Problem set 3 - assigment seminar 3 PDF

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INTRODUCTION TO ECONOMICS: ASSIGMENT 3**1. Angela is a farmer and Bruno owns a piece of land. Angela may use Bruno’s land in order to produce grain in exchange of some rent. In this context, we will define an allocation as ((gA,l),gB),where gA+gB =g and g represents the total grain produced by Angel...

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INTRODUCTION TO ECONOMICS: ASSIGMENT 3 1. Angela is a farmer and Bruno owns a piece of land. Angela may use Bruno’s land in order to produce grain in exchange of some rent. In this context, we will define an allocation as ((gA,l),gB),where gA+gB =g and • g represents the total grain produced by Angela. • gA represents the amount of grain that Angela keeps to herself. • gB represents the amount of grain that Bruno gets as rent. • l represents Angela’s free time, where l ∈ [0, 24]. a. Assume first that Angela has utility function UA (gA , l) = gA +

(l + 1). Her

production function is g = f(h) =  (h), where h represents daily hours worked. Bruno only cares about how much grain he gets in an allocation. Thus, you can think of his utility function as being UB(gB) = gB. i. If Angela got to keep all the grain to herself, what allocation would she optimally choose? The optimal choice for Angela where she will work less hours and get more units of grain will requiere the first order condition that maximises utility of Angela. In this case, as Angela keeps all grain the derivative of UA will have to be equal to the derivative of the production function g. "  ∂U = ∂g"

1 2 (l + 1)

=

1

which leads to the following result: ((gA,l),gB) = ((3.53,11.5),0)"

2 (24 − l )

ii. Assume that Angela’s reservation utility is 5 and that the her reservation indifference curve coincides with her biological constraint. Find the set of Pareto-efficient allocations in the following way. Start at an arbitrary indifference curve for Angela. Then, find the allocation in that indifference curve that maximises Bruno’s utility. Generalise this to obtain the full set of Pareto-efficient allocations. To find the Pareto-efficient allocations we must take into account the following considerations:" l =l*= 11.5h as optimality for Angela has to be maintained as maximises utility." gA + R = g(l) No grain can be wasted as otherwise someone could be better off. " R≥0 for Bruno to accept. " U(gA,l) ≥ U0 with U0 meaning working 0 hours and receiving the reservation grain. " Hence, as utility of reservation = 5:" - gA +  (11.5 + 1) ≥ 5; gA ≥ 1.46 units of grain. "

- As no grain can be wasted: gA + R = g(l) ; 1.46 + R = 

(24 − 11.5) where R = 2.07. "

In conclusion, the efficient set follows the following conditions: " - l=l*=11.5h" - gA≥1.46 units of grain. Until 3.53" - 0≤R≤2.07" - gA + R = 3.53 units of grain. "

- P = {((gA,l),gB) | l = 11.5, gB = sqrt(12.5)−gA, gA ∈ [5− sqrt(12.5),sqrt(12.5)]}."

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iii. Show that any Pareto-efficient allocation satisfies the condition MRS = MRT. Notice that MRT is the slope of the feasible frontier, in this case, the production function determines it. So MRT = 

1

"

2 (24 − l ) The MRS is the slope of the indifference curves that follow the utility function so in this case:"

1

MRS = 

"

2 (l + 1) We can clearly see that none of the formulas above include neither the gA nor the R (reservation). Hence, time will have to be constant to maintain the condition whereas gA or R can change. " iv. Assume now that the government passes a law whereby Angela can get a subsidy of 2 units of grain if she does not work at all. Bruno knows this and he will give Angela a take-it-or-leave-it offer of hours worked and gA. What will Bruno’s offer be? Will it be Pareto-efficient? Now U0 = 2+ (24 + 1) = 7;" If we proceed as in intem (ii): " gA +  (11.5 + 1) ≥ 7; Where gA ≥ 3.46 units of grain. " As no grain can be wasted: gA + R = g(l) ; 3.46 + R = 

