The question is this:

Suppose there is a production function of $Q=F(K, L)=\sqrt{K}\sqrt{L}$. Suppose that MPC=0.8 and the labor force is growing at a rate of 0.05 per year. Also labor force participation rate is 100%.

a. What is the steady state equilibrium capital labor ratio? b. Is this equilibrium stable? c. What is the corresponding level of consumption per worker?

Any help would be appreciated.


What is the depreciation rate ($\delta$)?


  • $Y_t = K_t^{\alpha} (A_tL_t)^{1-\alpha}$,
  • $\alpha = 0.5$ ($\alpha$ = capital income share),
  • $A_0 =1$ and $g = 0$ ($g$ = growth rate of technology),
  • $s = 0.2$ ($s$ = savings rate),
  • $n = 0.05$ ($n$ = growth rate of the labor force/population).

Capital in the next period is equal to the capital from the current period, minus the part that depreciates, plus the extra investment (savings rate times production): $$K_{t+1} = K_t(1 - \delta) + s K_t^{\alpha} (A_t L_t)^{1-\alpha}$$

Multiply and divide LHS by $A_{t+1}L_{t+1}$ and multiply and divide RHS by $A_{t}L_{t}$.

$$\frac{K_{t+1}}{A_{t+1}L_{t+1}} A_{t+1}L_{t+1} = A_t L_t \left[(1 - \delta) \frac{K_t}{A_t L_t} + s \frac{K_t^{\alpha} (A_t L_t)^{1-\alpha}}{A_t L_t} \right] $$

Then note that $A_t = A_0 (1+g)^t$ and $L_t = L_0 (1 + n)^t$ and capital per effective worker is $\tilde{k}_t = \frac{K_t}{A_t L_t}$. Using this, rewrite the above equation to obtain: $$\tilde{k}_{t+1} = \frac{(1 - \delta) \tilde{k}_t + s \tilde{k}^\alpha_t}{(1+g)(1+n)}$$

On the balanced growth path $\tilde{k}_t = \tilde{k}_{t+1} = \tilde{k}^\ast$. Using this, rewrite the above to get: $$\tilde{k}^\ast = \left( \frac{s}{g+n+gn+\delta} \right)^{ \frac{1}{1-\alpha} } $$

Since you implicitly assume that $A_t = 1, \forall t$ this balanced growth path capital per effective worker is equal to capital per worker:

$$\tilde{k}^\ast = \frac{K}{AL} = \frac{K}{L} = \left( \frac{s}{g+n+gn+\delta} \right)^{ \frac{1}{1-\alpha} } = \left( \frac{0.2}{0.05+\delta} \right)^{2} $$

As you can see this $\frac{K}{L}$ is stable.

For consumption per worker, we can consume what we don't invest: $$C = Y(1 - s)$$ $$\frac{C}{L} = \frac{Y}{L} (1 - s)$$

Plugging the $\tilde{k}^\ast$ from above into the production function gives us: $$\tilde{y}^\ast = \frac{Y}{L} = \left( \frac{s}{g+n+gn+\delta} \right)^{\frac{\alpha}{1-\alpha}} $$

So, $$\frac{C}{L} = \left( \frac{s}{g+n+gn+\delta} \right)^{\frac{\alpha}{1-\alpha}} (1 -s) = \frac{0.16}{0.05 + \delta}$$


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