Optimization of Cobb-Douglas Functions
1. Maximize the following Cobb Douglas production function subject to the given constraint by 
A. Forming the Lagrange function.
B. Finding the critical values for K and L. 
C. Test the second order conditions to ensure that you have found a maximum.
D. Evaluate the total function at the critical values.

With given PK=28, PL=10, and budget B=$4000.

Constrained Optimization in Economics
2. Minimize costs for a firm with a cost function 
Subject to the production quota 
A. Form the Lagrange function.
B. Find the critical values for X, Y and λ.
C. Evaluate the total function at the critical values.
D. Estimate additional costs if the production quota is increased to 40.


3.Optimization of Cobb-Douglas Functions
1. Maximize the following Cobb Douglas production function subject to the given constraint by 
A. Forming the Lagrange function.
B. Finding the critical values for K and L. 
C. Test the second order conditions to ensure that you have found a maximum.
D. Evaluate the total function at the critical values.

With given PK=28, PL=10, and budget B=$4000.

Constrained Optimization in Economics
2. Minimize costs for a firm with a cost function 
Subject to the production quota 
A. Form the Lagrange function.
B. Find the critical values for X, Y and λ.
C. Evaluate the total function at the critical values.
D. Estimate additional costs if the production quota is increased to 40.


3. A manufacturer of parts for a tricycle (3-wheel bicycle) industry sells 3 tires X for every bike frame Y. Thus, 

If the demand functions are:
Total costs are:
Find the profit maximizing level of a) output, b) price, and c) profit.
A. Form the profit function after finding Px and PY.
B. Form the Lagrange function.
C. Calculate the critical values for X and Y.
D. Evaluate the total function at the critical values.
E. Test the second order conditions to ensure that you have found a maximum.

Optimization
4. A firm produces two goods in pure competition and has the following total revenue and total cost functions: 
A. Write the profit function.
B. Calculate the critical values for X and Y.
C. Evaluate the total function at the critical values.
D. Test the second order conditions by taking the second derivatives and evaluating the signs to ensure that you found a maximum.

Interest Compounding
5. Find the value of $2500 at 10% interest for two years compounded 
A. Quarterly.
B. Annually.
C. Continuously.

Exponential Growth Functions
6. A firm with sales of $150,000 a year expects to grow continuously at 8% a year. Determine the expected level of sales in 6 years by using the formula .

7. The cost of food has been increasing by 3.6% a year. What can a family with current annual food expenditures of $7200 expected to pay for food in 5 years.

Break-Even Analysis
8. Given the total revenue function and total cost function below, find the breakeven point of output.
Supply and Demand Analysis
9. Given the following supply and demand equations find the equilibrium price and quantity.
Supply: 
Demand: 
Total costs are:
Find the profit maximizing level of a) output, b) price, and c) profit.
A. Form the profit function after finding Px and PY.
B. Form the Lagrange function.
C. Calculate the critical values for X and Y.
D. Evaluate the total function at the critical values.
E. Test the second order conditions to ensure that you have found a maximum.

Optimization
4. A firm produces two goods in pure competition and has the following total revenue and total cost functions: 
A. Write the profit function.
B. Calculate the critical values for X and Y.
C. Evaluate the total function at the critical values.
D. Test the second order conditions by taking the second derivatives and evaluating the signs to ensure that you found a maximum.

Interest Compounding
5. Find the value of $2500 at 10% interest for two years compounded 
A. Quarterly.
B. Annually.
C. Continuously.

Exponential Growth Functions
6. A firm with sales of $150,000 a year expects to grow continuously at 8% a year. Determine the expected level of sales in 6 years by using the formula .

7. The cost of food has been increasing by 3.6% a year. What can a family with current annual food expenditures of $7200 expected to pay for food in 5 years.

Break-Even Analysis
8. Given the total revenue function and total cost function below, find the breakeven point of output.

Supply and Demand Analysis
9. Given the following supply and demand equations find the equilibrium price and quantity.
Supply: 
Demand: