The Encyclopedia of Economic Principals

Chapter 93

Agrarian Households, Food Demand, and Adaptation

Missing markets, land rent, share contracts and lagged supply on the farm.

Four of the chapter's worked examples, made interactive: farm labor when the labor market is missing, refrigeration and the vegetable ring, share contracts that trade insurance for effort, and cobwebs that converge or explode. All values are the chapter's hypothetical teaching numbers.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Missing labor markets and farm labor

When do household preferences change how much a farm uses family labor?

With complete markets, production is separable from consumption: the farm hires up to the market wage. A missing market makes the internal shadow wage the margin, so transfers and preferences move output.

Equation, written in LaTeX: pf_L=\frac{U_{\ell}}{U_c}=w^s.

Equation, written in LaTeX: MR(L_f)=102-10L_f.

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MR is the marginal revenue product of a day of family labor L_f. The outside wage is 42. Without a labor market the household uses its shadow wage w^s, the value it places on forgone leisure.

Predict first. With the market open, does a transfer that raises the shadow wage from 22 to 32 change farm labor?

Your prediction

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Figure: Missing labor markets and farm labor. A falling marginal revenue line meets a horizontal wage line at 22, giving 8 days of family labor.
Labor market: Missing, Household shadow wage: 22
Constructed example: the book's hypothetical values are MR = 102 - 10 L_f, market wage 42 and shadow wages 22 and 32; a shadow wage of 42 (equal to the market wage) is added.

Calculated values

Labor market
Missing
Operative wage
22
Farm labor L_f (days)
8

Hypothetical teaching values, not data from a real farm or market. Here with no labor market the household's shadow wage of 22 sets the margin: 102 - 10 L_f = 22 gives L_f = 8 days. Without a market, anything that changes the household's valuation of leisure changes farm labor.

Worked steps

  1. Operative wage = 22 (shadow)
  2. 102 - 10 L_f = 22, so L_f = (102 - 22) / 10 = 8

Use the idea

Before reading a labor response to a transfer as a skill or health effect, check whether households can buy and sell labor at a market wage.

Where the conclusion applies

A linear marginal revenue product and a shadow wage taken as given in each case.

Check your understanding: With the market missing and a shadow wage of 32, how much family labor does the farm use?
102 - 10 L = 32, so L = 7 days.

Chapter 93 source: section "Agricultural-household nonseparability".

Demonstration 2 of 4

Refrigeration and the vegetable ring

How far does the vegetable ring reach when transport gets cheaper?

Each use bids its return net of delivery cost. The use with the higher bid takes each distance, so the boundary sits where the bids cross. Cheaper transport for the intensive crop pushes it outward, and the rent gain is capitalized into land.

Equation, written in LaTeX: R_v(d)=1{,}080-110d,

Equation, written in LaTeX: R_g(d)=620-20d,

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R_v and R_g are the rents vegetables and grain can bid at distance d from the market. Refrigeration cuts the vegetable transport slope from 110 to 55.

Predict first. Halving the vegetable slope from 110 to 55 moves the boundary from 5.11 to about what?

Your prediction

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Figure: Refrigeration and the vegetable ring. Two falling bid-rent lines: vegetables from 1,080 with slope 110 and grain from 620 with slope 20. They cross at 5.11; grain reaches zero at 31.00.
Vegetable transport slope: 110, Grain bid at the market: 620
Constructed example: the book's hypothetical values are bids 1,080 - 110d (or 55d) and 620 - 20d; a slope of 80 and a grain intercept of 700 are added.

Calculated values

Boundary d
5.11
Grain outer margin
31.00
Vegetable bid at d = 10
-20
Grain bid at d = 10
420
Use at d = 10
Grain

Hypothetical teaching values, not data from a real farm or market. With a vegetable transport slope of 110 and a grain bid of 620 at the market, the bids cross at d = 460 / 90 = 5.11: vegetables inside, grain outside to d = 31.00. At distance 10 vegetables bid -20 against grain 420, so the land goes to grain.

Worked steps

  1. 1,080 - 110d = 620 - 20d gives d = 460 / 90 = 5.11
  2. Grain margin: 620 - 20d = 0 gives d = 31.00
  3. At d = 10: vegetables 1,080 - 1,100 = -20; grain 620 - 200 = 420

Use the idea

To see how a transport or storage improvement reshapes land use, compare the bid-rent slopes of competing uses, not just their average returns.

Where the conclusion applies

Uniform land, linear bids and a single central market.

Check your understanding: With slope 80 and grain intercept 620, where is the boundary?
1,080 - 80d = 620 - 20d gives d = 460 / 60 = 7.67.

