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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.

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

Assumptions:

  • $Y_t = K_t^{\alpha} (A_tL_t)^{1-\alpha}$,
  • $\alpha = 0.5$ ($\alpha$ = 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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