3. First and Second Welfare TheoremsThere are two goods A and B and two inputs K and L. The production functions areAKLI-aBKLYa%3DY, =The production functions display the standard properties, including constant returnsto scale.A representative houschold has utilityU(Ca, Cs)where c; is the consumption of good i. The total supply of each factor is fixedKKa + K,L = La+ Loduetion Now assume a market system.i. Households own the labor and capital, which is rented to firms. The householdfaces a budget constraintwa La + wLb +raKa+ r¿K½ > PaCa + PbCbwhere wi is the real wage paid in industry i and r; is the rental rate inindustry i. The total supply of each factor is fixed, as described above. Solvethe household's maximization problem.ii. Firms maximize profits. A firm's profits in industry A areTa = PaFa(Ka, La) – waLa – raKaand a firm's profits in industry B areTy = PoF,(Kb, Lt) – wrL – r&K,Solve the maximization problem for each firm.iii. State the equilibrium conditions assuming households and firms are price-takers.

The production functions display the standard properties, including constant returns to scale. A representative household has utility U(ca, Cs) where c; is the consumption of good i. The total supply of each factor is fixed K = Ka+K, L = La+L

The production functions display the standard properties, including constant returns to scale. A representative household has utility U(ca, Cs) where c; is the consumption of good i. The total supply of each factor is fixed K = Ka+K, L = La+L

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

See similar textbooks

Related questions

Q: Find the conditional input demand for L and K for the following cases Production function Target…

A: The production function represents the functional relationship between output and input. If…

Q: A single product firm intends to produce 20 nits of output as cheaply as possible. By using K units…

A: The firm's problem is, Min K + 20L Subject to, K1/2 + L = 20 i) Since, firm intends to produce 20…

Q: 2) Assume a firm has the following production function 4 = K³L%¸ Assume w = 2, v = 16 and that K =…

A: In short-run, capital is kept fixed and labor can vary Profit maximizing quantity of labor occurs at…

Q: function: supply function and the input de functions for the f(x₁.7₂) = A(ar² + (1 -α)rPB/P produ

A: The CES (Constant Elasticity of Substitution) production function is a mathematical representation…

Q: the production function, the Technical rate of substitution, and the elasticity of scale

A: The production function of an organization or enterprise is an association between inputs used and…

Q: Given a demand function of P = 114 -7Qd and supply of P = 27 +6Qs, the utility function U =…

A: Demand:Demand is the desire of an individual ability and willingness to pay for a product. The…

Q: Yj-2x11 Y2=3x12 Q 4. ( A) Consider the following production functions: y2 are products and xj is the…

A: Given information Y1=2X11 Y2=2X12 Y are the products X are inputs Total use of X= 1/2Y1+1/3 Y2

Q: 3) Suppose a production function is y = x,2 x,1, and w, is the ice of x, and w, is the price of x,…

A: y = x11/2x21/3 y/x1 = MP1 = (1/2)x11/2- 1x21/3 MP1 = (1/2)x1- 1/2x21/3 y/x2 = MP2 =…

Q: a company can manufacture a product according to the production function Q=3K1/2L1/2, and capital is…

A: Given that, Q = 3 K0.5 L0.5K = 4 P = $10 W = $10

Q: The best optimal production of two goods is where the isovalue line is tangent to the labor…

A: The isovalue lines represent the relationships between two commodities that can be produced using…

Q: Find the elasticity of scale and the elasticity of substitution for the CES production function +…

A: CES function Elasticity of scale = 1 as all inputs rise by double, total output will also rise…

Q: Consider the following production function with inputs L and K: Q = (L 0.5 + K0.5)2. The input…

A: Firm wish to produce certain amount of output and it wishes to produce that output at lowest…

Q: A firm can produce output using three inputs and the following production 111 technology: q=x¹x¹x.…

A: A price-taking firm is one that cannot decide the price on its own. It takes the price as given.…

Q: Exercise: A firm produces output according to the following production function: q = A K1/4 L1/4 The…

A: Production function:Production is a function of land labor capital and entrepreneurship. It includes…

Q: Suppose that the production Is given as follow: Q(N) = 22VN, where N = units of labor employed The…

A: Production function : Q = 22(N)0.5 Price of output = 10 Price of input = 15 Cost = Input price *…

Q: If labor produces output according to Q = 8L^(1/2), labor costs $10, and output sells for $100, then…

A: Marginal product is the change in Total product due to an additional labor hired.=> MP = Change…

Q: A firm uses capital and labor to produce output according to the pproduction function q = 4pKL, for…

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: A firm has the production function F(L, K) = L^(3/4)K^(1/4) The price of labor is $9 and the price…

