.56 – 8,808) / 8,808 = 9.2253%
Question 3.2: α parameter (exponent of capital)
What is the α parameter (the exponent of capital) of the Cobb-Douglas production function?
Answer 3.2: 4.23% + α × 5.40% + (1 – α) × 3.27% = 9.2253% ➾
α × (5.40% – 3.27%) = (9.2253% – 3.27% – 4.23%) ➾
α = (9.2253% – 3.27% – 4.23%) / (5.40% – 3.27%) = 81.00%
Question 3.3: Elasticity of income with respect to capital
What is the elasticity of income with respect to capital?
Answer 3.3: 81% (= α)
Question 3.4: Elasticity of income with respect to labor
What is the elasticity of income with respect to labor?
Answer 3.4: 1 – 81% = 19%
Question 3.5: Percentage changes for factors of production
What are the percentage changes for A, K, and L from 20X2 to 20X3?
Answer 3.5: percentage change = (20X3 value – 20X2 value) / 20X2 value:
● technology level (A): (155.59 – 149.05) / 149.05 = 4.39%
● capital (K): (263.12 – 248.74) / 248.74 = 5.78%
● labor (L): (490.71 – 472.98) / 472.98 = 3.75%
Question 3.6: Percentage change for income
What is the percentage change for Y from 20X2 to 20X3?
Answer 3.6: 4.39% + 81% × 5.78% + 19% × 3.75% = 9.7843%
Question 3.7: Income
What is income (Y) in 20X3?
Answer 3.7: 9,620.56 × (1 + 9.7843%) = 10,561.86
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