OF SCHOOL PUBLIC POLICY UNIVERSITY OF MARYLAND PUAF 640: Microeconomics and Policy Analysis Fall 2008 PROBLEM SET 5 SOLUTIONS
Question 1: Bicycle manufacturers have roughly the following production function: X = 5*L*K2 where X is the number of bicycles produced (the output) L is the amount of labour hired (in worker days) K is the amount of capital used (in units of equipment) (a) Plot an isoquant map with units of K on the vertical axis and L on the horizontal axis. Bear in mind that each isoquant is associated with a particular level of output, and that isoquants never intersect each other.
Units of K 2.0
1.5
1.0
0.5
0.0 0 1 2 3 4 5 Units of L
Q =10
Q =15
Q =20
(b) Does this production function exhibit decreasing, constant or diminishing returns to scale? Give explanations and show calculations in support of your answer. Hint: Use K = 2 and L = 2 as your benchmark level of inputs.
1
Returns to scale summarises the proportion of change in output if all inputs are simultaneously change by a constant proportion/factor. At K0 = 2 and L0 = 2, X0 = 5*2*(2)2 = 40 bicycles Let all inputs be doubled such that at K1 = 4 and L1 = 2, X1 = 5*4*(4)2 = 320 bicycles So a doubling of inputs more than doubled the output. Hence this production function demonstrates increasing returns to scale. More generally, Given, X0 = 5*L0*K02 Let X1 = 5*(a*L0)*(a*K0)2 each input has been changed proportionately by a constant factor "a". X1 = 5*(a*L0)*(a*K0)2 X1 = 5*a*L0*a2*K02 X1 = 5*a3*L0*K02 X1 = a3*X0 Implying output changes by a factor a3 for a change in inputs by a factor a. So if a>1, this demonstrates IRS (c) Calculate the Marginal Physical Product of Labour (MPPL) of the given production function. Also compute the Marginal Physical Product of Capital (MPPK) of the given production function. MPPL MPPK = X/L [the partial derivative of X(L,K) with respect to L] = 5 * K2 = X/K [the partial derivative of X(L,K) with respect to K] = 5 * L * 2K = 10 * L * K
(d) What is MRTSKL at the point K = L = 2? MRTSKL = MPPL / MPPK (by definition) 2 = (5 * K ) / (10 * L * K) = K / 2L At the point K = L = 2, MRTSKL = 1/2 Suppose the firm manufacturing bicycles decides to spend at most $ 400,000. Labour wage rate, w = $20 and capital rental rate, r = $25 per unit of equipment. (e) Draw the isocost line that acts as a constraint for the firm. Make sure to denote its slope and intercepts.
2
Isocost line: I = w*L + r*K 400000 = 20*L + 25*K Rewriting, 25K = 400000 20L K = 16000 20/25X And hence a straight line with a vertical intercept of 16000 and a slope (Y/X) of 0.8
(f) What is the optimal number of bicycles this firm should manufacture to maximize profits? Remember, it does not want to spend any more than $400,000. Show your calculations. Hint: You could use the Lagrange function to solve this question. For this we use can either use the property of optimal point or the Lagrange equation.
3
Using the property of the optimal point: Only at the optimal point, the slope of the isocost line equals the slope of the isoquant. Slope of the isocost line = w/r (by definition) = 20/25 = 0.80 Slope of the isoquants = MRTSKL (by definition) = MPPL/MPPK (by definition) = K/2L Therefore at optimal point: K/2L = 0.80 K = 8L/5 Substituting this result into the isocost line equation: 400000 = 20L + 25*(8L/5) 400000 = 60L L* = 400000/60 = 6666.67 worker days K* = 8L*/5 = 10666.67 units of equipment
4
Question 2: A perfectly competitive, profit maximizing firm faces the following total cost curve. TC = 26Q - 4Q 2 + ( 1 3) Q 3 + 50 (a) Find the firm's marginal cost (MC), average variable cost (AVC), and average total cost (ATC) curves. e TC MC = = 26 - 8Q + Q 2 Q 1 1 AVC = Q - 4Q 2 + Q3 Q = 26 - 4Q + Q 2 26 3 3 TC 1 50 ATC = = 26 - 4Q + Q 2 + Q 3 Q (b) If the market price of the firm's output is $10, how much will this firm produce? What will the firm's profits be? If the firm produced, it would produce Q*= 4 with profits approximately equal to 71. But it will not produce at all since the market price is less than the AVC. According to the shut down rule, the firm will shut down. Thus, the correct answer is that output will be zero and profits will be zero. Give 5 points for correctly getting output and profits if the firm stayed open. Give 3 points for recognizing that this firm would shut down. This can be seen with the following algebra. The firm would set MR=MC=P=10. This implies that: MC = 26 - 8Q + Q 2 = 10 Q 2 - 8Q + 16 = 0
( Q - 4) ( Q - 4) = 0
Q* = 4 The profit would be: = P * Q * -TC ( Q *)
= 10 ( 4 ) - ( 4 ) - 4 42 + ( 1 3) 43 + 50 26 - 71 But since there is negative profit, we should check if the price is less than the AVC at the equilibrium quantity (i.e. where price is equal to MC). At Q* = 4: 16 AVC = 10 + f 10 = P * 3 Therefore, the firm should shut down since the price of the product is less than the average economic cost of production at the profit-maximizing output.
