In: Economics

Consider a producer making choices over two inputs, labour (l) and capital (k) with prices w = 3 and r = 1. The production technology is f(l, k) = l + 3k.

What is the marginal rate of technical substitution (MRTS)? Is there diminishing MRTS?

Find the input demands in long-run (as a function of output level)?

Find the long-run total cost, marginal cost, and average cost functions?

Do the properties of a typical cost function hold for the long run total cost function (show any two properties)?

Consider a producer making choices over two inputs, labour (l) and capital (k) with prices w = 3 and r = 1. The production technology is f(l, k) = l + 3k.

What is the marginal rate of technical substitution (MRTS)? Is there diminishing MRTS?

MRTS is the ratio of MPL and MPK. Here the two inputs have a fixed rate of substitution which is MRTS = 1/3 that is, 1 unit of capital is equally productive to three units of labor. MRTS is fixed and not diminishing

Find the input demands in long-run (as a function of output level)?

Equimarginal principle indicates that if MPL/w > MPK/r then the firm should used only labor if they are perfect substitutes. Here we see that MPL/w = 1/3 and MPK/r = 3. Since MPL/w < MPK/r, firm should use only capital in the long run given the prices

L* = 0, K* = Q/3 are the long run demand functions given the prices

For general prices w and r, we have the demand functions

L = Q, K = 0 , when MPL/w > MPK/r, that is, r/w > 3

L = 0, K = Q/3, if r/w < 3

Any combination that satisfies Q = L + 3K if r/w = 3

Find the long-run total cost, marginal cost, and average cost functions?

Cost function is C = wL + rK

C = wQ when r/w > 3 ..........(in this case with r/w = 1/3, cost function is C = 3Q)

C = rQ/3, when r/w < 3 .........(in this case with r/w = 1/3, cost function is C = Q/3)

C = wL + rK = wQ + rQ/3 ........(in this case with r/w = 1/3, cost function is C = 3Q + Q/3)

Do the properties of a typical cost function hold for the long run total cost function (show any two properties)?

Long run cost function has no fixed cost. Here there is no fixed cost so this property is satisfied. Long run ATC envelops the short run ATC. It is U-shaped as higher Q increases cost function

Consider a firm for which production depends on two normal
inputs, labour and capital, with prices w and r, respectively.
Initially the firm faces market prices of w = 6 and r = 4. The
price of capital (r) then shifts to r = 6 while w remains the same.
Use isocost-isoquant analysis to show and explain the following. A.
In which direction will the substitution effect change the firm’s
employment and capital stock? B. In which direction will the...

A firm produces output y using two factors of production
(inputs), labour L and capital K. The firm’s production function is
?(?,?)=√?+√?=?12+?12. The wage rate w = 6 and the rental price of
capital r = 2 are taken as parameters (fixed) by the firm. a. Show
whether this firm’s technology exhibits decreasing, constant, or
increasing returns to scale. b. Solve the firm’s long run cost
minimization problem (minimize long run costs subject to the output
constraint) to derive this...

Consider an individual making choices over two goods, x and y
with prices px = 3 and py = 4, and who has the income I = 120 and
her preferences can be represented by the utility function U(x,y) =
(x^2)(y^2). Suppose the government imposes a sales tax of $1 per
unit on good x:
(a) Calculate the substitution effect and Income effect (on good
x) after the price change. Also Illustrate on a graph.
(b) Find the government tax...

Assume that output is given by Q(L,K)=50 K^0.5 L^0.5 with price
of labour L = w and price of capital K = r
1.If capital in the short run is fixed at K what is the
short-run total cost?
2.Write the values for the derivatives of the Total cost with
respect to w and r. Does Shephard’s lemma hold in this case?

Consider a production function of two inputs, labor and capital,
given by Q = (√L + √K)2. Let w = 2 and r = 1. The marginal products
associated with this production function are as follows:MPL=(√L +
√K)L-1/2MPK=(√L + √K)K-1/2
a) Suppose the firm is required to produce Q units of output.
Show how the cost-minimizing quantity of labor depends on the
quantity Q. Show how the cost-minimizing quantity of capital
depends on the quantity Q.
b) Find the equation...

Consider the following production function using capital (K) and
labor (L) as inputs. Y = 10.K0.5L0.5. The marginal product of labor
is (MPL=) 5.K0.5/L0.5, and marginal product of capital (MPK) =
5.L0.5/K0.5.a. If K = 100 and L=100 what is the level of output
Y?b. If labor increases to 110 while K=100, what is the level of
output?c. If labor increases to 110 while K=100, what is the
marginal product of labor?d. If labor increases to 120 while K=100,
what...

Consider the following supply and demand equations in the market
for labour.
Supply: w=L
Demand: w= 500−L.
Use these equations to respond to the following questions.
(a) What is the market equilibrium price and quantity?
(b) Under a free market, what is the Total Surplus?
(c) Suppose that the government enacts a minimum wage of w= 400.
What is theTotal Surplus?
(d) What is the Surplus amount of labour under the minimum
wage?

Assume that output is given by Q(L,K)=50L^0.5K^0.5 with price of
labour L = w and price of capital K = r
a
Use the primal formulation of minimising costs to obtain the
demand for Labour L and capital K
2
b
Using the values of L & K obtained above, verify whether the
output Q equals the one given in the question by eliminating the
values of w and r. Are the primal and dual problems leading to the
same...

The KM Corporation builds widgets in Washington. It combines
capital (K) and labour (L) in the production function in the
following way:
Q(K,L) = K1/3L1/3 Labour cost: w =
$10 Capital rental cost: v =
$160
Illustrate this optimum McMansion production choice in an
isoquant diagram. Label everything.

A competitive firm uses two inputs, capital (?) and labour (?),
to produce one output, (?). The price of capital, ??, is $1 per
unit and the price of labor, ?? , is $1 per unit. The firm operates
in competitive markets for outputs and inputs, so takes the prices
as given. The production function is ?(?, ?) = 3? 0.25? 0.25. The
maximum amount of output produced for a given amount of inputs is ?
= ?(?, ?) units....

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