A: “Since you have posted a question with multiple sub-parts, we will provide the solution only to the…

Q: A manufacturer's production is modelled by the function f(x, y ) = 100x 34 y 14 where x у represents…

A: The objective of the question is to find the maximum production level given the cost constraints for…

Q: A firm has the following production function Y=(x_1^α+x_2^α )^(1/2α) Obtain the conditional demand…

A: Given production function Y=(X1a+X2a)1/2a Here X1 and X2 are inputs and We take W and r as input…

Q: A cost-minimising firm produces output (q) using capital (K) and labour (L) with the production…

A: The marginal product of labor (MPL) and capital (MPK) measures the additional output produced by…

Q: A firm produces its output using the following production function: f(x₁, x₂) = √√√x₁ - 1 + 4x2…

A: The production function is a concept in economics that describes the relationship between inputs…

Q: b) Firm ABC has the following production function Q = 6K³ + 4KL - L3 i. Determine the average…

Q: A firm operates with the production function Q=2L+4K. The firm has target production of 420 units of…

A: Answer: Given, Q=2L+4K Target production = 420 units K = 80 units (short-run) w (wage) = $100 r…

Q: wage, and rent: Find the cost minimizing amount of K for a producer to use when faced with the…

A: A firm minimizes cost at the following point:(MPL / MPK) = (w / r)where MPL is the marginal…

Q: A producer will supply 30 items at a price of $15 per item. For every $9 increase in price per item,…

A: A supply function is a tool utilized by economists to estimate the relationship between the price…

Q: A firm can produce output using two inputs: capital (K) and labor (L). The level of production that…

A: Given,The production function:And

Q: In industry, when the price of electricity goes up, the company employs more low-skilled workers.…

A: Price: It refers to the cost of the goods and services on which the people purchase goods and…

Q: *Find the elasticity of scale and the elasticity of substitution for the CES production function…

A: Let the production function be: Q = x1ρ+x2ρ1ρ

Q: A manufacturer's production is modelled by the function f(x, y ) = 100x 34 y 14 where x у represents…

A: The objective of the question is to find the maximum production level given the cost constraints for…

Q: Let a firm's production function be f(K, L) = min{2K, L}. a) Solve for the contingent demand…

A: The production function shows the connection between input land, labor, capital, and entrepreneur…

Q: A firm faces the production function Q = 6K0 L0.5. If it can buy input K at £32 a unit and input L…

A: Cost is minimized when marginal rate of technical substitution equals price ratio of inputs.…

Q: Say you were interested in maximizing U = U(x,y) with the usual budget constraint, M = Px*x+Py*Y.…

A: Given, Utility function : U=U(x, y)Budget Constraint : M=Px*x+Py*y

Q: iven the production function Y=3x+2x2-0.1x3 Compute The APP and MPP

A: Marginal physical product (MPP) measures the change in output due to a change in input. The average…

Q: nput either "increase" or "decrease" where relevant: A decrease in the price of a complementary…

A: At the point when the price of complementary goods increments, keeping other factors consistent,…

Q: Q = f(L) = 15L2 – L3 Price = 10 Wage Rate = 270 Graphically and mathematically show that the…

A: We know that total product is maximum when Marginal product is zero . And in second stage of…

Q: Input either "increase" or "decrease" where relevant: An increase in the prices of inputs will…

A: Price and supply vary in response to shifts in supply and demand conditions. Equilibrium is…

Q: Demand functions are usually assumed to be linear in the competitive market model because: O The…

A: The curve that slopes downward and shows the negative relationship between the price and the…

Q: The Constant Elasticity of Substitution (CES) production function is a flexible way to de- scribe…

A: The objective of the question is to formulate the firm's optimization problem. The firm aims to…

Q: 6. Optimize the Cobb-Douglas production function given the following parameters. The maximum about…

A: Answer Given : q= K0.4 L0.6 LPl + KPk = C Where C= Cost Lagrange function: La= q-a(Lpl+KPk-C)…

Question

3. First and Second Welfare Theorems
There are two goods A and B and two inputs K and L. The production functions are
AKLI-a
BKL
Ya
%3D
Y, =
The production functions display the standard properties, including constant returns
to scale.
A representative houschold has utility
U(Ca, Cs)
where c; is the consumption of good i. The total supply of each factor is fixed
K
Ka + K,
L = La+ Lo
duetion

Now assume a market system.
i. Households own the labor and capital, which is rented to firms. The household
faces a budget constraint
wa La + wLb +raKa+ r¿K½ > PaCa + PbCb
where wi is the real wage paid in industry i and r; is the rental rate in
industry i. The total supply of each factor is fixed, as described above. Solve
the household's maximization problem.
ii. Firms maximize profits. A firm's profits in industry A are
Ta = PaFa(Ka, La) – waLa – raKa
and a firm's profits in industry B are
Ty = PoF,(Kb, Lt) – wrL – r&K,
Solve the maximization problem for each firm.
iii. State the equilibrium conditions assuming households and firms are price-
takers.

