**A bicycle is assembled out of a bicycle frame and two wheels.**

**A firm has the production function Q **

**=**

*LK***. The firm initially faces input prices**

*w***=**

**$1 and**

*r***=**

**$1 and is required to produce**

*Q***=**

**100 units. Later the price of labor**

*w*goes up to $4. Find the optimal input combinations for each set of prices and use these to calculate the firm’s price elasticity of demand for labor over this range of prices.

Using the tangency condition, initially , implying that *K* = *L*. Since *KL* = 100, we get *K = L = *10.

Under the new prices, the tangency condition implies that *K*=4*L*. This means that the optimal input combination is (*L*, *K*) = (5, 20).

The percent change in price is (4 – 1)*100 = 300%. While the percent change in the demand for labor is [(5 – 10)/10]*100 = –50%. Therefore the price elasticity of demand over this range of prices is –50/300 = –1/6.

**A bicycle is assembled out of a bicycle frame and two wheels.****a) Write down a production function of a firm that produces bicycles out of frames and wheels. No assembly is required by the firm, so labor is not an input in this case. Sketch the isoquant that shows all combinations of frames and wheels that result in producing 100 bicycles.****b) Suppose that initially the price of a frame is $100 and the price of a wheel is $50. On the graph you drew for part (a), show the choices of frames and wheels that minimize the cost of producing 100 bicycles, and draw the isocost line through the optimal basket. Then repeat the exercise if the price of a frame rises to $200, while the price of a wheel remains $50.**

- a) The production function is
*Q*= min(*F*, ½*W*), where*F*denotes the number of frames and*W*denotes the number of wheels.

- b) To produce 100 bicycles in the least costly manner, the firm always needs to choose basket A, with 200 wheels and 100 frames.

Initially, when the price of a frame is $100 and the price of a wheel is $50, the isocost line is the lighter one shown in the graph; all points on the isocost line indicate an expenditure of $20,000.

Later, when the price of a frame is $200 and the price of a wheel is $50, the isocost line is the lighter one shown in the graph; all points on the isocost line indicate an expenditure of $30,000.

**A plant’s production function is***Q***=****2***KL***+***K***. For this production function,***MP*_{K}**=****2***L***+****1 and***MP*_{L}**=****2***K*. The price of labor services*w*is $4 and of capital services*r*is $5 per unit.**a) In the short run, the plant’s capital is fixed at***K***=****Find the amount of labor it must employ to produce***Q***=****45 units of output.****b) How much money is the firm sacrificing by not having the ability to choose its level of capital optimally?**

- a) Since , we get which implies that
*L*= 36/18 = 2. Therefore the firm’s total cost with this input combination is 4(2) + 5(9) = $53.

- b) If the firm could operate optimally, it would choose labor and capital to satisfy the tangency condition , implying that Also, Combining these two conditions, = 4.24 and
*L*= 4.8. Now the firm’s expenditure would be 4(4.24) + 5(4.8) = $41 approximately. Therefore, the firm loses about $12 because of its constraint on capital.