Question

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.) [4 marks] 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.

Answer 1

$$solution$$

A]constant return to scale

$Y = f (K,L)\\ \frac{Y}{L}=[\frac{1}{l}]F(k,l)=F(\frac{k}{l},\frac{l}{l})=F(\frac{k}{l},1)\\$

$Define\ Y ≡ Y /L \ and\ k ≡ K/L.\\ Then:\\ \ Y = F (k,1) = f (k)\\ Y=F(K,L)$

per-worker

$F (zK, zL) = A(zK) ^α (zL) ^{1−α}\\ F= z ^α z ^{1−α}AK^α L ^{1−α}\\ F= zAK^α L ^{1−α}\\ F= zF (K,L)$

Steady state

$ss:\Delta K=0\\ Investment\ equals \ depreciation$

$\Delta k=sf(k)-(n-\delta)k$

$Y=F(K,L)$

$0.05y=0.05(0.6\times 0.4\\ 0.05y=0.012\\ y=\frac{0.012}{0.05}\\ y=4.166$