( )
5
Question 3: Consider a firm that uses capital (K) and labor (L) as its inputs to produce bicycles. Suppose that this firm is producing its profit maximizing level of output. How will the optimal combination of inputs change (in the long run) if there is a decrease in the cost of capital? How will total cost change? Use a diagram to explain your answer. The firm is producing Q*, its profit maximizing level of output. Initially, the cost minimizing (or profit maximizing) way to do this is to use L*0 and K*0 units of input. At the initial wage rate and capital rent rate (w0 and v0), the firm can produce Q* by spending as little as TC0. We then decrease the rental rate of capital from v0 to v1 . Thus, the isocost line rotates out, such that it has a higher y-intercept. The new y-intercept is TC0/v1 . That is, with the new lower price of capital, the firm can hire more capital than it could before with the same total amount of expenditure. The new isocost line is also steeper; we have a new family of isocost lines. However, at this new lower price of capital, if the firm spends still TC0, then they can produce more than Q*. Thus, TC0 is no longer the cost minimizing way of producing Q*. The firm must find their new cost minimizing way of producing Q*, i.e. using their new family of isocost lines. The new cost minimizing amount, TC1 , is less than the initial cost (see the blue line). The new optimal input combination uses less labor and more capital. Intuitively, all of this makes sense. The cost of an input has decreased, so we can produce the same amount as before without spending as much. The input combination has changed such that we shifted toward the input that became relatively cheaper.
6
Question 4: An important principle of tax theory is that the effect a tax does not depend on the statutory incidence of the tax (i.e. who is legally liable) in a perfectly competitive market. The effective price paid by consumers and received by sellers is the same regardless of whether it is a sales tax or a production tax. This principle has a lot of applications, including whether employers or employees should `finance' health care benefits and import versus export taxes. Consider a sales tax ($t per unit sold) placed on all consumers in a perfectively competitive market. (a) Show that the effective price paid by consumers and producers is equivalent when a per unit excise tax ($t per unit sold) placed on consumers or producers. (Hint: Using the supply and demand framework, model these taxes as we did at the beginning of the semester. You do not have to use specific numbers.) (b) What happens to the market price? That is, while producer and consumer taxes yield the same effective prices, do they also yield the same market price? If not, how does the market price differ? When there is a tax on the consumer, we shift the demand curve down by the amount of the tax. When there is a tax on the producer, we shift the supply curve down by the amount of the tax. The below diagrams show that the effective prices faced by the consumer and producer are the same regardless of where the tax is levied. But, the market price differs. It is P1 for the consumer tax, and P1+t for the producer tax.
7
Extra Practice Question 1 (do not hand in): Please note that my answers highlight just some of the differences (those that come first to mind) between these markets and a perfectly competitive market. It is quite possible that there are other assumptions/market structure characteristics that are violated in each case. (a) Aircraft industry (e.g. companies like Boeing) This is far from a perfectly competitive industry. Firms are not price takers and they behave strategically, primarily because there are just a few, large firms (violates PC). Primarily just Boeing and Airbus. (b) Farm product, like potatoes Agricultural products to be as close as we get to a perfectly competitive industry. There are lots of small buyers (same as PC). There are lots of small sellers (same as PC). No barriers to entry. Homogeneity. However, this characterization can depend on how (where) we define the market. If we just consider the domestic market, then it's close to PC. But, if we consider the international market, there are certainly barriers for `nondomestic' farmers to enter the U.S. market. (c) Automobile industry There are a lot of small buyers but this not the case for the sellers. There are a handful of large sellers (violates PC). There is certainly some degree of substitutability across products, but cars are certainly not homogeneous (violates PC). (d) Cereal industry (e.g. companies like Kellogg) While there are many different types of cereals, there are just a few large prod...