Economics 331
Fall 2007
Exam #2: Key
Section #1:Section I: Short Answer (30 points):EACH QUESTION IS WORTH 7.5 POINTS AND THEN BEST 4.
1.If it is given that a person will choose to work fewer hours at a given wage rate the higher her income from non- wage sources, then we know that she will choose to work less if the wage rate increases.
FALSE. SINCE THE PERSON WILL WORK FEWER HOURS IF NON-WAGE INCOME RISES, WE KNOW LEISURE IS A NORMAL GOOD, SO IF THE WAGE RATE INCREASES, THE INCOME EFFECT WOULD LEAD THE PERSON TO WORK LESS. IF THE WAGE RATE INCREASES THE SUBSTITUTION EFFECT WOULD LEAD THE PERSON TO WORK MORE. ….. SINCE WE DON’T KNOW THE SIZE OF EACH EFFECT WE CAN’T TELL WHETHER THEY WILL WORK MORE OR LESS OVERALL.
2.The law of diminishing marginal returns does not apply to a production function that exhibits constant returns to scale.
FALSE:FOR EXAMPLE, SUPPOSE Q = K1/2L1/2. THIS HAS CONSTANT RETUNS TO SCALE SINCE THE SUM OF THE EXPONENTS IS EQUAL TO 1. THE MPL IS dQ/dL = ½ K1/2L-1/2. SINCE d(MPL)/dQ WHICH IS - ¼ K1/2L-3/2 < 0 THERE ARE DIMINISHING MARGINAL RETURNS TO LABOUR.
3.A competitive firm cannot be maximizing profits if the last worker hired adds more to total output than the average productivity of its total labour force.
TRUE. DIFFERENT WAYS TO EXPLAIN THIS, BUT THE EASIEST IS: IF THE FIRM IS PRODUCING WHERE THE MPL > APL, THEN THE FIRM IS PRODUCING WHERE IT’S MC CURVE IS DECREASING. A PROFIT MAXIMIZING FIRM WILL PRODUCE SOMEWHERE ALONG THE INCREASING PORTION OF ITS MARGINAL COST CURVE ABOVE MIN AVC.
4.Suppose economic knowledge is produced with blood (B), sweat (S) and tears (T) according to the production function, Q = B1/3 S1/3 T1/3. If the price of blood is twice that of sweat, and the price of sweat is twice that of tears, how many units of B, S and T would you need to produce 100 units of knowledge.
WE KNOW THAT MPB/MPS = PB/PS
MPB / MPS= (DQ/DB) /(DQ/DS)= (1/3) B-2/3 S1/3 T1/3 / (1/3) B1/3 S-2/3 T1/3 = S/B
SINCE PB = 2PS, PB/PS = 2
THEREFORE, S/B = 2 OR S = 2B.
FOLLOWING THE SAME LOGIC YOU SHOULD SHOW THAT:T = 4B,T = 2S
PLUGGING INTO THE BUDGET CONSTRAINT FOR S AND T.
Q = B1/3(2B1/3) (4B)1/3→Q= 21/3 41/3 B →Q = 2B: WITH Q = 100:B = 50
SINCE B = 50 AND S = 2B, S = 100, AND WITH T = 2S, T = 200.
5.Suppose that food is produced with inputs labour and land. If there were increasing marginal returns to labour and constant returns to scale, show whether it would be possible or not for one farm to feed the whole world from a flower pot (yes, the workers are really tiny).
THE PRODUCTION FUNCTION WOULD TAKE THE FORM: Q = LαDβ WHERE D IS LAND, α + β = 1, AND α > 1, AND β < 0. THE MPL = αLα-1DβAND THE MPD = βLαDβ-1. SINCE dMPL/dl >0, THE MARGINAL PRODUCT OF LABOUR INCREASES WITH L. ALSO, SINCE β < 0, THE MARGINAL PRODUCT OF LAND IS < 0. THIS IS SORT OF A BIG PROBLEM SINCE IT IMPLIES THAT BY INCREASING LAND, TOTAL OUTPUT WOULD GO DOWN.
