Questions (12 math problems) Equilibrium & France’s Workforce (Math Problem Sample)
Please answer all the questions.
Initially in this economy:
• Capital stock is $400 billion (so “K” = 400)NOTE: An important skill in economics is understanding how to work with, and express results in, different units (e.g. millions vs billions). In Parts I, II and III, I’ve provided hints as to what your answers should look like, but see if you can understand the logic behind those hints. In Part IV, you'll have to figure out the units for yourself.
Part I. Finding the initial equilibrium:
1. Solve for the equilibrium real hourly wage (W)
(Mint: Set Ls = LD and solve for the wage rate, W. If you do this correctly, your answer will be a two-digit integer)
2. Solve for the equilibrium employment (L).
(Hint: Substitute the numerical value for W into either the labor supply or labor demand equation to get equilibrium employment (L) ITie variable "L" represents millions of workers. So tins is asking you how many millions of workers will be employed. If you do this correctly, your answer will be a two-digit integer.)
3. Find equilibrium GDP (Y)
(Hint: Substitute the value for equilibrium employment, as well as the given value for K into the production function to get equilibrium output (and income) Y. Remember that Y is measured in billions. So this is asking you how many billions of base year dollars in goods and services the economy is producing. If you do this correctly, your answ er will be a 4-digit integer.
4. Sketch rough graphs for the labor market and the production function (i.e., don’t worry about accurate plotting or scale). Identify on your graphs the equilibrium numerical values for W, L, and Y. (Sketching all the graphs in this problem set will help you connect what’s happening here to the graphs you saw in the lectures.)
Part II. An Increase in the Capital Stock
Now suppose that during the current year (call it “Year 1”), planned investment spending (Ip) on new capital is $100 billion, and depreciation of the existing capital stock is $16 billion, with no other change. The following year (“Year 2”) assume all of the equations remain the same, except that (as a result of investment and depreciation during Year 1), the capital stock in Year 2 will be different. For Year 2:
1. Find the new equilibrium GDP (Y)
(Hint: Substitute the value for equilibrium employment, as well as the new value for K, into the production function to get equilibrium output (and income) Y. Remember that Y is measured in billions. So if you do all this correctly, your answer will be a 4-digit integer.
2. Add to your graph what has changed, and identify the new equilibrium GDP on your graph.Part 111. An Increase in the C apital Stock and an Increase in Labor Demand
With a greater capital stock in Year 2, w orkers will be more productive, and this will generally be accompanied by an increase in labor demand (i.e., firms w ill want to hire more workers at any given real wage rate), Suppose that in Year 2, with the new capital stock, firms want to hire 10 million more workers at each real wage than they wanted to hire in Year 1. Solve again for each of the following:
1. Solve for the new equilibrium real wage (W)
The following three equations provide a numerical version of the model in a fictional economy: that has only two types of resources: Labor and Capital.labor market: Ls = 20 + 0.1 W
LD = 40 - 0.3 W
production function: Y = [(100 x L x K)][Note: Note that we’re raising the part in brackets to the '/: power (taking the square root). So this says: multiply 100 x L x K then take the square root of the result. If you’re curious, this equation for the production function creates a curve that gets continually flatter as L increases, exhibiting "diminishing returns to labor" as we discussed in lecture ]
■ W is the real hourly wage rate ($ per hour in the constant dollars of some base year)
■ Ls and L° are labor supply and demand (equations giving us the number of millions of workers supplied and demanded at each real wage rate W.
■ L is employment (number of millions of workers)
[So, for example, if L = 10, that would represent 10 million workers, and you would use L = 10 whenever the model requires a value for L)
■ Y is annual real output and real income (S billions of base-year dollars per year)
[So, for example, if Y = 10, that would represent $10 billion dollars per year, and you would use Y = 10 whenever the model requires a value for YJ
■ K is the capital stock ($ billions)
[So, for example, if K = 10, that would represent $10 billion dollars worth of capital (e.g. buildings, machines, software, etc.), and you would use K = 10 whenever the model requires a value for K)\n
[Hint: What wall the new labor demand curve look like if firms want to hire 10 million more workers at each real wage?]
2. Solve for the new equilibrium employment (L).
[Hint:. If you do this correctly, your answer will be a number with two digits to the left of the decimal point, and one digit after the decimal point.]
3. Find equilibrium GDP (Y)
[Hint: If you do this correctly, you’ll get a 4 digit number with a lot of decimals after il Round your answer to the nearest 4-digit integer ]
4. Add to your graphs what has changed, and identify on your graph the new numbers for the wage rate, employment level and total output.Part IV: A hit more challen»in»
(io back to the initial assumptions and information in Part I. Suppose that each worker works only 5(X) hours per year (i.e., pretend it’s France).
1. What are the total annual labor earnings (i.e., the total wages paid to all workers combined)? Express your answer in $billions.
2. If labor and capital are the only two resources that households supply to firms in this economy, w hat are the total annual eamings of all capital owners combined? Express your answer in Sbillions.
[Hint: Remember that GDP is both total output and total income.]
PART 1: Finding the initial equilibrium
1. The equilibrium real hourly wage (w)
LS = 20 + 0.1 W
LD = 40 – 0.3 W
At equilibrium LS = LD, thus;
20 + 0.1 W = 40 – 0.3 W
20 – 40 = - 0.3 W – 0.1 W
-20 = -0.4 W
-20/-0.4 = -0.4 W/-0.4
W = $50 per hour
2. The equilibrium employment (L)
Given LS = 20 + 0.1 W and W = $50
Then, LS = 20 + 0.1 (50) = 25
But, LS = LD
Hence, L = 25 million workers
3. Equilibrium GDP (Y)
Y = [(100 * L * K)]1/2
Y = 10 * L 0.5 * K 0.5
L = 25 while K = 400
Hence, Y = 10 * 250.5 * 4000.5
Y = 1,000 units
Hence, Y = $ 1,000 billion per year
4 a). A graph on the on the changes in the labor market and the production function s
Part II: An increase in the Capital Stock
1 The new equilibrium GDP (Y)
Ip = $ 100 billion
Depreciation on capital stock = $
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