Chapter 93 source: section "Von Thünen land-use rings".

Demonstration 3 of 4

Share contracts trade insurance for effort

What tenant share induces effort, and how much weather risk comes with it?

A share contract insures the tenant against weather by passing part of the risk to the owner, but it also passes on part of the reward to effort. The owner chooses the share that balances the two.

Equation, written in LaTeX: s\Delta Y=0.50(48)=24.

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s is the tenant's share of output. Low effort yields 120 in good weather and 40 in bad; high effort adds Delta Y = 48 in either state at a cost of 27. Fixed rent is s = 1.

Predict first. At a 0.65 share, does the tenant choose high effort?

Your prediction

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Figure: Share contracts trade insurance for effort. Left: private gain 24.0 against effort cost 27. Right: tenant income spread 40 with a line at 80 for fixed rent.
Tenant share: 0.5, Effort cost: 27
Constructed example: the book's hypothetical values are outputs 120 and 40, gain 48, cost 27 and shares 0.50, 0.65 and 1; an effort cost of 35 is added.

Calculated values

Private gain s x 48
24.0
Effort
Low
Weather spread s x 80
40
Minimum share for high effort
0.562

Hypothetical teaching values, not data from a real farm or market. With share 0.50 the tenant keeps 0.50 x 48 = 24.0 of the extra output, less than the effort cost of 27, so effort is low. The tenant bears a weather spread of 40 against 80 under fixed rent: a larger share buys effort with risk.

Worked steps

  1. Private gain = 0.50 x 48 = 24.0
  2. 24.0 < 27: low effort
  3. Spread = 0.50 x (120 - 40) = 40
  4. Minimum share = 27 / 48 = 0.562

Use the idea

Judge any revenue-sharing contract by both the incentive it leaves on effort and the risk it loads on the person exerting it.

Where the conclusion applies

Two equally likely weather states and a binary effort choice with a fixed cost.

Check your understanding: With effort cost 35, what minimum share induces high effort?
s x 48 >= 35 needs s >= 0.729; 0.65 fails and fixed rent (1.00) works.

Chapter 93 source: section "Stiglitz sharecropping tradeoff".

Demonstration 4 of 4

Converging and exploding cobwebs

When does a lagged supply response settle down, and when does it explode?

Producers plan output on last period's price; the market then clears on current demand. If supply responds less steeply than demand (d < b), errors shrink; if more steeply, they grow.

Equation, written in LaTeX: a-bP_t=c+dP_{t-1}.

Equation, written in LaTeX: P_t-P^*=-\frac{d}{b}(P_{t-1}-P^*).

Equation, written in LaTeX: Q_t^s=24+3P_{t-1}.

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Demand is 108 - 2P_t; planned supply is 24 + d P_(t-1), set on last period's price. P* is the stationary price, and each deviation is -d/b times the previous one.

Predict first. With d = 3, do price deviations shrink or grow?

Your prediction

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Figure: Converging and exploding cobwebs. Left: demand and lagged supply with the cobweb path. Right: price by period from 40, around P* = 28.00.
Supply responsiveness d: 1, Starting price: 40
Constructed example: the book's hypothetical values are a = 108, b = 2, c = 24, d = 1 or 3 and starting prices 40 and 20; d = 2 is added.

Calculated values

P*
28.00
Deviation ratio -d/b
-0.50
First prices
22.00, 31.00, 26.50

Hypothetical teaching values, not data from a real farm or market. With supply 24 + 1P_(t-1) the stationary price is 28.00. Starting from 40, prices run 40.00, 22.00, 31.00, 26.50, 28.75, 27.62, 28.19. The deviation ratio is -1 / 2 = -0.50, so deviations shrink and price converges.

Worked steps

  1. P* = (108 - 24) / (2 + 1) = 28.00
  2. Supply = 24 + 1 x 40 = 64; P_1 = (108 - 64) / 2 = 22.00
  3. Supply = 24 + 1 x 22.00 = 46.00; P_2 = (108 - 46.00) / 2 = 31.00
  4. Ratio = -1 / 2 = -0.50: deviations shrink and price converges

Use the idea

Before predicting a hog or crop cycle, check whether output is fixed in advance on old prices and compare the supply and demand slopes.

Where the conclusion applies

Linear schedules, naive price expectations and no storage; the model stops when it would need a negative price.

Check your understanding: With d = 2 and starting price 40, what are P* and the first price?
P* = 84 / 4 = 21; supply 24 + 80 = 104, so P_1 = (108 - 104) / 2 = 2; the ratio is -1, a constant oscillation.

Chapter 93 source: section "Cobweb theorem".