# Solving for Steady-State in a Growth Model (Solow)

Please help with the attached problems, such as:

3. Suppose that with the following time-dependent Cobb-Douglas production function ... solve for steady-state values of key variables. E.g. output per capita, capital-output ratio. (see attached image file)

4. Similar problem, with a different production function (Y = K, which is a special case of Y = AK model where A=1).

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#### Solution Preview

3) I present a general solution and after working through it you will be able to solve all three parts by simply plugging in the values for parameters.

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First let's derive the capital accumulation equation: K(t+1) as a function of K(t)

The stock of capital tommorow will consist of all undepreciated capital tommorow plus the new capital. Amount of undepreciated capital is (1-delta)K(t) and amount of new capital is sY(t). Delta is the depreciation rate of capital, s is the savings rate.

So

K(t+1) = sY(t) + (1-delta)K(t)

= sy(t) + K(t) - deltaK(t)

now if I transfer K(t) to the left

K(t+1) - K(t) = sY(t) - deltaK(t)

Since steady state is defined to be the equilibrium where the stock of capital is not changing, we set K(t+1) = K(t) which means left side is zero.

0 = sY(t) - deltaK(t)

you can solve this equation for Y(t)/K(t) and you will get (delta/s). This is the steady state output per unit of capital. If the professor wants capital/output then you will ...

#### Solution Summary

This solution containts detailed explanation of finding steady-state variables in a growth model. Time-dependent Cobb-Douglas production function is used. There are two different questions that use two different specifications of CD production functions.

The solutions are general, so that student can see the process of arriving at the answer without getting boggled down with numbers. Of course, by substituting his or her numbers the student will find the required numerical values.