Consider the Solow growth model with government spending but without technological progress. Output \( Y \) is produced according to the production function \[ Y=F(K, L)=\sqrt[3]{K} \cdot \sqrt[3]{L^{2}} \] where \( K \) is the capital stock and \( L \) is the labour force. Labour force is equal to employment and total population and grows at the rate \( n \). Capital depreciates at the rate \( \delta \). The government spends ('consumes') resources without contributing to production or capital accumulation. Government purchases, \( G \), are financed with proportional income taxation. The government budget in per capita terms is balanced, \[ g=\tau y \] where \( y \equiv Y / L \) and \( g \equiv G / L \) are output and government purchases per capita, and \( \tau \in[0,1) \) denotes the proportional tax rate. Investment \( I \) and consumption \( C \) absorb, correspondingly, constant fractions \( s \) and \( (1-s) \) of disposable income. (a) [5 marks] Show that the function \( F(K, L) \) has constant returns to scale and diminishing marginal returns to capital.

Consider the Solow growth model with government spending but without technological progress. Output \( Y \) is produced according to the production function \[ Y=F(K, L)=\sqrt[3]{K} \cdot \sqrt[3]{L^{2}} \] where \( K \) is the capital stock and \( L \) is the labour force. Labour force is equal to employment and total population and grows at the rate \( n \). Capital depreciates at the rate \( \delta \). The government spends ('consumes') resources without contributing to production or capital accumulation. Government purchases, \( G \), are financed with proportional income taxation. The government budget in per capita terms is balanced, \[ g=\tau y \] where \( y \equiv Y / L \) and \( g \equiv G / L \) are output and government purchases per capita, and \( \tau \in[0,1) \) denotes the proportional tax rate. Investment \( I \) and consumption \( C \) absorb, correspondingly, constant fractions \( s \) and \( (1-s) \) of disposable income. (a) [5 marks] Show that the function \( F(K, L) \) has constant returns to scale and diminishing marginal returns to capital.

Calcular la proyección del vector \( \vec{u}=(-3,2,1) \) sobre el vector \( \vec{v}=(-1,2,4) \) a. \( \vec{w}=\left(-\frac{11}{21}, \frac{22}{21},-\frac{44}{21}\right) \) b. \( \vec{w}=\left(-\frac{11}{21}, \frac{22}{21}, \frac{44}{21}\right) \) c. \( \vec{w}=\left(-\frac{11}{21},-\frac{22}{21}, \frac{44}{21}\right) \) d. \( \vec{w}=\left(\frac{11}{21}, \frac{22}{21},-\frac{44}{21}\right) \)

15. (a) (i) Let \( \quad(L, \leq) \quad \) be a lattice. For any \( a, b \in L \), \( a \leq b \Leftrightarrow a \wedge b=a \Leftrightarrow a \vee b=b \).

1. Representar en un diagrama de Venn el conjunto \( [A-(A \cap B)]^{C} \). Considere a) \( A \) subconjunto de \( B \) b) \( B \) subconjunto de \( A \)

IX. Please determine whether the set of vectors \( \left\{\begin{array}{lll}\alpha_{1} & \alpha_{2} & \alpha_{3}\end{array}\right\} \) is linearly dependent or not, where \( \alpha_{1}=\left(\begin{array}{lll}1 & 1 & 1\end{array}\right)^{T}, \alpha_{2}=\left(\begin{array}{llll}2 & 1 & 1\end{array}\right)^{T}, \alpha_{3}=\left(\begin{array}{lll}3 & 2 & 1\end{array}\right)^{T} \). (Write the answer for the questions on your answer sheet. 10 points)