[12 marks] HINT: A similar example is discussed in the Course Notes. My firm produces a single
product, nails, but uses 2 plants, each of which requires inputs of labour and capital. Plant 1 has
production function F(K1,L1)=K1+2L1. Plant 2 has production function G(K2,L2)=3K2+4L2. Total
capital and labour are fixed at K and L respectively, but I can decide how to allocate each of the
factors between the two plants, maximizing total production subject to the resource constraints. (i)
Write down a formal statement of the resulting optimization problem. Say what the choice
variables and the parameters are. (ii) Write down the bagrangean function, and thereby obtain
first-order conditions for the problem. (iii) Obtain an expression for K1 in terms of the parameters.
HINT: You will need to use one of the two constraints. (iv) Obtain expressions for K2,L1,L2. (v)
Describe and obtain the extreme-value function M(K,L) for this problem. (vii) Interpret any
Lagrange multipliers in the problem. HINT: The question does not ask you to find the values of
these multipliers. [8 marks] A function f:RnR is strictly concave if for all xRn and yRn, where x=y,
and for all scalar (0,1), f(x+(1)y)>f(x)+(1)f(y). Using this definition, show that the function f(x)1xTx
is strictly concave. HINTS: For clarity and brevity, I recommend you use, as well as , the notation 1
. If you cannot manage (b) for general n, attempt to work it through for the special (scalar) case n=
1, where f(x)1x2. HINT: There is a similar problem among the Problems and Answers..
Russian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in Delhi
12 marks HINT A similar example is discussed in the Cours.pdf
1. [12 marks] HINT: A similar example is discussed in the Course Notes. My firm produces a single
product, nails, but uses 2 plants, each of which requires inputs of labour and capital. Plant 1 has
production function F(K1,L1)=K1+2L1. Plant 2 has production function G(K2,L2)=3K2+4L2. Total
capital and labour are fixed at K and L respectively, but I can decide how to allocate each of the
factors between the two plants, maximizing total production subject to the resource constraints. (i)
Write down a formal statement of the resulting optimization problem. Say what the choice
variables and the parameters are. (ii) Write down the bagrangean function, and thereby obtain
first-order conditions for the problem. (iii) Obtain an expression for K1 in terms of the parameters.
HINT: You will need to use one of the two constraints. (iv) Obtain expressions for K2,L1,L2. (v)
Describe and obtain the extreme-value function M(K,L) for this problem. (vii) Interpret any
Lagrange multipliers in the problem. HINT: The question does not ask you to find the values of
these multipliers. [8 marks] A function f:RnR is strictly concave if for all xRn and yRn, where x=y,
and for all scalar (0,1), f(x+(1)y)>f(x)+(1)f(y). Using this definition, show that the function f(x)1xTx
is strictly concave. HINTS: For clarity and brevity, I recommend you use, as well as , the notation 1
. If you cannot manage (b) for general n, attempt to work it through for the special (scalar) case n=
1, where f(x)1x2. HINT: There is a similar problem among the Problems and Answers.
![[12 marks] HINT: A similar example is discussed in the Course Notes. My firm produces a single
product, nails, but uses 2 plants, each of which requires inputs of labour and capital. Plant 1 has
production function F(K1,L1)=K1+2L1. Plant 2 has production function G(K2,L2)=3K2+4L2. Total
capital and labour are fixed at K and L respectively, but I can decide how to allocate each of the
factors between the two plants, maximizing total production subject to the resource constraints. (i)
Write down a formal statement of the resulting optimization problem. Say what the choice
variables and the parameters are. (ii) Write down the bagrangean function, and thereby obtain
first-order conditions for the problem. (iii) Obtain an expression for K1 in terms of the parameters.
HINT: You will need to use one of the two constraints. (iv) Obtain expressions for K2,L1,L2. (v)
Describe and obtain the extreme-value function M(K,L) for this problem. (vii) Interpret any
Lagrange multipliers in the problem. HINT: The question does not ask you to find the values of
these multipliers. [8 marks] A function f:RnR is strictly concave if for all xRn and yRn, where x=y,
and for all scalar (0,1), f(x+(1)y)>f(x)+(1)f(y). Using this definition, show that the function f(x)1xTx
is strictly concave. HINTS: For clarity and brevity, I recommend you use, as well as , the notation 1
. If you cannot manage (b) for general n, attempt to work it through for the special (scalar) case n=
1, where f(x)1x2. HINT: There is a similar problem among the Problems and Answers.](https://image.slidesharecdn.com/12markshintasimilarexampleisdiscussedinthecours-230320164001-773c36e6/85/12-marks-HINT-A-similar-example-is-discussed-in-the-Cours-pdf-1-320.jpg)
