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
Output per worker
2
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
Diminishing returns to capital
Constant returns to scale
Output per worker
1/3
Exercise 4
2/3
Diminishing returns to capital
Constant returns to scale
Output per worker
Exercise 5
Diminishing returns to capital
Constant returns to scale
Output per worker
0.25
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 (
plugging into the production function. ) by 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.
/
Exercise 4
2/3
Find the Steady State level of capital per worker. Show all your steps, including how you get to
the level of output per worker (which I give you). Start by writing out the definition of the steady
state.
Exercise 5
0.25
Find the Steady State level of capital per worker and output per worker. Start by writing out the
definition of the steady state.
The Power Rule of Calculus Suppose is a variable and and are constants. Then if a function is we find the “slope” of this function by “taking a derivative of y with respect to x” when we take the derivative (using the power rule of calculus), the exponent comes down to multiply the variable, and the new exponent becomes 1. 1. If the function is 2 we find the slope by taking the derivative, which is 2 1
2 2. If the function is the derivative is 2
2 2
2 3. Use this information to prove that if the function is /
6 . 4. Prove that if 3 , /
3
. 5. If 7 , then /
6. If , then /
7. If , then /
8. If 4 , then If , alternative notation is 3
, the derivative of with respect to is 9. If 10. If 5
, then /
, then /
5
Finding the Golden Rule Saving Rate General version Cobb‐Douglas, , version We know that if a country saves more, its capital‐per‐worker will be higher in the steady state. So we can find the optimal saving rate by finding the level of that maximizes steady‐state consumption‐per‐
worker. Steady‐state consumption‐per‐worker is Steady‐state consumption‐per‐worker, written as a function of steady‐state capital‐per‐worker, is /
We know that in the steady state, what is added to the capital stock through saving is equal to the capital that is lost to depreciation. Using this fact, we rewrite the formula for steady‐state consumption‐per‐worker to read /
/
which says, “consumption is equal to output minus the maintenance of capital”. This gives us a formula for / as a function of . 1. A country that has extremely little capital‐per‐worker in the steady state will have extremely low steady‐state output and consumption. 2. A country with extremely high steady‐state capital‐per‐worker will y*/N have very high output – but since this capital depreciates, enormous resources will have to be devoted to maintenance and max C*/N repairs (to fight off depreciation), little will be left over for consumption. 3. There’s a level of steady‐state capital‐per‐worker at which C*/N consumption is maximized. 4. At that level of , the / curve is flat – it’s slope is zero. k* We find the slope of the / function by taking the derivative of / with respect to (see the above worksheet on the Power Rule of Calculus). /
/
Now we need to find the spot where this slope = 0. /
0 /
0 So in the golden‐rule steady state, the marginal product of capital (the extra output produced by the extra unit of capital) is equal to the depreciation rate. , we can solve for k . If If we know the actual function Divide both sides by delta, and isolate capital: , then A
k
. 1 A
A
k
A
This is what has to be in the Golden Rule Steady State. In every steady state, we know, has to be We can put these two together to find out what the Golden Rule Saving Rate will be A
Notice that both sides are raised to the same exponents, so we can just delete them. A
We can also cancel out the A’s and the deltas “s” is the proportion of output that gets saved and turned into capital. Perhaps we can think of “s” as the contribution that output makes to capital accumulation. A is the engineering parameter that relates the amount of capital to the amount of production. It’s the contribution that capital makes to output. In the optimal, Golden Rule Steady State, these two are equal to each other.