1. Grapes must be harvested by hand. This production function is characterized by fixed proportions - each worker must have one pair of system clippers to produce any output. A skilled worker with clippers can harvest 50 kg of grapes per hour.

a. Sketch the grape production isoquants for q = 500, q = 1000, and q = 1500 and indicate where on these isoquants' firms are likely to operate.

b. Suppose a vineyard owner currently has 20 clippers. If the owner wishes to utilize fully these clippers, how many workers should be hired? What should grape output be?

c. Do you think the choices described in part b are necessarily profit maximizing? Why might the owner hire fewer workers than indicated in this part?

d. Ambidextrous harvesters can use two clippers - one in each hand - to produce 75 kg of grapes per hour. Draw an isoquant map (for q = 500, 1000 and 1500) for ambidextrous harvesters. Describe in general terms of considerations that would enter into an owner's decision to hire such harvesters.

2. The production function is given by

q=AKaLb,

(where A, a, b 0) is called Cobb-Douglas production function. This function is widely used in economic research. Using the function, show the following:

a. The production function in the equation is a special case of the Cobb-Douglas.

b. If a+b=1, a doubling of K and L will double q.

c. If a+b<1, a doubling of K and L, will less than double q.

d. If a+b>1, a doubling of K and L will more than double q.

e. Using the results from parts b - d, what can you say about the return of scale exhibited by the Cobb-Douglas function?

3. The production function for puffed rice is given by

q=100KL,

where q is the number of boxes produced per hour, K is the number of puffing guns used each hour, and L is the number of workers hired each hour.

a. Calculate q = 1000 isoquant for this production function and show it on a graph.

b. If K=10, how many workers are required to produce q = 1000? What is the average productivity of puffed-rice workers?

c. Suppose technical progress shifts the production function to q=200KL. Answer part a and b for this new situation.

d. Suppose technical progress proceeds continuously at a rate of 5 percent per year. Now the production function is given by

q=1.05t100KL,

where t is the number of years that have elapsed into the future. Now answer parts a and b for this production function.

(Note: Your answers should include terms in 1.05t. Explain the meaning of these terms.)