An economy can be described by the production function, 𝑌 = 𝐹(𝐾, 𝐿) = 𝐾 𝛼𝐿 1−𝛼 (a) Show that this production function exhibits constant returns to scale? (b) What is the per-worker production function?

Question 3 [20 marks]

An economy can be described by the production function,
𝑌 = 𝐹(𝐾, 𝐿) = 𝐾 𝛼𝐿 1−𝛼

(a) Show that this production function exhibits constant returns to scale?

(b) What is the per-worker production function?

 

(c) Assuming a version of the Solow growth model with population growth but no technological progress, find expressions for the steady-state capital-output ratio, capital stock per worker, output per worker, and consumption per worker, as a function of the saving rate (𝑠), the depreciation rate (𝛿), and the population growth rate (𝑛). (You may assume the condition that capital per worker evolves according to ∆𝑘 = 𝑠𝑓(𝑘) − (𝑛 + 𝛿)𝑘.)

Now consider a specific economy described by the production function,
𝑌 = 𝐹(𝐾, 𝐿) = 𝐾 0.6𝐿 0.4

The economy has no technological progress and has a depreciation rate of 5% per year. The economy starts in a steady state with growth in output (𝑌) of 5% per year. Further, the economy exhibits a capital-output ratio of 2 in this steady state.

(d) What is the saving rate for this economy? [4 marks]

(e) Suppose that the saving rate changes such that the economy transitions to the Golden-Rule steady state. What is the capital-output ratio at the Golden-Rule steady state? What is the new saving rate? [6 marks]

(f) Draw a diagram with time, 𝑡, on the horizontal axis, and consumption on the vertical axis to show how consumption per worker increases and/or decreases as the economy transitions from the starting steady state to the Golden-Rule steady state. Was this economy initially in a dynamically efficient or dynamically inefficient steady state? Explain your answer.|