Expert Solution

Step 1

Given information

Household utility function: U(c_a ,c_b)

There are 2 goods and 2 inputs given

Production function:

For good A

Y_a = A K_a^$α$L_a ^1-^α

For good B

Y_b = B K_b^$β$ L_b^{$1 - β$}⁻^β $1$

Step by step

Solved in 3 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

A firm face the production function:Q=120L+200K-L2-2K2It can buy L at N5 a unit and K at N8 a unit with a budget of N70. Use any method ofyour choice to find the maximum output the firm can produce.
Suppose a firm uses a single input to produce a single output according to a production function f(x) = 10√x where x is the number of units of input. The output initially sells for £120 per unit. The input costs £20 per unit. A change in the market causes the product price to increase from £120 per unit to £200 per unit, all else equal. How does this change in product price affect the firm's profit maximizing level of profits? a. Profits increase by £16,000 Ob. Profits do not change Profits increase by £8,000 d. None of the other answers is correct Profits increase by £9,000 f. Profits increase by £18,000 g. Profits increase by £50,000 h. Profits increase by £32,000 C. e.
A software firm has only two inputs to production: domestic programmers based in the firm's U.K. office and international programmers working from home in low-cost countries. The two types of programmers are perfect substitutes but domestic programmers are more productive due to better communication in the office. The production function is: S=2D + I Where S is the amount of software written, D is the number of domestic programmers and I is the number of international programmers. Programmers can work part-time, so hiring 0.3 of a programmer would be possible. (a) The firm must produce 10 pieces of software this year. Show the firm's isoquant in a suitably labelled graph. Put "domestic programmers" on the vertical axis and "international programmers" on the horizontal axis. Label each axis from 0 to 10. (b) A domestic programmer can be hired for £100,000 per year. An international programmer can be hired for £60,000 per year. On the same graph, show the different combinations of…
Consider the following production function: Q = 100L0.5K0.5 subject to the budget constraint 5L + 10K = 75; where L is labour and K is capital. Derive the capital to labour ratio.
For function Y= 10+0.3x-0.006x2 suppose that price of the output is $5 and the price of input is $3 what level of input use will maximize the total value of the product
As distance to the city center increases, an office firm’s willingness to pay for space [increases / decreases / remains] at a(n) [increasing / decreasing / constant] rate.
a) Derive the goods market demand curve in terms of the output (Y) and the exogenousvariables:c0,c1,b0,b1,g0,g1andT. Show your work for full credit. b)Draw the Goods Market Equilibrium. Be sure to label all curves, label the equilibrium point, and label the slope of each curve. c)Solve for the equilibrium output (Y) in terms of the exogenous variables:c0,c1,b0,b1,g0,g1andT. Show your work for full credit. d)Supposeg1increases, but stillc1+b1+g1<1. Using a graph of the goods market, show how we would represent an increase in the value ofg1on equilibrium output y. Be sure to label all axes, curves, and equilibrium points. e)Suppose instead,c1+b1+g1= 0. Is the equilibrium in the goods market still possible? If so, what is the equilibrium output? You must explain your answer to receive full credit.
The production function is given by Q=10L+2K. The wage for labor is $20 dollar per unit and price for capital is $5 per unit. The optimal bundle (L, K) to minimize cost is: (0, Q/2) (Q/20, Q/4) (Q/15, Q/3) (Q/10, 0)
Production Function: q=32n+2nl+16l n is the size of net l is the hours of labor employed Total Cost: C=14+2n2+4l2 The price of fish is $2 per unit of fish What is the first order condition with respect to l?
3. Optimal inputs: A firm has a production function, y = ln(l, k) in a market with prices Py = 20, pe = 4, Pk = 5. Find the firm's profit maximizing choice for y, l, and k. How much profit do they earn?
Consider the following marginal cost function MC= 5 + 2qi. (a) Does the production process exhibits increasing returns? decreasing returns? constant returns? (b) If the price is $23. What’s the optimal production level? (c) If the price is $31. What’s the optimal production level?
Let y = f(x1, x2)=x11/2 + x1x2 be a firm’s production function, where x1≥0, x2≥0. Write down the firm’s production possibility set, and its input requirement set. Is this production function concave, quasi-concave? Is this production function homogenous? Find its returns to scale when x1=1, and x2=1.