BUT, SUPPOSE WE DOUBLELABOURHOLDINGLAND FIXED. SINCE THERE ARE INCREASING MARGINAL RETURNS OUTPUT WOULD MORE THAN DOUBLE. LET’S SAY, FROM HERE WE HALVE BOTH LABOUR AND LAND. WE ARE BACK TO THE SAME AMOUNT OF LABOUR BUT
HALF AS MUCHLAND PRODUCING MORE THAN THE AMOUNT AT WHICH WE STARTED.
Section II: Calculation Problems (35 points): 12 POINTS PER QUESTION: ALL 3 COUNT (YES THIS DOES NOT ADD UP TO 35, BUT 11 2/3 POINTS PER QUESTION IS A PAIN TO WORK WITH)
1.Dan’s utility function over leisure (L) and other goods (C) is given by:U(L,C) = LC + C.
Dan buys other goods at a price of $1 per unit out of the income he earns from working at $w per hour. Dan has T hours per week to devote to either leisure or labour.
a.What is the number of leisure hours Dan would like to have?
Dan will maximize his utility where MUL/MUC = w/PC:
MUL = dU/dL = Cand MUC = dU/dC = L + 1
Therefore:MUL/MUC = w/PC→C/(L+1) = w/1→C = wL + w
Dan’s budget constraint is given by:wL + PCC = wT.
With PC = 1 and subst. in for C:wL + wL + w = wT
2L = T – 1
L* = ½ (T – 1)
b.What is Dan’s price elasticity of supply of labour? Explain.
The price elasticity of supply of labour measures how responsive the quantity of labour supplied is to the wage. Dan’s supply of labour, N, is equal to T – L* = T – ½ (T – 1) = ½ (T+1). Since his supply of labour does not depend on the wage, Dan’s labour supply curve is perfectly inelastic and the price elasticity of supply is equal to 0.
2.A firm owns two production plants that make gadgets. Each plant has the same production function, Qi = L1/2K1/2, i = 1, 2. The plants differ, however, in terms of the amount of capital equipment in place in the short run: plant 1 uses 25 units of capital and plant 2 uses 100 units of capital. The price of capital and labour are both $1.
a.As the plant manager, your job is to minimize the short run total cost of producing Q units of output. You need to decide how much to produce at plant 1, Q1, and how much to produce at plant 2, Q2: Q1 + Q2 = Q. What percentage of its total output should be produced at each plant?
We know that output should be allocated at each plant such that the marginal cost of the last unit produced at each plant is the same. So first, we need to find the marginal cost curves.
For either plant:C = wL + rK
Since w=r=1:C = L + K
In the SR w/ fixed capital:Q = L1/2K1/2→L = Q2/K
So at plant 1 (K=25)L1 = Q12/25
And at plant 1 (K=25)L2 = Q22/100
Subst back into cost:
Plant 1’s cost:C1 = Q12/25 + 25
Plant 2’s cost:C2 = Q22/100 + 100
Therefore, MC at:
Plant 1:MC1 = Q1/12.5
Plant 2:MC2 = Q2/50
Production should be set so:Q1/12.5 = Q2/50→Q1 = .25Q2
Since Q1+Q2 = Q:Q1 = .25 x (Q – Q1)→1.25Q1 = .25Q→Q1 = .20Q
Therefore:Produce 20% at plant #1 and 80% at plant #2.
b.When total output is optimally allocated between the two plants, calculate the short-run total cost curve, the average total cost curve and the marginal cost curve. What is the marginal cost of the 100th gadget? the 150th gadget?
Given part (a) this is trivial:C(Q) = C(Q1) + C(Q2) = Q12/25 + 25 + Q22/100 + 100
Subst Q1 = .20Q and Q2 = .80Q:C(Q) = (1/125)Q2 + 125
Marginal cost:MC(Q) = (2/125)Q
MC(100) = $1.60
MC(150) = $2.40
c.How would you allocate gadget production between the two plants in the long run? Find the long-run total cost curve, the average total cost curve and the marginal cost curve.
