Equilibrium with Production

Question 1

What is the simplest model of a production economy?

Answer 1

One input (labour L = T - l) with some endowment T and one output good (x) with one consumer who is also the sole producer with a convex production set i.e. concave production function i.e. decreasing returns to labour (Robinson Crusoe Model)
A CE consists of a price of x (p) and a price of leisure or cost of labour (wage rate w) such that when these prices are taken as given and the agents make their optimal choice both markets clear

Question 2

How would you represent Robinson Crusoe’s (acting as a single agent) optimal choice graphically?

Answer 2

The point on the highest indifference curve in the feasible production set which satisfies the tangency condition |MRS| = f’(L) = MPL
On an L-x plot the concave production function will be a plateauing increasing line and this will be tangent to an IC which slopes backwards as labour is a bad (utility increases towards top-left)

Question 3

What problem does Robinson Crusoe acting as a single agent solve?

Answer 3

max u(x, T - L) wrt L, c subject to f(L) = c

Question 4

What leads to a CE in the Robinson Crusoe Economy?

Answer 4

First the firm’s problem is solved: max (px - wL) subject to x ≤ f(L) (tangency between iso-profit line and f so MPL = w/p)
Then the consumer’s problem is solved: max u s.t. pc ≤ π* + wL (tangency |MRS| = w/p)

Question 5

What is the y-intercept of the optimal iso-profit line?

Answer 5

π*/p is the profit measured in units of x

Question 6

What is the relationship between the final allocation when Robinson Crusoe acts as one agent and as two?

Answer 6

They are the same

Question 7

How could the Robinson Crusoe model extend to more people?

Answer 7

The result would be the same if each firm produced where MPL = w/p and each individual consumes where |MRS| = w/p so that MPL = |MRS| (condition for Pareto-efficiency) as before

Question 8

What can be said about the welfare in a Robinson Crusoe economy?

Answer 8

The First Welfare Theorem applies so the CE is Pareto-efficient

Question 9

How would you show that Walras’ Law holds in the Robinson Crusoe economy?

Answer 9

From the budget constraint px_d(w, p) = π(w, p) + wL_s(w, p) Walras’ Law can be stated in the form p(x_d - x_s) + w(L_d - L_s) = 0

Question 10

What would happen to the Robinson Crusoe economy if there were constant returns to labour?

Answer 10

The equilibrium profit would be zero and the real wage would be equal to the MPL
If the iso-profit line were steeper than the production function then it is optimal to set L = 0 but this is an unreasonable optimum for C-D preferences
If the iso-profit line were less steep than the production function increasing L would always be better so there is no equilibrium

Question 11

What would happen to the Robinson Crusoe economy if there were increasing returns to labour?

Answer 11

Whatever the prices, the firm can always get more profit by increasing production

Question 12

What is in the Robinson Crusoe Friday model?

Answer 12

Two output goods (x and y), one inelastic (fixed) input (labour L), two agents (RC and F), one firm (with production function f)

Question 13

What is the significance of the PPF?

Answer 13

Efficient production will be on the PPF, otherwise not all resources are being used

Question 14

What is the production possibility set and how is it represented graphically?

Answer 14

The feasible combinations of x and y that can be produced given the available resources
Using a Production Possibility Frontier which is a straight line for constant returns and a concave curve for decreasing returns

Question 15

What is the slope of the PPF?

Answer 15

The MRT

Question 16

How is a Pareto-efficient allocation in the two goods case represented?

Answer 16

At some point on the PPF, an Edgeworth box can be drawn with boundaries determined by the production choice
The allocation will be Pareto-efficient if the MRS chosen in the box is equal to the MRT on the PPF

Question 17

What are the conditions for Pareto-efficiency in a two good production economy?

Answer 17

Production must be on the frontier, all produced goods need to be consumed, and for any pair of goods all consumers have the same MRS which is equal to the MRT

Question 18

How is CE found in a two good production economy?

Answer 18

Firm maximises profit (tangency between PPF and (x, y) iso-profit line) then consumers arrive at Pareto-efficient outcome with each MRS equal to the negative price ratio (so FWT holds)

Question 19

What can be said about an economy where different firms produce x and y?

Answer 19

The wage between industries must be the same
The separate profit maximisations lead to the same FOCs as profit maximising together so the equilibrium will be the same as the case of a single firm

Question 20

What additional assumptions are needed for the SWT in production economies?

Answer 20

Convex production technology in addition to convex preferences