Instructor: Akos Lada
Mid-Career MPA
Harvard Kennedy School
Summer Program 2014
Problem/Question of the Day
Distributed: 7/25/14
Due in Class: 7/28/14
Questions 4-6: OPTIONAL
This is a modeling exercise. As a result, I will not give you functions or exact numbers, but you should
write your own model for the problem. No two models are going to be exactly the same. When you
make your assumptions, state them clearly, and keep in mind that generality comes at the cost of
tractability. The best model is the most general one that still allows us to solve it. Try to write your
model down with parameters rather than concrete numbers (e.g. for a linear demand curve write
q(p)=a-bp rather than q(p)=4-2p). Please write equations algebraically, but graphs probably also help.
You would like to write a model for the choice of a country on production and consumption. Assume the
country has a limited number of resources it can devote to producing military equipment and nonmilitary consumption goods. Assume that if the country has absolutely no military equipment then it will
conquered (bad outcome), or if it has absolutely no non-military consumption goods, the population will
starve (bad outcome again…).
Instructor: Akos Lada
Harvard Kennedy School
Mid-Career MPA
Summer Program 2014
1. Model the production possibilities of the country. How many inputs are there? What is the
resource constraint(s)? What is the shape of the production possibilities frontier?
The inputs to the production possibilities equation will be (i) production of military equipment and (ii)
production of consumption goods.
The resource constraint will be determined by the potential total output of the economy. The output of
the economy will be a function of the country’s stocks of labour, capital, land and technology.
I assume diminishing marginal returns to scale for either of the two goods, therefore at the extremes, the
country will be less efficient at producing only military goods or only consumption goods. This implies a
convex production possibilities frontier.
Let M = production of military equipment, C = production of consumption goods, a,b,k are constants. a
and b should be > 1 as this is required for diminishing marginal returns to scale, k is > 0 as we can only
have positive output from the economy.
PPC k = M^a + C^b
Instructor: Akos Lada
Harvard Kennedy School
Mid-Career MPA
Summer Program 2014
2. Now model the consumption choice of the country. Assume some utility function. Why did you
choose this utility function? What shape do the indifference curves take?
Firstly we assume that the more of C or M we get, the greater utility, so U is a function of C and M.
Assuming that we have zero utility if the country starves or the country is conquered. Let PC = probability
of being conquered and PS = probability of starving. Let PC = 1-M^x/k and PS = 1-C^y/l. Where k,l,x,y are
positive constants and M^x<=k, C^y<=l for all M and C.
U= f(not conquered, not starving) = f((1-(1-M^x/k))* (1-(1-C^y/l))); where k and l are positive constants.
Therefore U = f((M^x/k),(C^y/l),C,M)). U = q * C^a * M^b*(M^x/k) * (C^y/l); where a,b,x,y,k,l and q are
positive constants. We have the following terms: q = utility multiplier, C^a = utility from C, M^b = utility
from M, M^x/k = probability of not being conquered and C^y/l = probability of not starving.
This intuitively makes sense, the more we spend on M the lower probability of being conquered, the
greater U. Equally the more we spend on C, the lower probability of starving, the greater U.
We can simplify the equation to U = K * C^s * M^t
This chart is going to be a hyperbola which makes sense intuitively. As military spending tends to zero,
we need a lot of consumption to make up for the very likely probability that we are conquered. So as M>0 C->∞ for any given U. Equally as C tends to zero we need a lot of military spending to make up for the
very high probability of starvation, so as C->0, M->∞ for any given U.
Instructor: Akos Lada
Harvard Kennedy School
Mid-Career MPA
Summer Program 2014
3. Assuming autarky, graphically illustrate the optimal production and consumption choices of the
country.
Instructor: Akos Lada
Harvard Kennedy School
Mid-Career MPA
Summer Program 2014
4. (OPTIONAL) Assuming autarky, derive algebraically the optimal production and consumption
choices of the country.
PPF = M^a + C^b
U = K * C^s * M^t
Constrained optimization for two variables
PPF’M = a M ^ (a-1)
PPF’C = b C ^ (b-1)
U’M = t * K * C^s * M^(t-1)
U’C = s * K * C^(s-1) * M^t
U’M = λ (PPF’M) = t K C^s M^(t-1) = λ aM^(a-1). -> λ = tKC^sM^(t-a)/a = t KC^sM^tM^-a/a = t(U)/aM^a
U’C = λ (PPF’C) = s K C^(s-1) M^t = λ bC^(b-1). -> λ = sKM^t C^(s-b) / b = skM^tC^sC^-b/b=s(U)/bC^b
t(U)/aM^a=s(U)/bC^b -> M^a = tbC^b/as
PPF=C^b+tb/as * C^b = ((as+tb)/as)*(C^b)->C^b= PPFas/(as+bt)
PPF=M^a +as/tbM^a = ((tb+as)/bt) * M^a -> M^a= PPFbt/(as+bt)
Instructor: Akos Lada
Harvard Kennedy School
Mid-Career MPA
Summer Program 2014
5. (OPTIONAL) Assume a technology breakthrough occurs in the military industry. How does this
breakthrough affect the production and the consumption of the two goods?
Technology breakthrough will change the production possibilities constraint. It will make more military
spending possible but will not change spending on consumption goods.
Graphically it does not change the PPC intercept on the x axis but changes the intercept on the y axis by
moving it higher. We still expect a convex graph.
Our PPC was expressed as PPC k = M^a + C^b. Algebraically, the technology breakthrough will reduce
the value of the constant a, meaning more military production is possible for any given value of C, ceteris
paribus (except at the point k = C^b). Consumption of goods and military spending should both increase,
and a new equilibrium will be found on a greater utility curve.
Instructor: Akos Lada
Harvard Kennedy School
Mid-Career MPA
Summer Program 2014
6. (OPTIONAL) Go back to the original model (i.e. before question 4). Assume the world has
become more peaceful, thus neighboring countries have reduced the size of their militaries.
Model one way of how this exogenous change affects the consumption and production choices
of our country.
This exogenous change is going to affect our probabilistic model for the military spending. Looking at our
original model it was U = q * C^a * M^b*(M^x/k) * (C^y/l) which we simplified to U = K * C^s * M^t.
Where M^b = Utility from military spending, M^x/k = probability of not being conquered.
The reduction in risk of war reduces probability of being conquered for any given spending on M, which
means for any given M, (M^x/k) (which is our probability of not being conquered) must be higher so x
increases. Utility from military spending (jobs created, infrastructure, technology etc.) does not change
so b remains constant. Our constant s and K also should not change.
So in our simplified equation U = K * C ^ s * M ^ t, t would increase and utility would increase at previous
optimal point.
This is counterintuitive as we expect optimal M to decrease, as the marginal utility of our military
spending would be expected to be going down due to our neighbors disarming. The increase in t due to
lower probability of being conquered for our given M shifts our optimal utility function towards M and
away from C. Remember that our optimal C^b= PPFas/(as+bt) and M^a= PPFbt/(as+bt)
Graphically represented as follows: