Cobb-Douglas Production function , 01 Diminishing returns to individual inputs 2󰇛2,󰇜 󰇛2󰇜2 󰆒22 Constant returns to scale 2󰇛2,2󰇜 󰆒󰇛2󰇜󰇛2󰇜2󰇛󰇜2 󰆒22󰇛󰇜2󰇛󰇜2󰇛󰇜 󰆒2 Output per worker    Exercises Show that the following functions exhibit • Diminishing returns to capital and • Constant returns to scale Then rewrite the function in output-per-worker terms Exercise 1 0.5 .. Diminishing returns to capital 󰆒󰇛2󰇜.. 2... 󰆒2.2 Constant returns to scale 2󰇛2,2󰇜 󰆒󰇛2󰇜.󰇛2󰇜. 2.󰇛󰇜.2.. 󰆒2.2.󰇛󰇜.. 2..󰇛󰇜.. 2󰇛󰇜.. 󰆒2 Output per worker ...... .. . Exercise 2 0.75 Diminishing returns to capital Constant returns to scale Output per worker Exercise 3 1/3 Diminishing returns to capital Constant returns to scale Output per workerExercise 4 2/3 Diminishing returns to capital Constant returns to scale Output per worker Exercise 5 0.25 Diminishing returns to capital Constant returns to scale Output per worker Finding the Steady State General version Cobb‐Douglas, , version The level of capital per worker changes in the steady state if what is added to the capital stock through investment (󰇛󰇜) is different to what is lost due to depreciation (). ∆󰇛󰇜 ∆󰇛󰇜 The “steady state” is the situation in which the level of capital per worker is steady. That is, in the steady state, ∆0 ∆0 Therefore, in the steady state, what is added to the capital stock through investment is equal to what is lost, due to depreciation. 󰇛󰇜 󰇛󰇜 To find the value of the steady‐state capital‐per‐worker, we solve for . 󰇛󰇜󰇛󰇜 󰇡󰇢󰇛󰇛󰇜󰇜 󰇡󰇢󰇛󰇜 Steady‐state level of capital‐per‐worker  is the only level of  that satisfies this equation: 󰇛󰇜 󰇡󰇢 Output is given by the production function. That is, the production function tells us how much output per worker is produced by any given level of capital per worker. 󰇛󰇜 󰇛󰇜 Since we know steady‐state capital‐per‐worker (), we can find steady‐state output‐per‐worker () by plugging  into the production function. 󰇛󰇜 󰇩󰇡󰇢󰇪 󰇡󰇢 󰇛󰇜󰇡󰇢 󰇛󰇜 󰇛󰇜󰇡󰇢 Exercise 1 Suppose that 0.5 find the Steady State level of capital per worker and output per worker. First, set out the definition of the steady state ∆󰇛󰇜0 which implies 󰇛󰇜. Then solve for . Divide both sides by 󰇛󰇜. and by . 󰇛󰇜. Apply Rule 3 of exponents 󰇛󰇜.󰇛󰇜. Raise both sides to the power of 1/ to cancel out the exponent on . /.󰇛󰇛󰇜.󰇜/. Move  to the left-hand-side and simplify the fraction .  You’ve solved for steady-state capital per worker. Now solve for steady-state output per worker by plugging  into the production function 󰇛󰇜. 󰇧󰇨. Simplify the exponents by applying rule 4.  Simplify the A’s by applying rule 2.  Properties of exponents Rule 1: 󰇛󰇜 Rule 2:  Rule 3:  Rule 4: 󰇛󰇜 Rule 5: 1 Exercise 2 0.75 Here I give you the values of steady state level of capital per worker and output per worker. Show all the steps. Start by writing out the definition of the steady state.  󰇡󰇢 Exercise 3 1/3 Find the Steady State level of output per worker. Show all of your steps. Start by writing out the definition of the steady state.