To find the total cost function we first must find the optimal demands for labour and capital:
MRTS = MPL/MPK = w/r:MRTS =½ L-1/2K1/2 = ½ L1/2K-1/2 = K/L
So:K/L = w/r = 1/1→K = L
Plug back into production:Q = L1/2L1/2→L* = Q
By symmetry:L* = Q
Plug back into cost:C = wL* + rK*→C = 1Q+ 1Q→C = 2Q
AC:AC = C/Q = 2
MC:dC/dQ = 2
3.Suppose you own a perfectly competitive fast food franchise and its total cost function is:
C = 15 + 3q + 0.05q2
where C is cost in dollars and Q is the number of meals.
a.Suppose that the price of a meal is $5. How many meals will you sell and how much profit will you earn?
SINCE THE FIRM IS PERFECTLY COMPETITIVE, IT WILL PRODUCE WHERE P = MC.
MC = dC/dq = 3 + 0.1q
P = MC:5 = 3 + 0.1q→q = 20
Π = $5 x 20 – (15 + 3 x 20 + 0.05 x 202) = 100 – 95 = $5
b.Suppose that the price of a meal falls to $3. How many meals will you sell and how much profit will you earn?
P = MC:3 = 3 + 0.1q→q = 0
FIRM WIL SHUT DOWN AND EARN AN ECONOMIC LOSS OF $15 (IT’S FIXED COST).
Section III: Discussion Questions: (35 points):18 POINTS PER QUESTION. COUNT BEST TWO
1.COUPLE OF THINGS YOU NEEDED TO REALIZE.
WITH A PROPORTIONAL TAX, THE TAX RATE IS CONSTANT NO MATTER HOW MUCH INCOME YOU MAKE. THEREFORE, THE BUDGET CONSTRAINT PIVOTS DOWN BUT HAS A CONSTANT SLOPE. WITH A PROGRESSIVE TAX, THE TAX RATE INCREASES AS YOU EARN MORE INCOME. NOW THE BUDGET CONSTRAINT PIVOTS DOWN BUT THE SLOPE GETS FLATTER AS YOU MOVE TOWARDS THE y-AXIS.
NOW SINCE THE RATES ARE SET SO THAT WORKERS ARE INDIFFERENT BETWEEN THE TWO TAXES, THE TWO NEW BUDGETS CONTRAINS SHOULD TOUCH THE SAME INDIFFERENCE CURVE. THEREFORE, THE INDIFFERENCE CURVE MUST BE TANGENT TO THE PROGESSIVE TAX BUDGET CONSTRAINT TO THE RIGHT OF THE TANGENCY POINT WITH THE PROPORTIONAL TAX BUDGET CONSTRAINT. THEREFORE, THERE IS MORE LEISURE WITH THE PROPORTIONAL TAX. SINCE THE GAP BETWEEN THE PROGRESSIVE TAX CONSTRAINT AND THE ORIGINAL CONSTRAINT IS SMALLER, THE GOVERNMENT ALSO RAISES LESS TAX REVENUE.
3.ASSUME THAT UP UNTIL 1995 THE INDUSTRY WAS OPERATING AT LONG RUN EQUILIBRIUM. BEGINNING IN 1996, THE DEMAND FOR WIDGETS INCREASED, WHICH LED TO A HIGHER PRICE AND MARKET OUTPUT. SINCE THE PRICE OF WIDGETS INCREASED, EACH FIRM IN THE INDUSTRY EACH FIRM INCREASED ITS PRODUCTION TO THE POINT WHERE THE NEW PRICE WAS EQUAL TO MARGINAL COST. THE INDUSTRY WAS IN A SHORT RUN SITUATION WERE EACH FIRM MADE AN ECONOMIC PROFIT. SINCE THERE ARE NO BARRIERS TO ENTRY, NEW FIRMS ENTERED. THIS LED TO AN INCREASE IN SUPPLY, AND PUT A DOWNWARD PRESSURE ON PRICE. ENTRY CONTINUED UNTIL THE END OF 1998, WHEN THE PRICE WAS BACK TO ITS ORIGINAL LEVEL. ALTHOUGH THE PRICE WAS BACK TO $2, TOTAL MARKET OUTPUT AND THE NUMBER OF FIRMS WAS GREATER THAN IN 1995, BUT EACH FIRM ONCE AGAIN MADE A NORMAL PROFIT.
YOUR GRAPH SHOULD LOOK LIKE THE ONE ON PAGE (XX).
I’LL POST #2 AND THE GRAPHS THIS WEEKEND.