Notes on:

Becker, G. S., Hubbard, W. H. J., & Murphy, K. M. (2010): Explaining the Worldwide Boom in Higher Education of Women

1 What?

2 Why?

Past studies focused on changes in average costs and benefits of education for men and women over time -> They implicitly relied on a model of indiviual choice. (Jacob 2002; Buchmann and DiPrete 2006; DiPrete and Buchmann 2006; Goldin, Katz, and Kuziemko 2006)

3 How?

3.1 Optimal investment in colleage education by an individual

3.1.1 Model specifications

Consider the optimal investment in college education by different individuals: \[ S = F(h, H, A_{c}, A_{n}) \] where:

• \(h\) is time spent at college
• \(H\) measure the stock of human capital prior to any investment in \(S\)
• \(A_{c}\) and \(A_{n}\) measure cognitive and non-cognitive skills

Assumption: \(F_{h}\), \(F_{H}\), \(F_{c}\), \(F_{n}\) is positive; \(F_{hh} < 0\).

Target function: the disctounted expected utility: \[ V = U_{1}(x_{1}, l_{1}, H) + p(S; H)\beta U_{2}(x_{2}, l_{2}, S; H) \]

• \(\beta\): discount rate
• \(p\): prob. of surviving to the end of the next period
• \(x\): consumption of goods
• \(l\): household time

Assumtion: Utility is increasing and concave in \(x\), \(l\), \(S\).

There is also the assumption of full annuity insurance – which implies that expected disctounted consumption is equal to expected discounted income. Then, the full wealth budget constraint over the two periods is written as: \[ x_{1} + \frac{px_{2}}{1 + r} + w_{1}l_{1} + \frac{pw_{2}(S, H)l_{2}}{1 + r} + T(h) + w_{1}h = \\\ w_{1} + \frac{pw_{2}(S, H)}{1 + r} + \frac{pM(S)}{1 + r} = W \]

• \(r\): fixed interest rate
• \(W\): expected full wealth
• \(w\): hourly earnings
• Total time in each period is normalized to one
• \(T\): tuition and fees
• \(w_{l}h\) the earning forgone from being incollege
• \(M(S)\): gain from marriage in the second period, which is increment to expected wealth, and \(M_{s} > 0\).

Assumption: \(w_{2}>0\) (education should raise hourly earnings)

3.1.2 Solving strategy

Here I summarize the solving strategy sufficiently enough only to highlight interesting mechanisms.

Find \(S\), \(H\) that maximizes \(V\) under thee budget constraint.

• The first-order condition (foc) for consumption \(x\):

\[ U_{1x} = \mu \text{ and } p\beta{}U_{2x} = \frac{\mu{}p}{1+r} \] under full annuity insurance, uncertainty is not taken into the model, so: \[ \beta U_{2x} = \frac{U_{1x}}{1 + r} \]

• The foc for time spent in the household \(l\):

\[ U_{1l} = \mu w_{1} \text{ and } p\beta U_{2l} = \frac{\mu pw_{2}}{1 + r} \]

• Use \(e_{2}\) to denote the hours worked in the second period. The focs for the optimal time spent in college, divide both sides by \(\mu\) and \(F_{h}\), is given by: \[ \frac{pe_{2}w_{2s}}{1 +r} + \frac{\beta p_{s}U_{2}}{U_{1x}} + \frac{p_{s}(e_{2}w_{2} - x_{2})}{1+r} + \frac{pM_{s} + p_{s}M}{1 +r} + \frac{p\beta U_{2s}}{U_{1x}} = \frac{w_{1} + T_{h}}{F_{h}} \]

Let’s delve into individual terms in the last equation:

\(\frac{pe_{2}w_{2s}}{1+r}\)

the discounted expected increase in earnings from greater college education. This is the term that dominate discussion of “rates of return” to education in education literature.

\(\frac{\beta p_{s} U_{2}}{U_{1x}} = \frac{p_{s}\beta U_{2}}{\beta(1+r)U_{2x}} = \frac{p_{s} U_{2}}{U_{2x}(1+r)}\)

The second term measures the utilities gained on the probability of surviving in the future. \(\frac{U_{2}}{U_{2x}(1+r)}\) is called “the statistical value of life”, estimated to be in \$3-\$7 for a young male in the us.

\(\frac{p_{s}(e_{2}w_{2} - x_{2})}{1+r}\)

The benefit from an increased probability of survival in the future if future earning exceed education.

3.2 Market for college graduates

4 And?

Main takeaways:

• Elasticity of supply of women and men to college depends on the amount of heterogeneity across women and across men in the cost of attending college.

  • If women have a higher elasticity of supply to college, then even for equal changes in the benefits of college for men and women, women can overtake men in college attainment.

• Women on average find school less difficult than men -> Increase the net benefit of college

• Inequality of total cost of education appears to be lower among women than men. -> Women supply to college was more elastic than men -> When demand for college graduates grown, more women responded in increasing college attention.