(24 − 11.5) where R = 0.076. "

In conclusion, the efficient set follows the following conditions: " - l=l*=11.5h" - gA≥3.46 units of grain. " - 0≤R≤0.076" - gA + R = 3.53 units of grain. " It will be Pareto-efficient because as we saw before as long as time is constant and equal to 11.5h will fulfil the condition MRT = MRS as it doesn’t depend on other parameters. "

b. Repeat items (i)-(iv) from above but with production function g = f(h) = h and utility function UA(gA,l) = gAl for Angela. In item (ii), assume that Angela’s reservation utility is 0 and that her reservation indifference curve coincides with her biological constraint. (i) OPTIMALITY FOR ANGELA: MRS = MRT

∂U/ ∂l ∂U/∂gA

= ∂g ;

gA = 1 so gA = l and all optimal allocations will follow this condition. " l

The allocations that will be minimally acceptable by Bruno where he gets 0 and Angela consumes all the grain she produces will follow that gA = g , hence :" g = gA ; 24 - l = l ; l* = 12 h and gA* = 12 units of grain. ((gA,l),gB) = ((12,12),0)" (ii) U(R) = 0; " As R is 0, gA will still be equal to g so the result will be the same as in item (i) as Bruno’s utility will be maximised at ((12,12),0)" (iii) In this case we can check in item (i) that MRT = MRS doesn’t depend on any parameter other than l so gA or R won’t affect either and as long as gA = l the allocations will be Pareto-efficient. "

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(iv) now Angela receives 2 units from the government without working, so U0 = 24 * 2 = 48. " Hence, U0 will be equal to the U when : l2 = 48 ; l* = 6.93 h. gA* = 6.93 units. " From this we can compute the maximum rent R as gA + R = g; R = (24-6.93) -6.93 = 10.14 u. "

So the set goes from ((gA,l),gB) = ((6.93,6.93),10.14) where Angela obtains less utility but Bruno has maximised rent R=10.14 until ((gA,l),gB) = ((12,12),0) where Angela obtains all the grain and Bruno no rent. " And, consequently, it will be Pareto-efficient as follows the condition gA = l "

2. Exercise 5.5: Changing conditions for production (from The Economy). Using Figure 5.4, explain how you would represent the effects of each of the following:

1. An improvement in growing conditions such as more adequate rainfall This two conditions would imply a modification of the feasible frontier that d ete rm i nes what is technically feasible and what isn’t feasible technically. So if we have to represent that, we would change the orange line more or less as above. 2. Angela having access to half the land that If Angela has access to only half of the land, she’ll amount of grain that she needs to survive. This means will change. The feasible frontier will also change and

3.

the availability to Angela of a better designed do the work of farming.

she had previously have to work more hours to get the that the biological survival constraint will decrease to one half.

hoe making it physically easier to

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These improvements will increase productivity of Angela and she will produce more in the same time compared with before. In addition, she will be less tired and she’ll need less bushels of grain to survive. Therefore, the biological survival constraint will shift right and change as follows: "

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Angela’s income is the amount she produces minus the land rent she pays to Bruno. 1.

Using Figure 5.7a, suppose Angela works 11 hours. Would her income (after paying land rent) be greater or less than when she works 8 hours? Suppose instead, she works 6 hours, how would her income compare with when she works 8 hours?

When working 11 hours, Angela will produce more bushels than when working 8 hours. However, when she pays the rent to Bruno, she is left withe left bushels. In addition, the extra amount of hours working will have a lower cost than food she loses so the point is not optimal and Angela will chose not to work. When working 6 hours, she will produce less bushels but she’ll also need less food to survive. However, it’s not a feasible option as once the rent is payed she won’t have the necessary amount of food to survive, so she’ll prefer not to work.

2. Explain in your own words why she will choose to work 8 hours. At exactly 8 hours working, the MRS = MRT. This means that after paying the rent to Bruno, she’s still in her reservation indifference curve. Therefore, 8 hours is the only number of hours that are worth working for Angela....