Midterm 2

• One good model.
• Production function, $Y=F(A,K,eL)$ where,
• $Y$ is the output amount (real GDP/flow variable),
• $A$ is the technology/knowledge "amount",
• $e$ is the quality/level of education,
• $L$ is the amount of labor (stock variable),
• $K$ is capital (stock variable).
• $F$ has decreasing marginal returns wrt. each variable.
• Capital depreciates at a linear rate, $0\lt\delta\lt 1$. The depreciation period $t$ is then $\delta\cdot K_{t}$.
• Capital and output measured with the same units.
• $\gamma$ savings rate (exogenous/given/a constant by assumption).
• $K_{t+1}=K_{t}+I_{t}-\delta\cdot K_{t}$.
• Investment is savings, $I_{t}=S_{t}=\gamma\cdot Y_{t}$, where $\gamma$ is the fraction of output that is saved (a constant by assumption of the model).

Consider the Solow Growth Model with a production function $F(K,L) = KL - K^2/100$. Assume that the savings rate is $\gamma = 0.4$ and the depreciation rate is $\delta = 0.3$. The current level of capital is $K = 20$ and labor is constant and equal to $L = 1$. All computations must be accurate up to two decimal points.

• Output $Y = F(K,L) = KL - K^2/100 = 20\cdot 1 - 20^2/100 = 20 - 4 = 16$
• Investment or Savings $I = S = \gamma \cdot Y = 0.4 \cdot 16 = 6.40$
• Depreciation $D = \delta \cdot K = 0.3 \cdot 20 = 6$
• Next period capital $K_{t+1} = K + I - D = 20 + 6.40 - 6 = 20.40$
• Steady state capital $K^*$: We solve investment = depreciation $\Rightarrow \gamma \cdot F(K^*,L) = \delta \cdot K^* \Rightarrow$
  $0.4(K^* - K^{*2}/100) = 0.3\,K^*$ $\Rightarrow$ $0.4 - 0.004\,K^* = 0.3$ $\Rightarrow$ $K^* = 25$

Consider a monopolist facing a demand curve $Q = 120 - 2P$ and so $MR = 60 - Q$ and a constant marginal cost $MC = 20$. Find the monopoly quantity, price, consumer surplus, producer surplus, and deadweight loss.

Using the monopolist from the previous question ($Q = 120 - 2P$, $MR = 60 - Q$, $MC = 20$), assume the government places a price ceiling of $\$30$. Obtain the new quantity, consumer surplus, producer surplus, and dead-weight loss.