1. Ronco has the following production function for widgets, y:
y = F(zi,z2) = 4(ziz2)1/2
Let zi be the number of workers, and z2 be robots. Suppose that we are in a 'short run' situation where the number of robots is fixed at 22. The unit cost of a robot is w2 = 36 and the unit cost of a worker is w1 = 16.
a. Find and draw the short-run production function in two cases (on the same diagram): = 4 and 22 = 9. Explain briefly in words how an increase in the number of robots affects the productivity of workers.
b. For each case (22 = 4 and z2 = 9), find the input demand for workers, zi (y). Use this to find TC(y) for each case. Draw, on the same diagram, the two TC(y) curves you have found (as accurately as possible).
For the rest of the question, you can assume that 22 = 4. Given wi = 16, the marginal cost is MC(0= y/2
c. Find AC(y), AVC(y), AFC(y). Draw these curves accurately, along with MC(y), on the same diagram. Your diagram should accurately show the level of y at which AG(y) is at a minimum (hint: MC = AC at this y).
d. Find the supply function, y(p).
e. Suppose p = 16. Calculate how much the firm supplies, and what its profits will be.
f. Now suppose p = 10. Calculate how much y the firm supplies. and what its profits will be.
g. Assuming that p = 10, find the producer's surplus (PS) of Ronco. (This is the area below the price, but above the supply curve.) Show that PS is equal to profits net of fixed costs (i.e. py - VC(y)).
Now assume that the price of a worker falls to wi = 8.
h. Redo parts (c)-(g) above (drawing new diagrams where necessary). Note: Marginal cost is now MC(y) = y/4.
Now assume again that w1 = 16, and there are initially 15 firms identical to Ronco (including Ronco) in a perfectly competitive market. Let market demand be Yd(p) = 576 - 2p.
i. Find the market supply curve, Ys(p). Draw this alongside market de-mand and calculate the market equilibrium price and quantity.
j. Calculate the output and profit of Ronco in this short run equilibrium.
k. Assuming that all firms have the same long run and short run costs (e.g. it is impossible to buy more robots), find how many firms will exist in a long equilibrium. Re-draw the market to show this situation.
1. Calculate total market CS and PS in the long-run equilibrium that you found in part (k).
2. A firm operates two plants which have different cost functions:
C1(91) = 3(91)2 + 400 and ACC1(%) = 6Y1
C2(y2) = 10(y2)2 + 100 and MC2(y2) = 20y2
where yi and y2 are the amounts of output produced at each plant and y = yi + y2 is the total output the firm produces. Show all the steps of your work.
a. If the firm wants to produce y = 26 units of output, how much should it produce at each plant?
b. Given your answer in part (a.), which plant has the highest average cost of production?
c. If the firm can sell its product for $100 a unit, how much total profit will it make when y = 26? How much profit if y = 39?
3. A firm's production function is given by y = zi(4 + z2). The T RS for production is (4 + z2)/z1. Let wi = 20 and w2 = 10. Suppose the firm is initially in a long-run situation.
a. Briefly explain why z2 is not necessary in production, and draw the isoquant for y = 18.
b. Show if the firm wants to produce y = 18 units of output, its total cost of production will be $80. Explain your answer.
c. Show that if the price of w2 rises to $40, and the firm still wants to produce y = 18, it will not use any z2. What is the total cost of production in this case? Show this situation on an isoquant-isocost diagram.
Now assume again that wi = 20 and w2 = 10, but z2 is fixed at 5 units in the short run.
d. Assuming that y = 18, how much zi will the firm want to hire? What is the short run total cost of production? Briefly explain why the total cost is higher now than in part (b).
4. MicDonald's Restaurant sells its famous burger - the BigMic - using beef patties (xi) and bread slices (z2). The BigMic production function is:
F(zi, z2) = min ta 2 ' z2)
The prices of the inputs are wi and w2.
a. Suppose MicDonald's is in a long run situation. Find the firm's demands for each input and use these to find long run total and average costs, LTC(y) and LAC(y). Draw the firm's input choices on an iso-cost / isoquant diagram when wi = 1 and w2 = 0.25.
b. Suppose now MicDonald's is in a short run situation where x2 is fixed at 60. Find SR average cost, AC(y) when wi = 1 and w2 = 0.25. Graph AC(y) on the same diagram as LAC(y), clearly showing the y at which they are the same.
c. [5 marks] MicDonald's decides to add cheese (x3) to the Big Mic, so that its production function becomes:
lx2i,z32,11
F(xi , x2, x3) =rnin
Assuming MicDonald's is in a long run situation, recalculate LTC(y). How much does it cost to produce 60 BigMics if wi = 1, w2 = 0.25, and w3 = 0.25?