February 21st, 2020 
Growth Rates of Capital and Output
Consider the following production function:
11 Yt =F(Kt,Lt)=Kt2Lt2
Assume that capital depreciates at rate and that savings is a constant proportion s of output: St = sYt
Assume that investment is equal to savings:
It = St
Finally, assume that the population is constant:
Lt =Lt+1 =L
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The production function above expresses output as a function of capital and labor (workers). Derive a function that expresses output per worker as a function of capital per worker (i.e. find yt = f(kt)).
Write down the capital accumulation equation in terms of capital per worker (i.e. an equation with only kt+1, kt, , and s.
Solve for the steady state level of capital per worker as a function of and s. Solve for the steady state level of output per worker as a function of and s. What is the steady state growth rate of output per worker?
What is the steady state growth rate of output?
Extra Credit!!
Getting the right answer will get you 5 extra credit points that will go towards your homework grade! To get full extra credit you MUST show all your work. Consider the following production
function:
11 Yt=F(Kt,Lt)=(Kt2 +Lt2)2
Assume that capital depreciates 5% each year and that households save 5% of their income. Assume that investment is equal to savings. Finally, assume that the population is growing 15% each year. Solve for the steady state level of output per worker.
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